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##  Appendix A Evaluation in Terms of Relative Absolute Error

While the tables presented in the main body of the paper report the results of
our experiments in terms of the absolute error (AE) measure, in this section
we present, for the sake of completeness, the results in terms of relative
absolute error (RAE). We do not comment on these results since the trends that
emerge from them are essentially the same as for the AE measure. Tables 12,
13, 14, 15, 16, 17, 18, are the RAE equivalents of Tables 3, 4, 5, 6, 7, 9,
11, respectively

Table 12: Values of $\operatorname{RAE}$ obtained in our experiments for
different amounts of shift using CC as the base quantifier. Notational
conventions are as in Table 3.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0
Emotions | 0.1728 | 0.1463${}^{\phantom{{\ddagger}}}$ | 0.1163 | 0.1529 | 0.3138 | 0.3180 | 0.2872 | 0.3523${}^{\phantom{{\ddagger}}}$ | 0.4024 | 0.4225${}^{\phantom{{\ddagger}}}$ | 0.3575 | 0.5395${}^{\phantom{{\ddagger}}}$
Scene | 0.3509 | 0.2647${}^{\phantom{{\ddagger}}}$ | 0.1803 | 0.1464 | 0.5922 | 0.4659 | 0.3044 | 0.2490${}^{\phantom{{\ddagger}}}$ | 2.7287 | 2.2747${}^{\phantom{{\ddagger}}}$ | 1.2898 | 1.3553‡
Yeast | 0.7124 | 0.4604${}^{\phantom{{\ddagger}}}$ | 0.2163 | 0.2254 | 0.7935 | 0.5856 | 0.3626 | 0.4000${}^{\phantom{{\ddagger}}}$ | 1.2682 | 1.0984${}^{\phantom{{\ddagger}}}$ | 0.9905 | 1.0610${}^{\phantom{{\ddagger}}}$
Birds | 0.2981 | 0.3766${}^{\phantom{{\ddagger}}}$ | 0.2842 | 0.3129 | 0.3435 | 0.4379 | 0.3649 | 0.3867${}^{\phantom{{\ddagger}}}$ | 0.4039 | 0.4758${}^{\phantom{{\ddagger}}}$ | 0.4244 | 0.4159†
Genbase | 0.0088 | 0.0254${}^{\phantom{{\ddagger}}}$ | 0.0718 | 0.0273 | 0.0101 | 0.0270 | 0.0755 | 0.0288${}^{\phantom{{\ddagger}}}$ | 0.0088 | 0.0257${}^{\phantom{{\ddagger}}}$ | 0.0704 | 0.0275${}^{\phantom{{\ddagger}}}$
Medical | 0.2073 | 0.1402${}^{\phantom{{\ddagger}}}$ | 0.3248 | 0.1403 | 0.2052 | 0.1673 | 0.4483 | 0.1673${}^{\phantom{{\ddagger}}}$ | 0.2908 | 0.3065† | 0.6873 | 0.3065†
tmc2007_500 | 1.1568 | 0.2116${}^{\phantom{{\ddagger}}}$ | 0.2620 | 0.2351 | 1.2240 | 0.2924 | 0.3263 | 0.2779${}^{\phantom{{\ddagger}}}$ | 1.1152 | 0.4701${}^{\phantom{{\ddagger}}}$ | 0.4691 | 0.3989${}^{\phantom{{\ddagger}}}$
Ohsumed | 0.5071 | 0.1982${}^{\phantom{{\ddagger}}}$ | 0.2452 | 0.1887 | 0.5857 | 0.2527 | 0.3014 | 0.2387${}^{\phantom{{\ddagger}}}$ | 0.7796 | 0.4588${}^{\phantom{{\ddagger}}}$ | 0.4709 | 0.4353${}^{\phantom{{\ddagger}}}$
Enron | 0.3524 | 0.3099${}^{\phantom{{\ddagger}}}$ | 0.3068 | 0.2732 | 0.4227 | 0.3670 | 0.3920 | 0.3345${}^{\phantom{{\ddagger}}}$ | 0.6222 | 0.5026${}^{\phantom{{\ddagger}}}$ | 0.5554 | 0.4635${}^{\phantom{{\ddagger}}}$
Reuters-21578 | 0.2352 | 0.0858${}^{\phantom{{\ddagger}}}$ | 0.1433 | 0.0911 | 0.2315 | 0.0987 | 0.1676 | 0.1051${}^{\phantom{{\ddagger}}}$ | 0.2648 | 0.0949${}^{\phantom{{\ddagger}}}$ | 0.1697 | 0.1054${}^{\phantom{{\ddagger}}}$
RCV1-v2 | 0.5356 | 0.1513${}^{\phantom{{\ddagger}}}$ | 0.1979 | 0.1523 | 0.6071 | 0.1915 | 0.2997 | 0.2015${}^{\phantom{{\ddagger}}}$ | 0.7414 | 0.3032${}^{\phantom{{\ddagger}}}$ | 0.5434 | 0.3291${}^{\phantom{{\ddagger}}}$
Mediamill | 6.4321 | 0.2278${}^{\phantom{{\ddagger}}}$ | 0.2798 | 0.2703 | 7.2338 | 0.2792 | 0.3815 | 0.3745${}^{\phantom{{\ddagger}}}$ | 7.9048 | 0.5052${}^{\phantom{{\ddagger}}}$ | 0.5427 | 0.5363${}^{\phantom{{\ddagger}}}$
Bibtex | 0.5653 | 0.3249${}^{\phantom{{\ddagger}}}$ | 0.3501 | 0.3220 | 0.4882 | 0.2830 | 0.3349 | 0.2833‡ | 0.4091 | 0.2313${}^{\phantom{{\ddagger}}}$ | 0.3320 | 0.2401${}^{\phantom{{\ddagger}}}$
Corel5k | 1.3528 | 0.2474‡ | 0.2471 | 0.2643 | 1.4460 | 0.2522 | 0.2588 | 0.2923${}^{\phantom{{\ddagger}}}$ | 1.6125 | 0.2798${}^{\phantom{{\ddagger}}}$ | 0.3262 | 0.3444${}^{\phantom{{\ddagger}}}$
Delicious | 4.9852 | 0.3564${}^{\phantom{{\ddagger}}}$ | 0.3032 | 0.3043 | 5.3796 | 0.4033 | 0.3431 | 0.3449‡ | 5.0129 | 0.4246${}^{\phantom{{\ddagger}}}$ | 0.3745 | 0.3805†
Average | 1.2509 | 0.2301${}^{\phantom{{\ddagger}}}$ | 0.2380 | 0.2083 | 1.1155 | 0.2916 | 0.3085 | 0.2665${}^{\phantom{{\ddagger}}}$ | 1.4849 | 0.7943${}^{\phantom{{\ddagger}}}$ | 0.6530 | 0.6209${}^{\phantom{{\ddagger}}}$
Rank Average | 3.6 | 2.1 | 2.3 | 2.0 | 3.4 | 2.0 | 2.5 | 2.1 | 3.3 | 2.1 | 2.5 | 2.1

Table 13: Values of $\operatorname{RAE}$ obtained in our experiments for
different amounts of shift using PCC as the base quantifier. Notational
conventions are as in Table 3.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0
Emotions | 0.3105 | 0.1720 | 0.1297 | 0.1318${}^{\phantom{{\ddagger}}}$ | 0.4089 | 0.2945 | 0.3116 | 0.2812${}^{\phantom{{\ddagger}}}$ | 0.4459 | 0.4191${}^{\phantom{{\ddagger}}}$ | 0.6205 | 0.4639${}^{\phantom{{\ddagger}}}$
Scene | 0.2023 | 0.1416 | 0.1530 | 0.1332${}^{\phantom{{\ddagger}}}$ | 0.4292 | 0.2893 | 0.2266 | 0.2171${}^{\phantom{{\ddagger}}}$ | 1.7304 | 1.1136‡ | 1.1121 | 1.3674${}^{\phantom{{\ddagger}}}$
Yeast | 1.4781 | 0.8704 | 0.2080 | 0.2024${}^{\phantom{{\ddagger}}}$ | 2.0566 | 1.0807 | 0.3571 | 0.3319${}^{\phantom{{\ddagger}}}$ | 2.9340 | 1.4858${}^{\phantom{{\ddagger}}}$ | 1.0029 | 0.9498${}^{\phantom{{\ddagger}}}$
Birds | 3.4391 | 3.7460 | 0.2908 | 0.2730${}^{\phantom{{\ddagger}}}$ | 3.5060 | 3.7643 | 0.3516 | 0.3316${}^{\phantom{{\ddagger}}}$ | 3.3622 | 2.7057${}^{\phantom{{\ddagger}}}$ | 0.4570 | 0.4704†
Genbase | 0.0376 | 0.0228 | 0.1290 | 0.0378${}^{\phantom{{\ddagger}}}$ | 0.0376 | 0.0243 | 0.1466 | 0.0401${}^{\phantom{{\ddagger}}}$ | 0.0360 | 0.0250${}^{\phantom{{\ddagger}}}$ | 0.1497 | 0.0422${}^{\phantom{{\ddagger}}}$
Medical | 0.8565 | 0.5073 | 0.2812 | 0.2437${}^{\phantom{{\ddagger}}}$ | 1.2243 | 0.5253 | 0.4063 | 0.3556${}^{\phantom{{\ddagger}}}$ | 2.0092 | 0.6634‡ | 0.8955 | 0.6556${}^{\phantom{{\ddagger}}}$
tmc2007_500 | 0.3455 | 0.2420 | 0.2443 | 0.1987${}^{\phantom{{\ddagger}}}$ | 0.5268 | 0.3260 | 0.3092 | 0.2507${}^{\phantom{{\ddagger}}}$ | 0.4738 | 0.3160${}^{\phantom{{\ddagger}}}$ | 0.4486 | 0.4338${}^{\phantom{{\ddagger}}}$
Ohsumed | 0.2628 | 0.2192 | 0.2356 | 0.1689${}^{\phantom{{\ddagger}}}$ | 0.3865 | 0.2698 | 0.2784 | 0.2200${}^{\phantom{{\ddagger}}}$ | 0.5300 | 0.3342${}^{\phantom{{\ddagger}}}$ | 0.4005 | 0.4537${}^{\phantom{{\ddagger}}}$
Enron | 5.5524 | 5.4773 | 0.3640 | 0.3452${}^{\phantom{{\ddagger}}}$ | 6.6795 | 5.7874 | 0.4409 | 0.4329${}^{\phantom{{\ddagger}}}$ | 6.3135 | 7.0035${}^{\phantom{{\ddagger}}}$ | 0.6331 | 0.5825${}^{\phantom{{\ddagger}}}$
Reuters-21578 | 0.4275 | 3.9328 | 0.1359 | 0.1292${}^{\phantom{{\ddagger}}}$ | 1.7934 | 3.9820 | 0.1844 | 0.1687${}^{\phantom{{\ddagger}}}$ | 2.6267 | 3.9425${}^{\phantom{{\ddagger}}}$ | 0.2238 | 0.1942${}^{\phantom{{\ddagger}}}$
RCV1-v2 | 0.5285 | 0.4427 | 0.1824 | 0.1652${}^{\phantom{{\ddagger}}}$ | 1.0813 | 0.8803 | 0.3071 | 0.2816${}^{\phantom{{\ddagger}}}$ | 1.5968 | 1.2673${}^{\phantom{{\ddagger}}}$ | 0.5635 | 0.5201${}^{\phantom{{\ddagger}}}$
Mediamill | 1.8713 | 1.8111 | 0.2798 | 0.2833${}^{\phantom{{\ddagger}}}$ | 3.4939 | 2.6875 | 0.3841 | 0.3854‡ | 4.6897 | 3.2192${}^{\phantom{{\ddagger}}}$ | 0.5216 | 0.5511${}^{\phantom{{\ddagger}}}$
Bibtex | 1.3104 | 0.7556 | 0.3709 | 0.3102${}^{\phantom{{\ddagger}}}$ | 1.4016 | 1.0460 | 0.3695 | 0.3492${}^{\phantom{{\ddagger}}}$ | 1.4682 | 1.3451${}^{\phantom{{\ddagger}}}$ | 0.3970 | 0.4261†
Corel5k | 9.8431 | 6.2210 | 0.3989 | 0.3997‡ | 10.3929 | 6.6651 | 0.4224 | 0.4054${}^{\phantom{{\ddagger}}}$ | 12.0719 | 7.4614${}^{\phantom{{\ddagger}}}$ | 0.5075 | 0.4682${}^{\phantom{{\ddagger}}}$
Delicious | 3.2613 | 2.5366 | 0.3042 | 0.2875${}^{\phantom{{\ddagger}}}$ | 4.2250 | 4.1529 | 0.3396 | 0.3240${}^{\phantom{{\ddagger}}}$ | 4.1574 | 4.4159${}^{\phantom{{\ddagger}}}$ | 0.3677 | 0.3478${}^{\phantom{{\ddagger}}}$
Average | 2.0194 | 1.8966 | 0.2510 | 0.2252${}^{\phantom{{\ddagger}}}$ | 1.9582 | 1.6894 | 0.3134 | 0.2791${}^{\phantom{{\ddagger}}}$ | 2.0662 | 1.4948${}^{\phantom{{\ddagger}}}$ | 0.6694 | 0.6728${}^{\phantom{{\ddagger}}}$
Rank Average | 3.7 | 2.8 | 2.1 | 1.3 | 3.7 | 2.9 | 2.2 | 1.2 | 3.5 | 2.5 | 2.1 | 1.8

