# Towards Meta-Algorithm Selection

Alexander Tornede, Marcel Wever, Eyke Hüllermeier
Heinz Nixdorf Institute and Department of Computer Science
Paderborn University, Germany
{alexander.tornede, marcel.wever<EMAIL_ADDRESS>

###### Abstract

Instance-specific algorithm selection (AS) deals with the automatic selection
of an algorithm from a fixed set of candidates most suitable for a specific
instance of an algorithmic problem class, where “suitability” often refers to
an algorithm’s runtime. Over the past years, a plethora of algorithm selectors
have been proposed. As an algorithm selector is again an algorithm solving a
specific problem, the idea of algorithm selection could also be applied to AS
algorithms, leading to a meta-AS approach: Given an instance, the goal is to
select an algorithm selector, which is then used to select the actual
algorithm for solving the problem instance. We elaborate on consequences of
applying AS on a meta-level and identify possible problems. Empirically, we
show that meta-algorithm-selection can indeed prove beneficial in some cases.
In general, however, successful AS approaches have problems with solving the
meta-level problem.

##  1 Introduction

In instance-specific algorithm selection (AS), we are faced with instances of
an algorithmic problem class, such as the Boolean satisfiability problem
(SAT), and a set of algorithms to choose for solving the instance. The goal of
AS is to select the algorithm that is most suitable for a given instance, in
the sense that it performs best with respect to a certain criterion. AS is
motivated by the observation that algorithms perform differently well on
different instances, so that choosing in a per-instance manner can improve the
overall performance [1] — quite in line with theoretical “no free lunch”
results [2].

Several AS approaches have been proposed over the last years, the majority of
which makes use of machine learning techniques in order to exploit properties
of the instances for selecting a good algorithm. As an AS algorithm is again
an algorithm solving a specific problem, the idea of algorithm selection could
also be applied to AS algorithms, leading to a meta-AS approach. Indeed, a
certain complementarity among AS algorithms can be observed (e.g. [3]) and the
resulting meta-AS problem was first mentioned by [4] and [1]. Besides, the
strategy of selecting among a set of meta-experts, who in turn select the
expert actually solving the problem, appears to be quite common in practice.
Consider, for example, the case of building a house. Very few people select
the handicraft companies actually building their house themselves, but rather
choose a general contractor, who in turn picks craftsmen for building the
requested house.

One of the few existing works is AutoFolio [5], suggesting combined selection
and configuration of algorithm selectors. However, it mainly focuses on the
configuration of algorithm selection approaches on a scenario-level, i.e., it
tries to find the best configuration of an algorithm selector for a collection
of instances, also called scenario. Extending the idea of AutoFolio to the
problem of automated machine learning, AS approaches optimizing design choices
regarding automated machine learning strategies were proposed recently [6, 7].
In this work, we seek to select among available algorithm selectors, which in
turn select an actual algorithm solving the problem, for every instance
individually, allowing for much more nuanced decisions potentially yielding
performance improvements.

##  2 The (Per-Instance) Algorithm Selection Problem

The per-instance algorithm selection problem, first introduced in [8],
comprises a problem instance space $\mathcal{I}$ and a set of candidate
algorithms $\mathcal{A}$. The goal is to find a mapping
$s:\mathcal{I}\longrightarrow\mathcal{A}$, called algorithm selector. Given a
problem instance $i\in\mathcal{I}$, the mapping should ideally return the
algorithm $a^{*}\in\mathcal{A}$ optimizing a performance measure
$m:\mathcal{I}\times\mathcal{A}\longrightarrow\mathbb{R}$. This performance
measure is usually costly to evaluate, rendering an enumeration over the set
of algorithms infeasible. Accordingly, the virtual best solver (VBS) aka.
oracle, i.e., the optimal algorithm selector, is defined as

$s^{*}(i)=\arg\min_{a\in\mathcal{A}}\mathbb{E}\left[m(i,a)\right]$ (1)

for all problem instances $i\in\mathcal{I}$, where the expectation accounts
for the potential randomness in the application of an algorithm. The strategy
always selecting the algorithm which is best on average, i.e., averaged over
instances $i\in\mathcal{I}$, is called the single best solver (SBS).

###  2.1 Common Algorithm Selection Solutions

To compensate for the costly to evaluate performance measure $m$, most AS
approaches employ machine learning techniques for learning surrogate models of
the form (or a similar one)
$\widehat{m}:\mathcal{I}\times\mathcal{A}\longrightarrow\mathbb{R}$
approximating the original performance measure, while being fast to evaluate.
Such surrogates allow one to specify an algorithm selector
$s:\mathcal{I}\longrightarrow\mathcal{A}$ as

$s(i)\mathrel{\vbox{\hbox{\footnotesize.}\hbox{\footnotesize.}}}=\arg\min_{a\in\mathcal{A}}\widehat{m}(i,a)\,.$
(2)

In order to learn such surrogate models, we assume the existence of training
data in the form of a set of instances $\mathcal{I}_{D}\subset\mathcal{I}$ for
which the performance $m(i,a)$ is available for some algorithms
$a\in\mathcal{A}$. In addition, we assume that instances can be represented as
$d$-dimensional feature vectors using a feature function
$f:\mathcal{I}\longrightarrow\mathbb{R}^{d}$. Examples of such features for an
instance of the Boolean satisfiability problem (SAT) are the number of clauses
or the number of variables. Note that the computation of these features costs
time, which is important to consider when the measure $m$ to optimize is
(related to) runtime. This means that algorithm selection approaches NOT
requiring such features can be advantageous if the feature computation is time
consuming.

A simple example approach inferring such a surrogate model in the form of a
regression function was introduced in [9] (PAReg), where the authors suggest
to learn a regression function
$\widehat{m}_{a}:\mathcal{I}\longrightarrow\mathbb{R}$ for each algorithm
$a\in\mathcal{A}$, predicting the performance of algorithm $a$ on a given
instance. Hence a selector (2) can be defined by
$\widehat{m}(i,a)\mathrel{\vbox{\hbox{\footnotesize.}\hbox{\footnotesize.}}}=\widehat{m}_{a}(i)$.
Another natural idea is to tackle the problem through multiclass-
classification (MCC) methods where each algorithm corresponds to one class
[3]. Other approaches, such as SUNNY [10] or ISAC [11], estimate the unknown
performance of an instance/algorithm pair by falling back to similar
instances, for which evaluations are available, using k-nearest neighbor or
clustering techniques. More recently, an effective decision-theoretic approach
based on survival analysis was presented in [3], called Run2Survive (R2SPAR10
/ R2SExp), explicitly modeling censored information present in the training
data. A more recent version of [9], called SATzilla’11 [12], leverages
pairwise comparisons of algorithms by training a cost-sensitive decision
forest for each pair of algorithms, predicting which one most likely performs
better for a given instance. The selected algorithm is determined via majority
voting. Following the idea of pairwise comparisons, [13] investigate how
combined ranking and regression