Table 14: Values of $\operatorname{RAE}$ obtained in our experiments for
different amounts of shift using ACC as the base quantifier. Notational
conventions are as in Table 3.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0
Emotions | 0.2491 | 0.2991 | 0.1579${}^{\phantom{{\ddagger}}}$ | 0.1069${}^{\phantom{{\ddagger}}}$ | 0.4297 | 0.4301 | 0.3294 | 0.2672 | 0.4519${}^{\phantom{{\ddagger}}}$ | 0.4690 | 0.6290${}^{\phantom{{\ddagger}}}$ | 0.4285${}^{\phantom{{\ddagger}}}$
Scene | 0.2035 | 0.1676 | 0.1792${}^{\phantom{{\ddagger}}}$ | 0.1554${}^{\phantom{{\ddagger}}}$ | 0.3371 | 0.3863 | 0.2778 | 0.2576 | 1.3324‡ | 1.5409 | 1.3054${}^{\phantom{{\ddagger}}}$ | 1.4589${}^{\phantom{{\ddagger}}}$
Yeast | 2.2331 | 0.5355 | 0.2116${}^{\phantom{{\ddagger}}}$ | 0.2121† | 2.4014 | 0.6702 | 0.3733 | 0.3834 | 2.7847${}^{\phantom{{\ddagger}}}$ | 1.4287 | 0.9628${}^{\phantom{{\ddagger}}}$ | 1.1162${}^{\phantom{{\ddagger}}}$
Birds | 3.3221 | 3.0772 | 0.3130† | 0.3060${}^{\phantom{{\ddagger}}}$ | 3.2554 | 3.0525 | 0.3861 | 0.3320 | 2.5615${}^{\phantom{{\ddagger}}}$ | 2.3043 | 0.4546${}^{\phantom{{\ddagger}}}$ | 0.3517${}^{\phantom{{\ddagger}}}$
Genbase | 0.0266 | 0.0081 | 0.1277${}^{\phantom{{\ddagger}}}$ | 0.0058${}^{\phantom{{\ddagger}}}$ | 0.0266 | 0.0076 | 0.1440 | 0.0065 | 0.0253${}^{\phantom{{\ddagger}}}$ | 0.0081 | 0.1381${}^{\phantom{{\ddagger}}}$ | 0.0065${}^{\phantom{{\ddagger}}}$
Medical | 0.2321 | 0.6153 | 0.3447${}^{\phantom{{\ddagger}}}$ | 0.2579${}^{\phantom{{\ddagger}}}$ | 0.2192 | 0.9236 | 0.5121 | 0.3689 | 0.2408${}^{\phantom{{\ddagger}}}$ | 1.3156 | 0.8683${}^{\phantom{{\ddagger}}}$ | 0.6021${}^{\phantom{{\ddagger}}}$
tmc2007_500 | 0.3560 | 0.3457 | 0.2564${}^{\phantom{{\ddagger}}}$ | 0.2597† | 0.5111 | 0.4541 | 0.3327 | 0.3252 | 0.4769${}^{\phantom{{\ddagger}}}$ | 0.4228 | 0.5063† | 0.4963†
Ohsumed | 0.2620 | 0.2467 | 0.2381${}^{\phantom{{\ddagger}}}$ | 0.1935${}^{\phantom{{\ddagger}}}$ | 0.3216 | 0.2863 | 0.2820 | 0.2437 | 0.4047${}^{\phantom{{\ddagger}}}$ | 0.3372 | 0.4122${}^{\phantom{{\ddagger}}}$ | 0.4397${}^{\phantom{{\ddagger}}}$
Enron | 7.3850 | 1.8492 | 0.3285‡ | 0.3278${}^{\phantom{{\ddagger}}}$ | 8.1460 | 2.1083 | 0.4251 | 0.4155 | 9.3550${}^{\phantom{{\ddagger}}}$ | 3.3057 | 0.5961${}^{\phantom{{\ddagger}}}$ | 0.6283${}^{\phantom{{\ddagger}}}$
Reuters-21578 | 2.1867 | 0.1421 | 0.1320${}^{\phantom{{\ddagger}}}$ | 0.1491${}^{\phantom{{\ddagger}}}$ | 2.1224 | 0.1381 | 0.1672 | 0.1798 | 1.5822${}^{\phantom{{\ddagger}}}$ | 0.1984 | 0.1723${}^{\phantom{{\ddagger}}}$ | 0.2147${}^{\phantom{{\ddagger}}}$
RCV1-v2 | 0.2434 | 1.0216 | 0.1758${}^{\phantom{{\ddagger}}}$ | 0.1609${}^{\phantom{{\ddagger}}}$ | 0.3070 | 1.0289 | 0.2919 | 0.2450 | 0.4303${}^{\phantom{{\ddagger}}}$ | 0.9287 | 0.5660${}^{\phantom{{\ddagger}}}$ | 0.5090${}^{\phantom{{\ddagger}}}$
Mediamill | 2.2137 | 0.9835 | 0.2969${}^{\phantom{{\ddagger}}}$ | 0.2527${}^{\phantom{{\ddagger}}}$ | 3.7688 | 1.1194 | 0.4184 | 0.3713 | 5.3468${}^{\phantom{{\ddagger}}}$ | 1.3355 | 0.5751${}^{\phantom{{\ddagger}}}$ | 0.5984‡
Bibtex | 3.6761 | 3.8536 | 0.3600${}^{\phantom{{\ddagger}}}$ | 0.3549${}^{\phantom{{\ddagger}}}$ | 3.7272 | 4.4074 | 0.3696 | 0.3863 | 4.1667${}^{\phantom{{\ddagger}}}$ | 5.0247 | 0.4294${}^{\phantom{{\ddagger}}}$ | 0.4587‡
Corel5k | 9.8232 | 0.2167 | 0.2457${}^{\phantom{{\ddagger}}}$ | 0.1865${}^{\phantom{{\ddagger}}}$ | 10.2793 | 0.1994 | 0.2622 | 0.2165 | 10.7374${}^{\phantom{{\ddagger}}}$ | 0.1848 | 0.3477${}^{\phantom{{\ddagger}}}$ | 0.3049${}^{\phantom{{\ddagger}}}$
Delicious | 5.0943 | 1.9548 | 0.3215${}^{\phantom{{\ddagger}}}$ | 0.2694${}^{\phantom{{\ddagger}}}$ | 5.7915 | 2.2703 | 0.3597 | 0.3020 | 5.9927${}^{\phantom{{\ddagger}}}$ | 2.5536 | 0.3907${}^{\phantom{{\ddagger}}}$ | 0.3355${}^{\phantom{{\ddagger}}}$
Average | 2.6225 | 1.1079 | 0.2468${}^{\phantom{{\ddagger}}}$ | 0.2167${}^{\phantom{{\ddagger}}}$ | 1.9959 | 1.1025 | 0.3264 | 0.2793 | 1.8528${}^{\phantom{{\ddagger}}}$ | 1.1373 | 0.7049${}^{\phantom{{\ddagger}}}$ | 0.6926${}^{\phantom{{\ddagger}}}$
Rank Average | 3.5 | 3.0 | 2.1 | 1.3 | 3.5 | 3.0 | 2.1 | 1.4 | 2.9 | 2.7 | 2.3 | 2.1

Table 15: Values of $\operatorname{RAE}$ obtained in our experiments for
different amounts of shift using PACC as the base quantifier. Notational
conventions are as in Table 3.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0
Emotions | 0.3105 | 0.1720 | 0.1297 | 0.1318${}^{\phantom{{\ddagger}}}$ | 0.4089 | 0.2945 | 0.3116 | 0.2812${}^{\phantom{{\ddagger}}}$ | 0.4459 | 0.4191${}^{\phantom{{\ddagger}}}$ | 0.6205 | 0.4639${}^{\phantom{{\ddagger}}}$
Scene | 0.2023 | 0.1416 | 0.1530 | 0.1332${}^{\phantom{{\ddagger}}}$ | 0.4292 | 0.2893 | 0.2266 | 0.2171${}^{\phantom{{\ddagger}}}$ | 1.7304 | 1.1136‡ | 1.1121 | 1.3674${}^{\phantom{{\ddagger}}}$
Yeast | 1.4781 | 0.8704 | 0.2080 | 0.2024${}^{\phantom{{\ddagger}}}$ | 2.0566 | 1.0807 | 0.3571 | 0.3319${}^{\phantom{{\ddagger}}}$ | 2.9340 | 1.4858${}^{\phantom{{\ddagger}}}$ | 1.0029 | 0.9498${}^{\phantom{{\ddagger}}}$
Birds | 3.4391 | 3.7460 | 0.2908 | 0.2730${}^{\phantom{{\ddagger}}}$ | 3.5060 | 3.7643 | 0.3516 | 0.3316${}^{\phantom{{\ddagger}}}$ | 3.3622 | 2.7057${}^{\phantom{{\ddagger}}}$ | 0.4570 | 0.4704†
Genbase | 0.0376 | 0.0228 | 0.1290 | 0.0378${}^{\phantom{{\ddagger}}}$ | 0.0376 | 0.0243 | 0.1466 | 0.0401${}^{\phantom{{\ddagger}}}$ | 0.0360 | 0.0250${}^{\phantom{{\ddagger}}}$ | 0.1497 | 0.0422${}^{\phantom{{\ddagger}}}$
Medical | 0.8565 | 0.5073 | 0.2812 | 0.2437${}^{\phantom{{\ddagger}}}$ | 1.2243 | 0.5253 | 0.4063 | 0.3556${}^{\phantom{{\ddagger}}}$ | 2.0092 | 0.6634‡ | 0.8955 | 0.6556${}^{\phantom{{\ddagger}}}$
tmc2007_500 | 0.3455 | 0.2420 | 0.2443 | 0.1987${}^{\phantom{{\ddagger}}}$ | 0.5268 | 0.3260 | 0.3092 | 0.2507${}^{\phantom{{\ddagger}}}$ | 0.4738 | 0.3160${}^{\phantom{{\ddagger}}}$ | 0.4486 | 0.4338${}^{\phantom{{\ddagger}}}$
Ohsumed | 0.2628 | 0.2192 | 0.2356 | 0.1689${}^{\phantom{{\ddagger}}}$ | 0.3865 | 0.2698 | 0.2784 | 0.2200${}^{\phantom{{\ddagger}}}$ | 0.5300 | 0.3342${}^{\phantom{{\ddagger}}}$ | 0.4005 | 0.4537${}^{\phantom{{\ddagger}}}$
Enron | 5.5524 | 5.4773 | 0.3640 | 0.3452${}^{\phantom{{\ddagger}}}$ | 6.6795 | 5.7874 | 0.4409 | 0.4329${}^{\phantom{{\ddagger}}}$ | 6.3135 | 7.0035${}^{\phantom{{\ddagger}}}$ | 0.6331 | 0.5825${}^{\phantom{{\ddagger}}}$
Reuters-21578 | 0.4275 | 3.9328 | 0.1359 | 0.1292${}^{\phantom{{\ddagger}}}$ | 1.7934 | 3.9820 | 0.1844 | 0.1687${}^{\phantom{{\ddagger}}}$ | 2.6267 | 3.9425${}^{\phantom{{\ddagger}}}$ | 0.2238 | 0.1942${}^{\phantom{{\ddagger}}}$
RCV1-v2 | 0.5285 | 0.4427 | 0.1824 | 0.1652${}^{\phantom{{\ddagger}}}$ | 1.0813 | 0.8803 | 0.3071 | 0.2816${}^{\phantom{{\ddagger}}}$ | 1.5968 | 1.2673${}^{\phantom{{\ddagger}}}$ | 0.5635 | 0.5201${}^{\phantom{{\ddagger}}}$
Mediamill | 1.8713 | 1.8111 | 0.2798 | 0.2833${}^{\phantom{{\ddagger}}}$ | 3.4939 | 2.6875 | 0.3841 | 0.3854‡ | 4.6897 | 3.2192${}^{\phantom{{\ddagger}}}$ | 0.5216 | 0.5511${}^{\phantom{{\ddagger}}}$
Bibtex | 1.3104 | 0.7556 | 0.3709 | 0.3102${}^{\phantom{{\ddagger}}}$ | 1.4016 | 1.0460 | 0.3695 | 0.3492${}^{\phantom{{\ddagger}}}$ | 1.4682 | 1.3451${}^{\phantom{{\ddagger}}}$ | 0.3970 | 0.4261†
Corel5k | 9.8431 | 6.2210 | 0.3989 | 0.3997‡ | 10.3929 | 6.6651 | 0.4224 | 0.4054${}^{\phantom{{\ddagger}}}$ | 12.0719 | 7.4614${}^{\phantom{{\ddagger}}}$ | 0.5075 | 0.4682${}^{\phantom{{\ddagger}}}$
Delicious | 3.2613 | 2.5366 | 0.3042 | 0.2875${}^{\phantom{{\ddagger}}}$ | 4.2250 | 4.1529 | 0.3396 | 0.3240${}^{\phantom{{\ddagger}}}$ | 4.1574 | 4.4159${}^{\phantom{{\ddagger}}}$ | 0.3677 | 0.3478${}^{\phantom{{\ddagger}}}$
Average | 2.0194 | 1.8966 | 0.2510 | 0.2252${}^{\phantom{{\ddagger}}}$ | 1.9582 | 1.6894 | 0.3134 | 0.2791${}^{\phantom{{\ddagger}}}$ | 2.0662 | 1.4948${}^{\phantom{{\ddagger}}}$ | 0.6694 | 0.6728${}^{\phantom{{\ddagger}}}$
Rank Average | 3.7 | 2.8 | 2.1 | 1.3 | 3.7 | 2.9 | 2.2 | 1.2 | 3.5 | 2.5 | 2.1 | 1.8

Table 16: Values of $\operatorname{RAE}$ obtained in our experiments for
different amounts of shift using SLD as the base quantifier. Notational
conventions are as in Table 3.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0 |  BC+BA |  MLC+BA |  BC+MLA |  MLC+MLA0
Emotions | 0.5117 | 0.1310 | 0.1646 | 0.1199 | 0.6961 | 0.2455 | 0.3179${}^{\phantom{{\ddagger}}}$ | 0.2907† | 0.5526 | 0.2769 | 0.4625 | 0.3885
Scene | 0.1519 | 0.1914 | 0.1432 | 0.1847 | 0.2518 | 0.3744 | 0.2587† | 0.3693${}^{\phantom{{\ddagger}}}$ | 1.0904 | 2.1677 | 1.3721 | 1.9117
Yeast | 1.3664 | 0.3688 | 0.2111 | 0.2210 | 1.3507 | 0.4597 | 0.3471${}^{\phantom{{\ddagger}}}$ | 0.3409${}^{\phantom{{\ddagger}}}$ | 1.8907 | 0.9233 | 1.0121 | 0.8924
Birds | 0.9407 | 0.4495 | 0.2837 | 0.4340 | 1.1251 | 0.5193 | 0.3447${}^{\phantom{{\ddagger}}}$ | 0.5015${}^{\phantom{{\ddagger}}}$ | 1.7344 | 0.5495 | 0.3750 | 0.5064
Genbase | 0.0273 | 0.0052 | 0.1146 | 0.0081 | 0.0282 | 0.0053 | 0.1344${}^{\phantom{{\ddagger}}}$ | 0.0082${}^{\phantom{{\ddagger}}}$ | 0.0268 | 0.0051 | 0.1430 | 0.0084
Medical | 0.2526 | 0.1661 | 0.3077 | 0.1611 | 0.2617 | 0.2028 | 0.4439${}^{\phantom{{\ddagger}}}$ | 0.1968${}^{\phantom{{\ddagger}}}$ | 0.4769 | 0.3791 | 0.7228 | 0.3722
tmc2007_500 | 0.2548 | 0.1979 | 0.2212 | 0.1987 | 0.3232 | 0.2751 | 0.2939${}^{\phantom{{\ddagger}}}$ | 0.2479${}^{\phantom{{\ddagger}}}$ | 0.3446 | 0.2761 | 0.4381 | 0.3597
Ohsumed | 0.2232 | 0.1701 | 0.1998 | 0.1693 | 0.2495 | 0.2079 | 0.2464${}^{\phantom{{\ddagger}}}$ | 0.2072${}^{\phantom{{\ddagger}}}$ | 0.2801 | 0.3330 | 0.3193 | 0.3374
Enron | 1.1351 | 0.2639 | 0.3135 | 0.3362 | 1.2907 | 0.3181 | 0.3931${}^{\phantom{{\ddagger}}}$ | 0.4285${}^{\phantom{{\ddagger}}}$ | 1.8511 | 0.4268 | 0.5422 | 0.5888
Reuters-21578 | 0.1398 | 0.0810 | 0.1320 | 0.0869 | 0.3150 | 0.0821 | 0.2248${}^{\phantom{{\ddagger}}}$ | 0.0927${}^{\phantom{{\ddagger}}}$ | 1.6152 | 0.0974 | 0.2528 | 0.1060
RCV1-v2 | 0.2199 | 0.1282 | 0.1624 | 0.1328 | 0.4811 | 0.1456 | 0.2821${}^{\phantom{{\ddagger}}}$ | 0.1724${}^{\phantom{{\ddagger}}}$ | 0.7695 | 0.2318 | 0.5706 | 0.2990
Mediamill | 10.8687 | 0.4144 | 0.2649 | 0.2549 | 11.3338 | 0.7821 | 0.4033${}^{\phantom{{\ddagger}}}$ | 0.3748${}^{\phantom{{\ddagger}}}$ | 11.1308 | 1.2462 | 0.5882 | 0.5481
Bibtex | 3.7700 | 0.3307 | 0.3830 | 0.3284 | 3.7014 | 0.2854 | 0.4529${}^{\phantom{{\ddagger}}}$ | 0.2913${}^{\phantom{{\ddagger}}}$ | 3.4731 | 0.2393 | 0.6482 | 0.2614
Corel5k | 5.3304 | 0.2248 | 0.5850 | 0.6043 | 5.4527 | 0.2321 | 0.6448${}^{\phantom{{\ddagger}}}$ | 0.6184${}^{\phantom{{\ddagger}}}$ | 5.6702 | 0.2668 | 0.7806 | 0.7129
Delicious | 7.8086 | 0.4778 | 0.3098 | 0.3587 | 8.6288 | 0.5295 | 0.3578${}^{\phantom{{\ddagger}}}$ | 0.4013${}^{\phantom{{\ddagger}}}$ | 7.8740 | 0.5120 | 0.3911 | 0.4135
Average | 2.3798 | 0.2376 | 0.2562 | 0.2396 | 1.7450 | 0.3066 | 0.3154 | 0.2835 | 1.5997 | 0.7513 | 0.6837 | 0.7033
Rank Average | 3.7 | 2.0 | 2.4 | 1.9 | 3.7 | 1.9 | 2.6 | 1.8 | 3.3 | 1.9 | 2.7 | 2.1

Table 17: Values of $\operatorname{RAE}$ obtained for different multi-label
classifiers using PCC as the base quantifier in MLC+BA.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  ML-knn0 |  DT |  RF |  CChain |  CLEMS |  LSC |  SG |  BC+BA |  ML-knn0 |  DT |  RF |  CChain |  CLEMS |  LSC |  SG |  BC+BA |  ML-knn0 |  DT |  RF |  CChain |  CLEMS |  LSC |  SG
Emotions | 0.104† | 0.511 | 0.160 | 0.106 | 0.129 | 0.512 | 0.625 | 0.104 | 0.366 | 0.589 | 0.423 | 0.388 | 0.373 | 0.590 | 0.670 | 0.342 | 0.543 | 0.790 | 0.548 | 0.556${}^{\phantom{{\ddagger}}}$ | 0.593 | 0.788 | 0.932${}^{\phantom{{\ddagger}}}$ | 0.504${}^{\phantom{{\ddagger}}}$
Scene | 0.407${}^{\phantom{{\ddagger}}}$ | 0.494 | 0.194 | 0.178 | 0.175 | 0.495 | 0.556 | 0.171 | 0.742 | 0.565 | 0.441 | 0.447 | 0.376 | 0.567 | 0.643 | 0.416 | 3.491 | 2.277 | 2.571 | 2.851${}^{\phantom{{\ddagger}}}$ | 2.205 | 2.262 | 2.387‡ | 2.710${}^{\phantom{{\ddagger}}}$
Yeast | 0.929${}^{\phantom{{\ddagger}}}$ | 0.509 | 0.251 | 0.210 | 0.251 | 0.495 | 0.723 | 0.205 | 1.038 | 0.654 | 0.475 | 0.441 | 0.449 | 0.647 | 0.831 | 0.430 | 1.607 | 1.389 | 1.225 | 1.199† | 1.172 | 1.393 | 1.515${}^{\phantom{{\ddagger}}}$ | 1.202†
Birds | 0.550${}^{\phantom{{\ddagger}}}$ | 0.386 | 0.315 | 0.246 | 0.326 | 0.386 | 0.389 | 0.342 | 0.684 | 0.388 | 0.359 | 0.318 | 0.417 | 0.388 | 0.391 | 0.409 | 1.007 | 0.397 | 0.441 | 0.384${}^{\phantom{{\ddagger}}}$ | 0.541 | 0.397 | 0.401${}^{\phantom{{\ddagger}}}$ | 0.469${}^{\phantom{{\ddagger}}}$
Genbase | 0.047${}^{\phantom{{\ddagger}}}$ | 0.406 | 0.030 | 0.064 | 0.025 | 0.406 | 0.400 | 0.022 | 0.051 | 0.402 | 0.032 | 0.072 | 0.027 | 0.402 | 0.398 | 0.024 | 0.053 | 0.427 | 0.031 | 0.075${}^{\phantom{{\ddagger}}}$ | 0.026 | 0.425 | 0.408${}^{\phantom{{\ddagger}}}$ | 0.023${}^{\phantom{{\ddagger}}}$
Medical | 0.173‡ | 0.396 | 0.172 | 0.216 | 0.205 | 0.395 | 0.415 | 0.212 | 0.218 | 0.389 | 0.201 | 0.313 | 0.265 | 0.388 | 0.405 | 0.283 | 0.397 | 0.547 | 0.311 | 0.629${}^{\phantom{{\ddagger}}}$ | 0.465 | 0.537 | 0.567${}^{\phantom{{\ddagger}}}$ | 0.534${}^{\phantom{{\ddagger}}}$
tmc2007_500 | 1.939${}^{\phantom{{\ddagger}}}$ | 0.419 | 0.007 | 0.093 | 0.191 | 0.420 | 0.464 | 0.197 | 2.192 | 0.521 | 0.009 | 0.162 | 0.299 | 0.522 | 0.575 | 0.304 | 2.402 | 0.899 | 0.012 | 0.327${}^{\phantom{{\ddagger}}}$ | 0.524 | 0.895 | 0.973${}^{\phantom{{\ddagger}}}$ | 0.546${}^{\phantom{{\ddagger}}}$
Ohsumed | 1.912${}^{\phantom{{\ddagger}}}$ | 0.427 | 0.248 | 0.213 | 0.169 | 0.430 | 0.447 | 0.193 | 2.326 | 0.484 | 0.357 | 0.352 | 0.228 | 0.486 | 0.506 | 0.266 | 2.961 | 1.031 | 0.825 | 0.946${}^{\phantom{{\ddagger}}}$ | 0.481 | 1.033 | 1.064${}^{\phantom{{\ddagger}}}$ | 0.548${}^{\phantom{{\ddagger}}}$
Enron | 0.621${}^{\phantom{{\ddagger}}}$ | 0.370 | 0.369 | 0.308 | 0.284 | 0.372 | 0.386 | 0.282 | 0.741 | 0.408 | 0.431 | 0.401 | 0.376 | 0.408 | 0.428 | 0.373 | 1.146 | 0.537 | 0.641 | 0.662${}^{\phantom{{\ddagger}}}$ | 0.631 | 0.530 | 0.570${}^{\phantom{{\ddagger}}}$ | 0.631${}^{\phantom{{\ddagger}}}$
Reuters-21578 | 0.910${}^{\phantom{{\ddagger}}}$ | 0.171 | 0.146 | 0.192 | 0.095 | 0.171 | 0.175 | 0.091 | 1.184 | 0.215 | 0.171 | 0.283 | 0.110 | 0.215 | 0.220 | 0.106 | 1.040 | 0.298 | 0.191 | 0.292${}^{\phantom{{\ddagger}}}$ | 0.105 | 0.297 | 0.302${}^{\phantom{{\ddagger}}}$ | 0.100${}^{\phantom{{\ddagger}}}$
RCV1-v2 | 2.028${}^{\phantom{{\ddagger}}}$ | 0.253 | 0.264 | 0.282 | 0.166 | 0.253 | 0.262 | 0.160 | 2.525 | 0.303 | 0.432 | 0.560 | 0.227 | 0.300 | 0.316 | 0.218 | 2.970 | 0.550 | 0.752 | 0.997${}^{\phantom{{\ddagger}}}$ | 0.379 | 0.545 | 0.566${}^{\phantom{{\ddagger}}}$ | 0.351${}^{\phantom{{\ddagger}}}$
Mediamill | 7.677${}^{\phantom{{\ddagger}}}$ | 0.222 | 0.355 | 0.343 | 0.295 | 0.231 | 0.243 | 0.290 | 8.616 | 0.287 | 0.522 | 0.528 | 0.438 | 0.302 | 0.327 | 0.444 | 8.941 | 0.553 | 0.831 | 0.839${}^{\phantom{{\ddagger}}}$ | 0.748 | 0.563 | 0.605${}^{\phantom{{\ddagger}}}$ | 0.726${}^{\phantom{{\ddagger}}}$
Bibtex | 1.669${}^{\phantom{{\ddagger}}}$ | 0.275 | 0.392 | 0.365 | 0.336 | 0.275 | 0.274 | 0.323 | 1.643 | 0.259 | 0.364 | 0.370 | 0.297 | 0.259 | 0.258 | 0.336 | 1.513 | 0.278 | 0.294 | 0.345${}^{\phantom{{\ddagger}}}$ | 0.246 | 0.277 | 0.276${}^{\phantom{{\ddagger}}}$ | 0.335${}^{\phantom{{\ddagger}}}$
Corel5k | 2.284${}^{\phantom{{\ddagger}}}$ | 0.173 | 0.297 | 0.292 | 0.266 | 0.177 | 0.176 | 0.220 | 2.511 | 0.163 | 0.330 | 0.324 | 0.284 | 0.166 | 0.166 | 0.226 | 2.973 | 0.173 | 0.397 | 0.403${}^{\phantom{{\ddagger}}}$ | 0.334 | 0.175 | 0.175${}^{\phantom{{\ddagger}}}$ | 0.266${}^{\phantom{{\ddagger}}}$
Delicious | 6.631${}^{\phantom{{\ddagger}}}$ | 0.234 | 0.356 | 0.341 | 0.281 | 0.215 | 0.225 | 0.325 | 7.474 | 0.235 | 0.410 | 0.412 | 0.318 | 0.194 | 0.215 | 0.387 | 7.327 | 0.279 | 0.439 | 0.464${}^{\phantom{{\ddagger}}}$ | 0.352 | 0.194 | 0.235${}^{\phantom{{\ddagger}}}$ | 0.430${}^{\phantom{{\ddagger}}}$
Average | 1.990${}^{\phantom{{\ddagger}}}$ | 0.330 | 0.248 | 0.242 | 0.216 | 0.329 | 0.358 | 0.211 | 1.931 | 0.415 | 0.332 | 0.364 | 0.299 | 0.415 | 0.450 | 0.301 | 2.421 | 1.025 | 0.931 | 1.077${}^{\phantom{{\ddagger}}}$ | 0.854${}^{\phantom{{\ddagger}}}$ | 1.020 | 1.087${}^{\phantom{{\ddagger}}}$ | 0.952${}^{\phantom{{\ddagger}}}$
Rank Average | 6.7 | 4.6 | 4.3 | 4.3 | 3.1 | 4.8 | 5.6 | 2.5 | 6.9 | 4.4 | 4.3 | 4.7 | 3.1 | 4.5 | 5.3 | 2.9 | 6.9 | 4.2 | 4.2 | 5.3 | 3.1 | 3.8 | 5.0 | 3.5

Table 18: Values of $\operatorname{RAE}$ obtained for different multi-label
aggregation methods in BC+MLA.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  RakEL-LPQ |  kMeans-LPQ0 |  Ridge-RQ |  RF-RQ |  SVR-RQ |  BC+BA |  RakEL-LPQ |  kMeans-LPQ0 |  Ridge-RQ |  RF-RQ |  SVR-RQ |  BC+BA |  RakEL-LPQ |  kMeans-LPQ0 |  Ridge-RQ |  RF-RQ |  SVR-RQ
Emotions | 0.1045 | 0.1119 | 0.1028 | 0.1202 | 0.1179 | 0.1454${}^{\phantom{{\ddagger}}}$ | 0.3657 | 0.3507${}^{\phantom{{\ddagger}}}$ | 0.3490 | 0.2843 | 0.2700 | 0.2935 | 0.5430 | 0.5707${}^{\phantom{{\ddagger}}}$ | 0.5265${}^{\phantom{{\ddagger}}}$ | 0.4284${}^{\phantom{{\ddagger}}}$ | 0.6379${}^{\phantom{{\ddagger}}}$ | 0.4726${}^{\phantom{{\ddagger}}}$
Scene | 0.4068 | 0.1540 | 0.1612 | 0.1426 | 0.1382 | 0.1519${}^{\phantom{{\ddagger}}}$ | 0.7417 | 0.3454${}^{\phantom{{\ddagger}}}$ | 0.3696 | 0.2340 | 0.1695 | 0.2515 | 3.4909 | 2.1628${}^{\phantom{{\ddagger}}}$ | 2.3644${}^{\phantom{{\ddagger}}}$ | 1.1922${}^{\phantom{{\ddagger}}}$ | 1.2952† | 1.0550${}^{\phantom{{\ddagger}}}$
Yeast | 0.9286 | 0.2037 | 0.2049 | 0.2256 | 0.2469 | 0.2084${}^{\phantom{{\ddagger}}}$ | 1.0377 | 0.4173${}^{\phantom{{\ddagger}}}$ | 0.4187 | 0.4065 | 0.3798 | 0.3816 | 1.6070 | 1.1721‡ | 1.1786‡ | 1.0485${}^{\phantom{{\ddagger}}}$ | 0.9558${}^{\phantom{{\ddagger}}}$ | 1.0586${}^{\phantom{{\ddagger}}}$
Birds | 0.5500 | 0.3923 | 0.3263 | 0.3348 | 0.2642 | 0.2646† | 0.6844 | 0.4967${}^{\phantom{{\ddagger}}}$ | 0.4161 | 0.4264 | 0.3353 | 0.3372 | 1.0065 | 0.6907${}^{\phantom{{\ddagger}}}$ | 0.5887${}^{\phantom{{\ddagger}}}$ | 0.4836${}^{\phantom{{\ddagger}}}$ | 0.4117${}^{\phantom{{\ddagger}}}$ | 0.3643${}^{\phantom{{\ddagger}}}$
Genbase | 0.0472 | 0.0155 | 0.0351 | 0.0209 | 0.2265 | 0.0874${}^{\phantom{{\ddagger}}}$ | 0.0506 | 0.0166${}^{\phantom{{\ddagger}}}$ | 0.0362 | 0.0217 | 0.3069 | 0.0947 | 0.0530 | 0.0161${}^{\phantom{{\ddagger}}}$ | 0.0358${}^{\phantom{{\ddagger}}}$ | 0.0217${}^{\phantom{{\ddagger}}}$ | 0.3450${}^{\phantom{{\ddagger}}}$ | 0.0864${}^{\phantom{{\ddagger}}}$
Medical | 0.1725 | 0.1879 | 0.1487 | 0.3360 | 0.3186 | 0.3463${}^{\phantom{{\ddagger}}}$ | 0.2177 | 0.2553${}^{\phantom{{\ddagger}}}$ | 0.1803 | 0.4714 | 0.6118 | 0.5229 | 0.3974 | 0.4843${}^{\phantom{{\ddagger}}}$ | 0.3328${}^{\phantom{{\ddagger}}}$ | 0.7950${}^{\phantom{{\ddagger}}}$ | 1.5663${}^{\phantom{{\ddagger}}}$ | 0.8961${}^{\phantom{{\ddagger}}}$
tmc2007_500 | 1.9392 | 0.1892 | 0.1735 | 0.2462 | 0.2906 | 0.2626${}^{\phantom{{\ddagger}}}$ | 2.1917 | 0.2964† | 0.2709 | 0.2702 | 0.3068 | 0.3226 | 2.4020 | 0.5277${}^{\phantom{{\ddagger}}}$ | 0.4822${}^{\phantom{{\ddagger}}}$ | 0.3361${}^{\phantom{{\ddagger}}}$ | 0.3852${}^{\phantom{{\ddagger}}}$ | 0.4660${}^{\phantom{{\ddagger}}}$
Ohsumed | 1.9122 | 0.1687 | 0.1876 | 0.2185 | 0.2554 | 0.2285${}^{\phantom{{\ddagger}}}$ | 2.3257 | 0.2275${}^{\phantom{{\ddagger}}}$ | 0.2519 | 0.2563 | 0.2858 | 0.2736 | 2.9609 | 0.4925${}^{\phantom{{\ddagger}}}$ | 0.5201${}^{\phantom{{\ddagger}}}$ | 0.3393${}^{\phantom{{\ddagger}}}$ | 0.3995${}^{\phantom{{\ddagger}}}$ | 0.3872${}^{\phantom{{\ddagger}}}$
Enron | 0.6213 | 0.2876 | 0.3000 | 0.3429 | 0.3391 | 0.3272${}^{\phantom{{\ddagger}}}$ | 0.7415 | 0.3847${}^{\phantom{{\ddagger}}}$ | 0.3972 | 0.4201 | 0.4164 | 0.4117 | 1.1457 | 0.6518${}^{\phantom{{\ddagger}}}$ | 0.6627${}^{\phantom{{\ddagger}}}$ | 0.6278${}^{\phantom{{\ddagger}}}$ | 0.4975${}^{\phantom{{\ddagger}}}$ | 0.5685${}^{\phantom{{\ddagger}}}$
Reuters-21578 | 0.9101 | 0.0890 | 0.0936 | 0.1469 | 0.2540 | 0.1389${}^{\phantom{{\ddagger}}}$ | 1.1838 | 0.1091${}^{\phantom{{\ddagger}}}$ | 0.1125 | 0.1754 | 0.2732 | 0.1740 | 1.0398 | 0.1091${}^{\phantom{{\ddagger}}}$ | 0.1145${}^{\phantom{{\ddagger}}}$ | 0.2379${}^{\phantom{{\ddagger}}}$ | 0.2457${}^{\phantom{{\ddagger}}}$ | 0.1616${}^{\phantom{{\ddagger}}}$
RCV1-v2 | 2.0283 | 0.1480 | 0.2005 | 0.2038 | 0.3375 | 0.1896${}^{\phantom{{\ddagger}}}$ | 2.5246 | 0.2026${}^{\phantom{{\ddagger}}}$ | 0.2687 | 0.3146 | 0.5212 | 0.2836 | 2.9699 | 0.3350${}^{\phantom{{\ddagger}}}$ | 0.4316${}^{\phantom{{\ddagger}}}$ | 0.5659${}^{\phantom{{\ddagger}}}$ | 0.9107${}^{\phantom{{\ddagger}}}$ | 0.5063${}^{\phantom{{\ddagger}}}$
Mediamill | 7.6767 | 0.3203 | 0.3214 | 0.4169 | 0.3768 | 0.2708${}^{\phantom{{\ddagger}}}$ | 8.6165 | 0.4931${}^{\phantom{{\ddagger}}}$ | 0.4933 | 0.5394 | 0.4639 | 0.3722 | 8.9409 | 0.7837${}^{\phantom{{\ddagger}}}$ | 0.7821${}^{\phantom{{\ddagger}}}$ | 0.6751${}^{\phantom{{\ddagger}}}$ | 0.5796${}^{\phantom{{\ddagger}}}$ | 0.5320${}^{\phantom{{\ddagger}}}$
Bibtex | 1.6686 | 0.3097 | 0.3183 | 0.3959 | 0.4328 | 0.3532${}^{\phantom{{\ddagger}}}$ | 1.6432 | 0.3180${}^{\phantom{{\ddagger}}}$ | 0.3320 | 0.3897 | 0.4883 | 0.3297 | 1.5127 | 0.3186${}^{\phantom{{\ddagger}}}$ | 0.3338${}^{\phantom{{\ddagger}}}$ | 0.3796${}^{\phantom{{\ddagger}}}$ | 0.6234${}^{\phantom{{\ddagger}}}$ | 0.3046${}^{\phantom{{\ddagger}}}$
Corel5k | 2.2844 | 0.2205 | 0.2941 | 0.2996 | 0.2903 | 0.2299${}^{\phantom{{\ddagger}}}$ | 2.5112 | 0.2337${}^{\phantom{{\ddagger}}}$ | 0.3164 | 0.3265 | 0.3091 | 0.2458 | 2.9733 | 0.2669${}^{\phantom{{\ddagger}}}$ | 0.3760${}^{\phantom{{\ddagger}}}$ | 0.3956${}^{\phantom{{\ddagger}}}$ | 0.3686${}^{\phantom{{\ddagger}}}$ | 0.2898${}^{\phantom{{\ddagger}}}$
Delicious | 6.6314 | 0.3309 | 0.3353 | 0.3425 | 0.3633 | 0.3012${}^{\phantom{{\ddagger}}}$ | 7.4738 | 0.4034${}^{\phantom{{\ddagger}}}$ | 0.4165 | 0.3628 | 0.3635 | 0.3357 | 7.3272 | 0.4569${}^{\phantom{{\ddagger}}}$ | 0.4822${}^{\phantom{{\ddagger}}}$ | 0.3565‡ | 0.3459${}^{\phantom{{\ddagger}}}$ | 0.3599†
Average | 1.9897 | 0.2101 | 0.2188 | 0.2598 | 0.2965 | 0.2366${}^{\phantom{{\ddagger}}}$ | 1.9312 | 0.3035${}^{\phantom{{\ddagger}}}$ | 0.3058 | 0.3213 | 0.3705 | 0.3043 | 2.4206 | 0.8541${}^{\phantom{{\ddagger}}}$ | 0.8796${}^{\phantom{{\ddagger}}}$ | 0.6423‡ | 0.7820${}^{\phantom{{\ddagger}}}$ | 0.6270${}^{\phantom{{\ddagger}}}$
Rank Average | 5.3 | 1.9 | 2.5 | 3.9 | 4.1 | 3.2 | 5.6 | 2.5 | 3.1 | 3.3 | 3.5 | 2.9 | 5.5 | 3.3 | 3.7 | 2.7 | 3.4 | 2.5

##  Appendix B Exploring other Regressors in RQ

We here report additional experiments that extend the ones presented in
Section 6.5; the present experiments concern the use of regression algorithms
other than ridge regression (Ridge-RQ), random forest regression (RF-RQ), and
linear SVR (SVR-RQ), which were the only regression algorithms we considered
in Section 6.5. Tables 19 and 20 report the results of these experiments in
terms of AE and RAE, respectively, obtained by optimizing the hyperparameters
shown in Table 21. These results show that there are no substantial
differences in performance for the low-shift regime, while these differences
are instead noticeable in the mid-shift and (especially) in the high-shift
regimes. These results are well correlated with the results we obtained in the
validation phase, on which we relied upon for choosing ridge regression,
random forest regression, and linear SVR, as representative regression models.

Table 19: $\operatorname{AE}$ for different regressors for multi-label
quantifiers

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  Ridge (Ridge-RQ) |  MultitaskLasso |  RandomForest (RF-RQ)  |  StackedLinearSVR |  ChainedLinearSVR0 |  LinearSVR (SVR-RQ) |  BC+BA |  Ridge (Ridge-RQ) |  MultitaskLasso |  RandomForest (RF-RQ)  |  StackedLinearSVR |  ChainedLinearSVR0 |  LinearSVR (SVR-RQ) |  BC+BA |  Ridge (Ridge-RQ) |  MultitaskLasso |  RandomForest (RF-RQ)  |  StackedLinearSVR |  ChainedLinearSVR0 |  LinearSVR (SVR-RQ)
Emotions | .0418 | .0500 | .0633 | .0484${}^{\phantom{{\ddagger}}}$ | .0687 | .0574 | .0586${}^{\phantom{{\ddagger}}}$ | .0685 | .0626 | .0811 | .0707 | .1038 | .1153 | .0720${}^{\phantom{{\ddagger}}}$ | .0923 | .0782${}^{\phantom{{\ddagger}}}$ | .1029 | .1235${}^{\phantom{{\ddagger}}}$ | .1295 | .1767 | .0951
Scene | .0842 | .0350 | .0440 | .0303${}^{\phantom{{\ddagger}}}$ | .0363 | .0359 | .0384${}^{\phantom{{\ddagger}}}$ | .1140 | .0454 | .1110 | .0306 | .0859 | .0868 | .0459${}^{\phantom{{\ddagger}}}$ | .1427 | .0581${}^{\phantom{{\ddagger}}}$ | .1764 | .0507‡ | .1251 | .1260 | .0499
Yeast | .1756 | .0477 | .0491 | .0561${}^{\phantom{{\ddagger}}}$ | .0457 | .0464 | .0480${}^{\phantom{{\ddagger}}}$ | .1900 | .0920 | .0943 | .0956 | .0982 | .1004 | .0913${}^{\phantom{{\ddagger}}}$ | .2206 | .1374${}^{\phantom{{\ddagger}}}$ | .1424 | .1517${}^{\phantom{{\ddagger}}}$ | .1517 | .1542 | .1361
Birds | .0286 | .0213 | .0230 | .0185${}^{\phantom{{\ddagger}}}$ | .0217 | .0213 | .0181${}^{\phantom{{\ddagger}}}$ | .0328 | .0289 | .0299 | .0251 | .0297 | .0280 | .0243${}^{\phantom{{\ddagger}}}$ | .0440 | .0405${}^{\phantom{{\ddagger}}}$ | .0424 | .0382${}^{\phantom{{\ddagger}}}$ | .0436 | .0427 | .0351
Genbase | .0011 | .0015 | .0137 | .0103${}^{\phantom{{\ddagger}}}$ | .0168 | .0177 | .0022${}^{\phantom{{\ddagger}}}$ | .0011 | .0014 | .0175 | .0148 | .0245 | .0246 | .0025${}^{\phantom{{\ddagger}}}$ | .0010 | .0013${}^{\phantom{{\ddagger}}}$ | .0173 | .0172${}^{\phantom{{\ddagger}}}$ | .0364 | .0347 | .0023
Medical | .0127 | .0230 | .0186 | .0202${}^{\phantom{{\ddagger}}}$ | .0207 | .0194 | .0191${}^{\phantom{{\ddagger}}}$ | .0146 | .0306 | .0263 | .0393 | .0315 | .0265 | .0279${}^{\phantom{{\ddagger}}}$ | .0169 | .0365${}^{\phantom{{\ddagger}}}$ | .0314 | .0651${}^{\phantom{{\ddagger}}}$ | .0439 | .0380 | .0351
tmc2007_500 | .1108 | .0226 | .0260 | .0211${}^{\phantom{{\ddagger}}}$ | .0312 | .0272 | .0213‡ | .1154 | .0274 | .0436 | .0224 | .0528 | .0508 | .0292${}^{\phantom{{\ddagger}}}$ | .1008 | .0282${}^{\phantom{{\ddagger}}}$ | .0571 | .0237${}^{\phantom{{\ddagger}}}$ | .0726 | .0714 | .0321
Ohsumed | .1004 | .0193 | .0220 | .0183‡ | .0231 | .0230 | .0183${}^{\phantom{{\ddagger}}}$ | .1087 | .0208 | .0363 | .0184 | .0433 | .0428 | .0209${}^{\phantom{{\ddagger}}}$ | .1177 | .0221${}^{\phantom{{\ddagger}}}$ | .0472 | .0201${}^{\phantom{{\ddagger}}}$ | .0601 | .0591 | .0215
Enron | .0347 | .0192 | .0187 | .0174${}^{\phantom{{\ddagger}}}$ | .0180 | .0189 | .0169${}^{\phantom{{\ddagger}}}$ | .0397 | .0250 | .0256 | .0219 | .0267 | .0269 | .0227${}^{\phantom{{\ddagger}}}$ | .0439 | .0287${}^{\phantom{{\ddagger}}}$ | .0300 | .0212${}^{\phantom{{\ddagger}}}$ | .0332 | .0325 | .0253
Reuters-21578 | .0167 | .0048 | .0072 | .0073${}^{\phantom{{\ddagger}}}$ | .0082 | .0082 | .0049${}^{\phantom{{\ddagger}}}$ | .0243 | .0070 | .0114 | .0109 | .0181 | .0182 | .0070‡ | .0370 | .0103${}^{\phantom{{\ddagger}}}$ | .0143 | .0135${}^{\phantom{{\ddagger}}}$ | .0267 | .0279 | .0088
RCV1-v2 | .0456 | .0095 | .0129 | .0125${}^{\phantom{{\ddagger}}}$ | .0147 | .0147 | .0093${}^{\phantom{{\ddagger}}}$ | .0533 | .0144 | .0245 | .0208 | .0295 | .0295 | .0146${}^{\phantom{{\ddagger}}}$ | .0654 | .0221${}^{\phantom{{\ddagger}}}$ | .0366 | .0370${}^{\phantom{{\ddagger}}}$ | .0472 | .0472 | .0215
Mediamill | .1697 | .0184 | .0169 | .0158${}^{\phantom{{\ddagger}}}$ | .0167 | .0176 | .0157${}^{\phantom{{\ddagger}}}$ | .1736 | .0259 | .0266 | .0229 | .0315 | .0321 | .0251${}^{\phantom{{\ddagger}}}$ | .1806 | .0316${}^{\phantom{{\ddagger}}}$ | .0321 | .0294${}^{\phantom{{\ddagger}}}$ | .0440 | .0442 | .0322
Bibtex | .0354 | .0097 | .0105 | .0104${}^{\phantom{{\ddagger}}}$ | .0106 | .0111 | .0092${}^{\phantom{{\ddagger}}}$ | .0374 | .0125 | .0154 | .0147 | .0168 | .0157 | .0116${}^{\phantom{{\ddagger}}}$ | .0423 | .0140${}^{\phantom{{\ddagger}}}$ | .0198 | .0199${}^{\phantom{{\ddagger}}}$ | .0224 | .0197 | .0129
Corel5k | .0582 | .0074 | .0080 | .0087${}^{\phantom{{\ddagger}}}$ | .0073 | .0091 | .0075${}^{\phantom{{\ddagger}}}$ | .0585 | .0084 | .0088 | .0091 | .0088 | .0102 | .0082${}^{\phantom{{\ddagger}}}$ | .0594 | .0094${}^{\phantom{{\ddagger}}}$ | .0098 | .0100${}^{\phantom{{\ddagger}}}$ | .0106 | .0117 | .0087
Delicious | .1420 | .0093 | .0096 | .0093${}^{\phantom{{\ddagger}}}$ | .0100 | .0109 | .0093${}^{\phantom{{\ddagger}}}$ | .1417 | .0108 | .0134 | .0102 | .0139 | .0147 | .0109${}^{\phantom{{\ddagger}}}$ | .1238 | .0102${}^{\phantom{{\ddagger}}}$ | .0151 | .0092${}^{\phantom{{\ddagger}}}$ | .0163 | .0172 | .0104
Average | .0677 | .0180 | .0206 | .0186${}^{\phantom{{\ddagger}}}$ | .0211 | .0207 | .0177${}^{\phantom{{\ddagger}}}$ | .0761 | .0288 | .0401 | .0300 | .0440 | .0444 | .0288${}^{\phantom{{\ddagger}}}$ | .1012 | .0454‡ | .0781 | .0529${}^{\phantom{{\ddagger}}}$ | .0788 | .0831 | .0452
Rank Average | 5.8 | 3.1 | 4.3 | 3.0 | 4.6 | 4.8 | 2.3 | 5.9 | 2.3 | 4.2 | 2.7 | 5.2 | 5.5 | 2.3 | 5.8 | 2.3 | 3.9 | 3.1 | 5.5 | 5.4 | 2.0

Table 20: $\operatorname{RAE}$ for different regressors for multi-label
quantifiers.

| low shift | mid shift | high shift
---|---|---|---
|  BC+BA |  Ridge (Ridge-RQ) |  MultitaskLasso |  RandomForest (RF-RQ)  |  StackedLinearSVR |  ChainedLinearSVR0 |  LinearSVR (SVR-RQ) |  BC+BA |  Ridge (Ridge-RQ) |  MultitaskLasso |  RandomForest (RF-RQ)  |  StackedLinearSVR |  ChainedLinearSVR0 |  LinearSVR (SVR-RQ) |  BC+BA |  Ridge (Ridge-RQ) |  MultitaskLasso |  RandomForest (RF-RQ)  |  StackedLinearSVR |  ChainedLinearSVR0 |  LinearSVR (SVR-RQ)
Emotions | 0.104 | 0.120 | 0.149 | 0.118 | 0.165 | 0.149${}^{\phantom{{\ddagger}}}$ | 0.145${}^{\phantom{{\ddagger}}}$ | 0.366 | 0.284${}^{\phantom{{\ddagger}}}$ | 0.360${}^{\phantom{{\ddagger}}}$ | 0.270 | 0.454 | 0.559 | 0.293 | 0.543 | 0.428${}^{\phantom{{\ddagger}}}$ | 0.543${}^{\phantom{{\ddagger}}}$ | 0.638${}^{\phantom{{\ddagger}}}$ | 0.716 | 1.026 | 0.473${}^{\phantom{{\ddagger}}}$
Scene | 0.407 | 0.143 | 0.220 | 0.138 | 0.173 | 0.170${}^{\phantom{{\ddagger}}}$ | 0.152${}^{\phantom{{\ddagger}}}$ | 0.742 | 0.234${}^{\phantom{{\ddagger}}}$ | 0.578${}^{\phantom{{\ddagger}}}$ | 0.169 | 0.455 | 0.443 | 0.252 | 3.491 | 1.192${}^{\phantom{{\ddagger}}}$ | 4.040${}^{\phantom{{\ddagger}}}$ | 1.295† | 2.857 | 2.743 | 1.055${}^{\phantom{{\ddagger}}}$
Yeast | 0.929 | 0.226 | 0.217 | 0.247 | 0.220 | 0.228${}^{\phantom{{\ddagger}}}$ | 0.208${}^{\phantom{{\ddagger}}}$ | 1.038 | 0.407${}^{\phantom{{\ddagger}}}$ | 0.403‡ | 0.380 | 0.438 | 0.455 | 0.382 | 1.607 | 1.049${}^{\phantom{{\ddagger}}}$ | 1.081† | 0.956${}^{\phantom{{\ddagger}}}$ | 1.199 | 1.238 | 1.059${}^{\phantom{{\ddagger}}}$
Birds | 0.550 | 0.335 | 0.346 | 0.264 | 0.353 | 0.305${}^{\phantom{{\ddagger}}}$ | 0.265† | 0.684 | 0.426${}^{\phantom{{\ddagger}}}$ | 0.435${}^{\phantom{{\ddagger}}}$ | 0.335 | 0.462 | 0.405 | 0.337 | 1.007 | 0.484${}^{\phantom{{\ddagger}}}$ | 0.510${}^{\phantom{{\ddagger}}}$ | 0.412${}^{\phantom{{\ddagger}}}$ | 0.595 | 0.609 | 0.364${}^{\phantom{{\ddagger}}}$
Genbase | 0.047 | 0.021 | 0.266 | 0.226 | 0.283 | 0.336${}^{\phantom{{\ddagger}}}$ | 0.087${}^{\phantom{{\ddagger}}}$ | 0.051 | 0.022${}^{\phantom{{\ddagger}}}$ | 0.355${}^{\phantom{{\ddagger}}}$ | 0.307 | 0.411 | 0.445 | 0.095 | 0.053 | 0.022${}^{\phantom{{\ddagger}}}$ | 0.369${}^{\phantom{{\ddagger}}}$ | 0.345${}^{\phantom{{\ddagger}}}$ | 0.652 | 0.675 | 0.086${}^{\phantom{{\ddagger}}}$
Medical | 0.173 | 0.336 | 0.305 | 0.319 | 0.297 | 0.226${}^{\phantom{{\ddagger}}}$ | 0.346${}^{\phantom{{\ddagger}}}$ | 0.218 | 0.471${}^{\phantom{{\ddagger}}}$ | 0.445${}^{\phantom{{\ddagger}}}$ | 0.612 | 0.449 | 0.320 | 0.523 | 0.397 | 0.795${}^{\phantom{{\ddagger}}}$ | 0.693${}^{\phantom{{\ddagger}}}$ | 1.566${}^{\phantom{{\ddagger}}}$ | 0.919 | 0.711 | 0.896${}^{\phantom{{\ddagger}}}$
tmc2007_500 | 1.939 | 0.246 | 0.377 | 0.291 | 0.331 | 0.279${}^{\phantom{{\ddagger}}}$ | 0.263${}^{\phantom{{\ddagger}}}$ | 2.192 | 0.270${}^{\phantom{{\ddagger}}}$ | 0.495${}^{\phantom{{\ddagger}}}$ | 0.307 | 0.526 | 0.463 | 0.323 | 2.402 | 0.336${}^{\phantom{{\ddagger}}}$ | 0.900${}^{\phantom{{\ddagger}}}$ | 0.385${}^{\phantom{{\ddagger}}}$ | 1.092 | 0.975 | 0.466${}^{\phantom{{\ddagger}}}$
Ohsumed | 1.912 | 0.219 | 0.332 | 0.255 | 0.254 | 0.250${}^{\phantom{{\ddagger}}}$ | 0.228${}^{\phantom{{\ddagger}}}$ | 2.326 | 0.256${}^{\phantom{{\ddagger}}}$ | 0.456${}^{\phantom{{\ddagger}}}$ | 0.286 | 0.444 | 0.435 | 0.274 | 2.961 | 0.339${}^{\phantom{{\ddagger}}}$ | 0.953${}^{\phantom{{\ddagger}}}$ | 0.400${}^{\phantom{{\ddagger}}}$ | 1.186 | 1.168 | 0.387${}^{\phantom{{\ddagger}}}$
Enron | 0.621 | 0.343 | 0.346 | 0.339 | 0.338 | 0.328‡ | 0.327${}^{\phantom{{\ddagger}}}$ | 0.741 | 0.420${}^{\phantom{{\ddagger}}}$ | 0.443${}^{\phantom{{\ddagger}}}$ | 0.416 | 0.455 | 0.428 | 0.412 | 1.146 | 0.628${}^{\phantom{{\ddagger}}}$ | 0.691${}^{\phantom{{\ddagger}}}$ | 0.498${}^{\phantom{{\ddagger}}}$ | 0.741 | 0.651 | 0.569${}^{\phantom{{\ddagger}}}$
Reuters-21578 | 0.910 | 0.147 | 0.261 | 0.254 | 0.227 | 0.232${}^{\phantom{{\ddagger}}}$ | 0.139${}^{\phantom{{\ddagger}}}$ | 1.184 | 0.175† | 0.291${}^{\phantom{{\ddagger}}}$ | 0.273 | 0.389 | 0.382 | 0.174 | 1.040 | 0.238${}^{\phantom{{\ddagger}}}$ | 0.246${}^{\phantom{{\ddagger}}}$ | 0.246${}^{\phantom{{\ddagger}}}$ | 0.393 | 0.437 | 0.162${}^{\phantom{{\ddagger}}}$
RCV1-v2 | 2.028 | 0.204 | 0.329 | 0.338 | 0.324 | 0.324${}^{\phantom{{\ddagger}}}$ | 0.190${}^{\phantom{{\ddagger}}}$ | 2.525 | 0.315${}^{\phantom{{\ddagger}}}$ | 0.585${}^{\phantom{{\ddagger}}}$ | 0.521 | 0.638 | 0.643 | 0.284 | 2.970 | 0.566${}^{\phantom{{\ddagger}}}$ | 0.889${}^{\phantom{{\ddagger}}}$ | 0.911${}^{\phantom{{\ddagger}}}$ | 1.060 | 1.068 | 0.506${}^{\phantom{{\ddagger}}}$
Mediamill | 7.677 | 0.417 | 0.407 | 0.377 | 0.331 | 0.361${}^{\phantom{{\ddagger}}}$ | 0.271${}^{\phantom{{\ddagger}}}$ | 8.616 | 0.539${}^{\phantom{{\ddagger}}}$ | 0.524${}^{\phantom{{\ddagger}}}$ | 0.464 | 0.523 | 0.561 | 0.372 | 8.941 | 0.675${}^{\phantom{{\ddagger}}}$ | 0.649${}^{\phantom{{\ddagger}}}$ | 0.580${}^{\phantom{{\ddagger}}}$ | 0.810 | 0.874 | 0.532${}^{\phantom{{\ddagger}}}$
Bibtex | 1.669 | 0.396 | 0.434 | 0.433 | 0.425 | 0.427${}^{\phantom{{\ddagger}}}$ | 0.353${}^{\phantom{{\ddagger}}}$ | 1.643 | 0.390${}^{\phantom{{\ddagger}}}$ | 0.490${}^{\phantom{{\ddagger}}}$ | 0.488 | 0.523 | 0.477 | 0.330 | 1.513 | 0.380${}^{\phantom{{\ddagger}}}$ | 0.567${}^{\phantom{{\ddagger}}}$ | 0.623${}^{\phantom{{\ddagger}}}$ | 0.661 | 0.544 | 0.305${}^{\phantom{{\ddagger}}}$
Corel5k | 2.284 | 0.300 | 0.296 | 0.290 | 0.320 | 0.423${}^{\phantom{{\ddagger}}}$ | 0.230${}^{\phantom{{\ddagger}}}$ | 2.511 | 0.327${}^{\phantom{{\ddagger}}}$ | 0.316${}^{\phantom{{\ddagger}}}$ | 0.309 | 0.362 | 0.453 | 0.246 | 2.973 | 0.396${}^{\phantom{{\ddagger}}}$ | 0.374${}^{\phantom{{\ddagger}}}$ | 0.369${}^{\phantom{{\ddagger}}}$ | 0.460 | 0.549 | 0.290${}^{\phantom{{\ddagger}}}$
Delicious | 6.631 | 0.343 | 0.369 | 0.363 | 0.371 | 0.415${}^{\phantom{{\ddagger}}}$ | 0.301${}^{\phantom{{\ddagger}}}$ | 7.474 | 0.363${}^{\phantom{{\ddagger}}}$ | 0.462${}^{\phantom{{\ddagger}}}$ | 0.364 | 0.478 | 0.532 | 0.336 | 7.327 | 0.357‡ | 0.567${}^{\phantom{{\ddagger}}}$ | 0.346${}^{\phantom{{\ddagger}}}$ | 0.648 | 0.720 | 0.360†
Average | 1.990 | 0.260 | 0.319 | 0.296 | 0.302 | 0.307 | 0.237${}^{\phantom{{\ddagger}}}$ | 1.931 | 0.321${}^{\phantom{{\ddagger}}}$ | 0.458${}^{\phantom{{\ddagger}}}$ | 0.370 | 0.483 | 0.478 | 0.304 | 2.421 | 0.642‡ | 1.358${}^{\phantom{{\ddagger}}}$ | 0.782${}^{\phantom{{\ddagger}}}$ | 1.276 | 1.260 | 0.627${}^{\phantom{{\ddagger}}}$
Rank Average | 5.9 | 3.0 | 4.9 | 3.8 | 4.3 | 4.1 | 2.1 | 6.1 | 2.7 | 4.4 | 2.6 | 5.3 | 4.9 | 2.1 | 6.0 | 2.3 | 3.9 | 3.0 | 5.5 | 5.3 | 2.0

Table 21: Hyperparameters explored during model selection for different
regressors. Only the hyperparameters which are specific to the regressor are
listed. All methods are deployed with an LR classifier; for hyperparameters
$C$ and _ClassWeight_ , we explore in the ranges
$\\{10^{-1},\ldots,10^{2},10^{3}\\}$ and {Balanced, None}, respectively.

Regressor | Hyperparameter | Description | Values
---|---|---|---
Ridge | _Alpha_ | regularization strength | {$10^{-3},\ldots,10^{2},10^{3}$}
MultitaskLasso | _Alpha_ | regularization strength | {$10^{-3},\ldots,10^{2},10^{3}$}
RandomForest | _N_estimators_ | number of trees | {10, 100, 200}
StackedLinearSVR | $C$ | inverse of regularization strength in Linear SVR | {$10^{-1},\ldots,10^{2},10^{3}$}
ChainedLinearSVR0 | $C$ | inverse of regularization strength in Linear SVR | {$10^{-1},\ldots,10^{2},10^{3}$}
LinearSVR | $C$ | inverse of regularization strength in Linear SVR | {$10^{-1},\ldots,10^{2},10^{3}$}


$a^{\prime}+c^{\prime}+b^{\prime}=k$ true. We call $(V,E)$ a _relaxed edge-
magic total matching partition_ of $[1,p+q]$.

$\ast$ Find all possible relaxed edge-magic total matching partitions $(V,E)$
of $[1,p+q]$.

  2. Set-2.

From Definition 10: Select two subsets $V,E\subset[1,2q-1]$ with
$E=[1,2p-1]^{o}$ such that there is a constant $k$, each $c\in E$ corresponds
two distinct $a,b\in V$ holding $a+b+c=k$. We call $(V,E)$ an _odd-edge-magic
matching partition_ of $[1,2q-1]$.

$\ast$ Find all possible odd-edge-magic matching partitions $[1,2q-1]$.

  3. Set-3.

From Definition 11: Selecting two subsets $V,E\subset[1,2q-1]$ with
$E=[1,2p-1]^{o}$ holds: (i) each $c\in E$ corresponds $a,b\in V$ to form an
_ev-matching_ $(acb)$; (ii) each $c\in E$ corresponds to $z\in E$ with the ev-
matching $(xzy)$ such that $c=|x-y|$; (iii) each $c\in E$ with the ev-matching
$(acb)$ corresponds to $c^{\prime}\in E$ with the ev-matching
$(a^{\prime}c^{\prime}b^{\prime})$ such that
$(|a-b|-c)+(|a^{\prime}-b^{\prime}|-c^{\prime})=0$. We call $(V,E)$ an _ee-
difference odd-edge-magic matching partition_ of $[1,2q-1]$.

$\ast$ Find all possible ee-difference odd-edge-magic matching partitions
$(V,E)$ of $[1,2q-1]$. Here, each ev-matching $(acb)$ corresponds an edge $c$
of a graph, where the edge $c$ has two ends $a,b$.

  4. Set-4.

From Definition 12: Partition $[1,p+q]$ into two subsets $V,E$ satisfy: (i)
each $c\in E$ corresponds $a,b\in V$ to form an _ev-matching_ $(acb)$; (ii)
(e-magic) each $c\in E$ with the ev-matching $(acb)$ hold $c+|a-b|=k$; (iii)
(ee-difference) each $c\in E$ corresponds to $z\in E$ with the ev-matching
$(xzy)$ such that $c=|x-y|$; (iv) (ee-bandwiden) each $c\in E$ with the ev-
matching $(acb)$ corresponds to $c^{\prime}\in E$ with the ev-matching
$(a^{\prime}c^{\prime}b^{\prime})$ such that
$(|a-b|-c)+(|a^{\prime}-b^{\prime}|-c^{\prime})=0$; (iv) (EV-ordered) $\max
V<\min E$ (or $\max V>\min E$); (v) (ve-matching) each ev-matching $(acb)$
matches with another ev-matching $(uwv)$ such that $a+w=k^{\prime}$ or
$b+w=k^{\prime}$, $k^{\prime}$ is a constant, except the _singularity_
$\lfloor\frac{p+q+1}{2}\rfloor$. We call $(V,E)$ a _6C-partition_ of
$[1,p+q]$.

For a given 6C-partition $(V,E)$ of $[1,p+q]$, if there exists another
6C-partition $(V^{\prime},E^{\prime})$ of $[1,p+q]$ such that
$V\setminus(V\cap V^{\prime})=E^{\prime}$, $E=V^{\prime}\setminus(V\cap
V^{\prime})$ for $V\cap V^{\prime}=\\{\lfloor\frac{p+q+1}{2}\rfloor\\}$, we
get a partition $(V\cup E^{\prime},E\cup V)$, and call it a _6C-complementary
matching partition_ of $[1,p+q]$.

$\ast$ Find all possible 6C-partitions $(V,E)$ of $[1,p+q]$, and all possible
6C-complementary matching partitions $(V\cup E^{\prime},E\cup V)$.

  5. Set-5.

From Definition 13: Partition $[0,p+q-1]$ into two subsets $V,E$ with
$E=[1,q]$ and $V\subseteq[0,q-1]$ satisfy: (i) each $c\in E$ corresponds
$a,b\in V$ to form an _ev-matching_ $(acb)$; (ii) (ee-difference) each $c\in
E$ corresponds to $z\in E$ with the ev-matching $(xzy)$ such that $c=|x-y|$;
(iii) (ee-bandwiden) each $c\in E$ with the ev-matching $(acb)$ corresponds to
$c^{\prime}\in E$ with the ev-matching $(a^{\prime}c^{\prime}b^{\prime})$ such
that
$(|a-b|-c)+(|a^{\prime}-b^{\prime}|-c^{\prime})=|a-b|+|a^{\prime}-b^{\prime}|-(c+c^{\prime})=0$;
(iv) there exists a constant $k$ such that each $c\in E$ with its ev-matching
$(acb)$ holds $c+|a-b|=k$; (v) each $c\in E$ corresponds another
$c^{\prime}\in E$ with its ev-matching $(a^{\prime}c^{\prime}b^{\prime})$ such
that $c+a^{\prime}=p$ or $c+b^{\prime}=p$. We call $(V,E)$ an _ee-difference
graceful-magic matching partition_ of $[0,p+q-1]$.

$\ast$ Find all possible ee-difference graceful-magic matching partitions of
$[0,p+q-1]$.

  6. Set-6.

From Definition 15: Partition $[1,p+q]$ into two disjoint subsets $V$ and $E$
with $V\cup E=[1,p+q]$ such that each $c\in E$ corresponds to distinct $a,b\in
V$ holding $c=|a-b|$, and there exists a constant $k$ satisfying $a+c+b=k$ for
each $c\in E$ which corresponds to distinct $a,b\in V$. We call $(V,E)$ an
_edge-magic graceful matching partition_ of $[1,p+q]$. If $(E,V)$ is another
edge-magic graceful matching partition of $[1,p+q]$, we say $(V,E)$ (resp.
$(E,V)$) to be a _ve-exchanged matching partition_ of $[1,p+q]$.

$\ast$ Find all possible edge-magic graceful matching partitions of $[1,p+q]$,
and all possible ve-exchanged matching partitions.

  7. Set-7.

If there are two sets $V\subseteq[0,q]^{2}$ (or $[0,2q-1]^{2}$) and
$E\subseteq[1,q]$ (or $[1,2q-1]$) such that each $c\in E$ with its ev-matching
$(acb)$ holds $c=|a-b|$, where $a\in A\in V$ and $b\in B\in V$ with $A\cap
B=\emptyset$, then we call $(V,E)$ a _v-set e-proper graceful (or odd-
graceful) matching partition_.

  8. Set-8.

From the twin odd-graceful/odd-elegant labellings: Partition $[0,2q]$ into two
subsets $S_{1},S_{2}$ such that $S_{1}\subset[0,2q-1]$, $S_{2}\subset[1,2q]$,
$|S_{1}\cap S_{2}|=1$ and $S_{1}\cup S_{2}=[0,2q]$. For
$E_{1}=E_{2}=[1,2q-1]^{o}$, each $c_{i}\in E_{i}$ corresponds two numbers
$a_{i},b_{i}\in S_{i}$ holding $c_{i}=|a_{i}-b_{i}|$ (or
$c_{i}=a_{i}+b_{i}~{}(\bmod~{}2q)$) with $i=1,2$. We call $(S_{1},S_{2})$ a
_twin odd-graceful (or odd-elegant) matching partition_ of $[0,2q]$.

$\ast$ Characterize twin odd-graceful (or odd-elegant) matching partitions,
and find them.

  9. Set-9.

From Definitions 17 and 18: Select a subset $E\subset[0,p-1]$ such that each
$c\in E$ corresponds two distinct $a,b\in V=[0,p-1]$ to hold $c=|a-b|$ (or
$c=a+b~{}(\bmod~{}|E|)$), we call $f_{E}=(V,E)$ a _graph matching partition_ ,
and call $S_{um}(G,f_{E})=\sum_{c\in E}|a-b|$ a _difference-sum number_ (or
$F_{um}(G,f_{E})=\sum_{c\in E}(a+b)~{}(\bmod~{}|E|)$ is a _felicitous-sum
number_).

$\ast$ Determine $\max_{f_{E}}S_{um}(G,f_{E})$ (profit) and
$\min_{f_{E}}S_{um}(G,f_{E})$ (cost) over all difference-sum matching
partitions $f_{E}=(V,E)$ of $[0,p-1]$. Find $\max_{f_{E}}F_{um}(G,f_{E})$ and
$\min_{f_{E}}F_{um}(G,f_{E})$ over all felicitous-sum matching partitions
$f_{E}=(V,E)$ of $[0,p-1]$.

It may be interesting to consider such algebraic groups on the above set
partition problems.

###  V-B Matching partitions of algebraic matrices

We introduce an algebraic expression of a $(p,q)$-graph $G$ as follows:

######  Definition 22.

A _Topsnut-matrix_ $A_{vev}(G)$ of a $(p,q)$-graph $G$ is defined as
$A_{vev}(G)=(X,W,Y)^{-1}$ with

${\begin{split}&X=(x_{1}~{}x_{2}~{}\cdots~{}x_{q}),W=(e_{1}~{}e_{2}~{}\cdots~{}e_{q})\\\
&Y=(y_{1}~{}y_{2}~{}\cdots~{}y_{q}),\end{split}}$ (33)

and $G$ has another _Topsnut-matrix_ $A_{vv}(G)$ defined as
$A_{vv}(G)=(X,Y)^{-1}$, where $X,Y$ are called _vertex-vectors_ , $W$ is
called _edge-vector_. $\Box$

So, $A_{vev}(G)$ is a $(3\times q)$-matrix, and $A_{vv}(G)$ is a $(2\times
q)$-matrix. Clearly, such $(3\times q)$-matrices and $(2\times q)$-matrices
are easy saved in computer, and produce quickly text-based passwords. For
example, we have the following four Topsnut-matrices $A_{1},A_{2},B_{1}$ and
$B_{2}$ from Fig.37 to Fig.40:

Figure 37:  A $(3\times 11)$-matrix $A_{1}$ for $G_{a}$ in the perfect Max-min
difference-sum matching partition $A$ shown in Fig.25.
Figure 38:  A $(3\times 11)$-matrix $A_{2}$ for $G_{d}$ in the perfect Max-min
difference-sum matching partition $A$ shown in Fig.25.
Figure 39:  A $(3\times 7)$-matrix $B_{1}$ for $F_{b}$ in the perfect Max-min
felicitous-sum matching partition $B$ shown in Fig.25.
Figure 40:  A $(3\times 7)$-matrix $B_{1}$ for $F_{c}$ in the perfect Max-min
felicitous-sum matching partition $B$ shown in Fig.25.

We point out: (i) A $(3\times q)$-matrix $A_{vev}(G)$ is not unique for
expressing a Topsnut-gpw $G$, in other words, a Topsnut-gpw $G$ many have two
or more $(3\times q)$-matrices; (ii) Topsnut-matrices differ from popular
algebraic matrices, since Topsnut-matrices are only the expression of joining
vertices by edges. Clearly, we need algebraic operations on Topsnut-matrices.

Let $D(A)$ be the matrix of the perfect Max-min difference-sum matching
partition $A=\odot_{12}\langle G_{a},G_{d}\rangle$ shown in Fig.25. So, $D(A)$
is a $(3\times 22)$-matrix, denoted directly as $D(A)=\odot_{12}\langle
A_{1},A_{2}\rangle$ (see Fig.37 and Fig.38), called a _matrix matching
partition_. Similarly, the matrix $D(B)$ of the perfect Max-min felicitous-sum
matching partition $B=\odot_{9}\langle F_{b},F_{c}\rangle$ shown in Fig.25 is
a $(3\times 16)$-matrix, and we have another _matrix matching partition_
$D(B)=\odot_{8}\langle B_{1},B_{2}\rangle$ (see Fig.39 and Fig.40).

Along the orange line in the matrix $B_{1}$, we can get a text-based password

$T_{ext}(B_{1})=077088066033473385561572$

and another text-based password

$T_{ext}(B_{2})=731734825701611001143323$

along the orange line in the matrix $B_{2}$. Obviously, it is not easy to
reconstruct the perfect Max-min felicitous-sum matching partition $B$ shown in
Fig.25 from $T_{ext}(B_{1})$ and $T_{ext}(B_{2})$, even it is impossible if
Topsnut-gpws with large numbers of vertices and edges.

By the vertex-split and vertex-identifying operations, as well as the edge-
split and edge-identifying operations, we can define algebraic operations on
$(3\times q)$-matrices of $(p,q)$-graphs that are topological structures of
Topsnut-gpws, such as $D(A)=\odot_{12}\langle A_{1},A_{2}\rangle$ and
$D(B)=\odot_{8}\langle B_{1},B_{2}\rangle$ obtained by the _vertex-identifying
operation of $(3\times q)$-matrices_.

###  V-C Topsnut-matchings made by graphic groups

Let $T^{odd}_{group}$ be a set of odd-graceful Topsnut-groups. We define a
labelling $f:V(G)\rightarrow T^{odd}_{group}$ for a $(p,q)$-graph $G$, and set
$f(uv)=\odot_{k}\langle f(u),f(v)\rangle$ to be a matching of two odd-graceful
Topsnut-groups $G^{odd}_{i}$ and $G^{odd}_{j}$, here, each $T_{i}\in
G^{odd}_{i}$ matches with $T_{j}\in G^{odd}_{j}$ such that $\odot_{k}\langle
T_{i},T_{j}\rangle$ is just an odd-graceful Topsnut-matching, and vice versa.

For encrypting a network by graphic groups we show a simple example in Fig.41,
Fig.42 and Fig.43. We have an operation defined by

$f_{i}(x)+f_{j}(x)-f_{k}(x)=f_{i+j-k~{}(\bmod~{}13)}(x)$ (34)

for each element $x\in V(G)\cup E(G)$ shown in Fig.41. We can see that there
are many ways to realize a network encrypted by a graphic group, since there
are two or more ways to join $G_{i}$ with $G_{j}$ by an edge (allow by two or
more edges). Thereby, we have obtained many encrypted networks.

Figure 41:  A graphic group based on a path $G$ and the edge-magic graceful
labelling, each $G_{i}$ admits a pan-edge-magic graceful labelling $f_{i}$
under modulo 13.
Figure 42:  A tree admits: (a) a graceful group labelling based on the zero
$G_{1}$ shown in Fig.41; (b) an odd-graceful group labelling based on the zero
$G_{9}$ shown in Fig.41.
Figure 43:  A network encrypted by a graphic group shown in Fig.41.

###  V-D Algebraic groups from Topsnut-gpws, Topsnut-matrices and text-based
passwords

We have know that a Topsnut-gpw $G$ has its Topsnut-matrix $A(G)$, and $A(G)$
induces a text-based password $D(G)$. So $G$, $A(G)$ and $D(G)$ can produce
three Abelian additive groups, we call them _Topsnut-group, Topsnut-matrix
group and Text-pw group_ , respectively. If a Topsnut-gpw $G$ matches with
another Topsnut-gpw $H$, so two Topsnut-groups induced by $G$ and $H$ match to
each other. More results on such groups can be found in [24].

##  VI Researching problems

For further researching Topsnut-matchings we propose the following problems:

  1. Pro-1.

(A complete graph obtained from labeled trees) Given disjoint trees
$T_{1},T_{2},\dots,T_{m}$ with
$\sum^{m}_{i=1}|E(T_{i})|\geq\frac{1}{2}n(n-1)$. Can we find a labelling
$f_{i}$ for each tree $T_{i}$ such that $f_{i}:V(T_{i})\rightarrow[0,n-1]$ and

$\\{|f_{i}(u)-f_{i}(v)|:uv\in E(T_{i})\\}=[1,|E(T_{i})|],$

and identify the vertices of $\bigcup^{m}_{i=1}V(T_{i})$ having the same label
into one, the resulting graph is just $K_{n}=\odot\langle
T_{i}\rangle^{m}_{1}$, or $K_{n}=\ominus(T_{i})^{m}_{1}$, or
$K_{n}=\bigcup^{m}_{i=1}T_{i}$?

  2. Pro-2.

For any odd-graceful graph $H$ of $p$ vertices, does there exists a pan-odd-
graceful Topsnut-matching team $\odot_{1}\langle H,T_{i}\rangle^{p}_{1}$? Find
conditions for the perfect odd-gracefully Topsnut-matching team
$\odot_{1}\langle H,T_{i}\rangle^{p}_{1}$ with $T_{i}\cong T_{j}$ for $i\neq
j$. Find nontree graphs which induce pan-odd-graceful Topsnut-matching teams.

  3. Pro-3.

Plant the pan-odd-graceful Topsnut-matching team $\odot_{1}\langle
H,T_{i}\rangle^{p}_{1}$ to other graph labellings.

  4. Pro-4.

For a given lobster $T$, find another lobster $T^{\prime}$ such that
$\odot_{k}\langle T,T^{\prime}\rangle$ admits a twin odd-graceful labelling
(or a twin odd-elegant labelling) ([27, 9]).

  5. Pro-5.

For a given $(p,q)$-tree $G$ admitting a 6C-labelling $f$, find all possible
$(p,q)$-tree $H$ admits a 6C-labelling $g$ such that $\odot\langle G,H\rangle$
are 6C-complementary matchings.

  6. Pro-6.

Find conditions for a connected graph $G$ to be a multiple-tree matching
partition $G=\oplus_{F}\langle T_{i}\rangle^{m}_{1}$ with $m\geq 2$.

  7. Pro-7.

Find all possible odd-graceful Topsnut-matchings $\odot_{k}\langle G,H\rangle$
for a given $(p,q)$-graph $G$ admitting.

  8. Pro-8.

Determine v-set e-proper graceful/odd-graceful labellings of Euler graphs.

  9. Pro-9.

Determine the conditions for $A_{M}$ and $B_{q}$ defined in Definition 21 such
that the sequence type of labellings defined in Definition 21 exist.

  10. Pro-10.

Find a connected graph $G$ such that for any integer $M$ holding

$\min_{f}S_{um}(G,f)<M<\max_{f}S_{um}(G,f)$

there exists a difference-sum labelling $h$ of $G$ with $M=S_{um}(G,h)$.

  11. Pro-11.

Find a connected graph $G$ such that for any integer $M$ holding

$\min_{f}F_{um}(G,f)<M<\max_{f}F_{um}(G,f)$

there exists a felicitous-sum labelling $h$ of $G$ with $M=F_{um}(G,h)$.

  12. Pro-12.

If $G$ is a generalized sun-graph or a lobster, compute the exact values of
$\min_{f}S_{um}(G,f)$ and $\max_{f}S_{um}(G,f)$.

  13. Pro-13.

Find all graphs $H$ for forming set-ordered matching graphs $G\ominus H$ with
$H\not\cong G$.

  14. Pro-14.

Find all matching graphs $H$ such that $\odot_{k}\langle G,H\rangle$ admitting
twin odd-graceful labellings.

  15. Pro-15.

Consider other v-set e-proper $\varepsilon$-labellings of a complete graph
$K_{n}$, where $\varepsilon\in\\{$edge-magic total labelling, odd-elegant
labelling, harmonious labelling, the labellings defined in this paper$\\}$.
For example:

(i) A v-set e-proper felicitous labelling $(F,f)$ of a $(p,q)$-graph $G$ is
defined as: $F:V(G)\rightarrow[0,q-1]^{2}$ with $F(x)\cap F(y)=\emptyset$ for
distinct $x,y\in V(G)$, and $f:E(G)\rightarrow[0,q-1]$ holding
$f(E(G))=[0,q-1]$ and $f(uv)=f^{\prime}(u)+f^{\prime\prime}(v)~{}(\bmod~{}q)$
with $f^{\prime}(u)\in F(u)$ and $f^{\prime\prime}(v)\in F(v)$.

Does $K_{n}$ admits a v-set e-proper felicitous labelling?

(ii) A v-set e-proper edge-magic total labelling $(F,f)$ of a $(p,q)$-graph
$G$ is defined by $F:V(G)\rightarrow[1,M]^{2}$ with $p+q\leq M$ and $F(x)\cap
F(y)=\emptyset$ for distinct $x,y\in V(G)$, and $f:E(G)\rightarrow[1,M]$ with
$f(uv)\neq f(xy)$ for any two edges $uv,xy\in E(G)$, and there exists a
constant $k$ such that $f^{\prime}(u)+f(uv)+f^{\prime\prime}(v)=k$ for any
edge $uv\in E(G)$ with $f^{\prime}(u)\in F(u)$ and $f^{\prime\prime}(v)\in
F(v)$.

Does $K_{n}$ admits a v-set e-proper edge-magic total labelling? Find the
parameter $E_{mt}(G)=\min_{(F,f)}\\{M\\}$ over all v-set e-proper edge-magic
total labellings of $G$.

  16. Pro-16.

If we can split a connected graph having a v-set e-proper graceful labelling
into a tree, then characterize this graph and its possible v-set e-proper
graceful labellings.

  17. Pro-17.

For each $p\geq 2$, find a $(p,q)$-graph $G=\odot_{f}\langle
G_{i}\rangle^{m}_{1}$ defined in definition 8, such that $q$ is the largest
edge number on such $(p,q)$-graphs.

  18. Pro-18.

A $(p,q)$-graph $G$ and a $(q,p)$-graph $H$ admit two edge-magic graceful
labellings $f$ and $g$, respectively, and $f$ and $g$ are reciprocal inverse
because $f(E(G))=g(V(H))\setminus X^{*}$ and $f(V(G))\setminus X^{*}=g(E(H))$
for $X^{*}=f(V(G))\cap g(V(H))$. Find such pairs of graphs $G$ and $H$, and
characterize them.

  19. Pro-19.

Find reciprocal complementary (reciprocal-inverse matching)
$G=\odot_{k}\langle T,G\rangle$ for a fixed graph $T$, where $T$ and $G$ admit
reciprocal-inverse labellings $f$ and $g$, respectively, such that

$f(E(T))=g(V(G))\setminus X^{*}\textrm{ and }f(V(T))\setminus X^{*}=g(E(G))$

for $X^{*}=f(V(T))\cap g(V(G))$.

  20. Pro-20.

If a total coloring $g$ of a graph $G$ arrives at
$B_{tol}(G,g)=\min_{f}B_{tol}(G,f)$, is there
$\chi^{\prime\prime}(G)=|\\{g(x):x\in V(G)\cup E(G)\\}|$?

  21. Pro-21.

For any connected subgraph $H$ of a connected graph $G$, does there exists
$\max_{g}S_{um}(H,g)\leq\max_{f}S_{um}(G,f)$?

  22. Pro-22.

Determine connected graphs having a group of consecutive difference proper
vertex colorings, or a group of consecutive sum proper vertex colorings.

  23. Pro-23.

Find connected graphs admitting one of the edge-magic proper total coloring
and the equitably proper total coloring.

  24. Pro-24.

Let $f:V(G)\cup E(G)\rightarrow[1,\chi^{\prime\prime}(G)]$ be a proper total
coloring of a graph $G$, and let $f^{*}(E(G))=\\{f(u)+f(uv)+f(v):uv\in
E(G)\\}$. Characterize $G$ if $f^{*}(E(G))=[a,b]$ is a consecutive integer
set.

##  VII Conclusion

We have known that Topsnut-matching is a larger topic in researching Topsnut-
gpws, _nature-inspired passwords_. Results and techniques of graph theory are
proven to be powerful in designing and researching Topsnut-gpws, since there
are no polynomial algorithms for many of these results and techniques. Many of
the graph labellings introduced here match with mathematical conjectures, so
they may provide computationally unbreakable for our Topsnut-gpws. It is
hopeful to let more people using Topsnut-gpws and PTopsnut-gpws (allow label
vertices and edges with non-mathematical elements) for protecting their
information and profits in networks ( [23], [26]). We have seen there are over
200 graph labellings introduced in [2]. More new graph labellings emerge
everyday. It is time to consider _Graph Labelling Analysis_ as a subbranch of
graph theory. So, we try doing some exploring work here, although we have two
hands empty on this topic.

Matching can help us to design Topsnut-gpws for one public key vs one private
key, one public key vs two or more private keys, and more public keys vs more
private keys. Matching opens a window for us to understand something new in
cryptography. It is very important that matching is just one of mathematical
principles. Almost mathematical operations have their own matching operations.
The graph labellings first defined or introduced here match with other
existing graph labelllings, and can be shown to be related with mathematical
conjectures, or open problems.

Researching Topsnut-matching can derive two interesting topics: one is _set
matching partition_ to number theory, and another is about labeled graphs for
constructing large scale of graphs with labellings, which differs from finding
labellings to unlabeled graphs. We have listed possible researching problems
for further studying works on Topsnut-gpws, and hope to find more something
new and to do more theoretical works on Topsnut-gpws. We try to use Topsnut-
groups to build up so-called network passwords for encrypting a network with
thousand and thousand vertices. So we have investigated one of Topsnut-groups,
called _Abelian additive graphic group_ (_graphic group_ for short). This type
of graphic groups based on addition operation processes a particular property:
every element in a graphic group can be regarded as “zero” of the graphic
group, so we can call it an _every-zero graphic group_. Unfortunately, we do
not discover graphic group based on multiplication operation. It may be a way
to find more graph labellings of a graph from connection between two or more
graphic groups on the graph.

Several new colorings and new parameters on proper total coloring have been
introduced and investigated. We have found that the difference-sum labelling
(_extremal labelling_) can be admitted by every graph, so then it breaks down
the case of no labelling admitted by each graph. Thereby, we are motivated
from the difference-sum labelling and know that there are many _extremal
labellings_ like the difference-sum labelling, which mean that we may touch a
new subbranch of graph labellings.

The above research works on two different areas motivate us to think of the
biological combination of man and AI machine in current development of the
world, rather than AI machines only that will take a long time to success. An
application project supported by mathematics like passwords depends on
mathematics going deep into and continuous improvement, how long will this
support last, how far can the project go.

## Acknowledgment

The author, _Bing Yao_ , is delight for supported by the National Natural
Science Foundation of China under grants 61163054, 61363060 and 61662066;
Scientific research project of Gansu University under grants 2016A-067,
2017A-047 and 2017A-254. Bing Yao, also, thank every member of _Topological
Graphic Passwords Symposium_ in the second semester of 2017-2018 academic year
for their effective discussion and constructive suggestions.

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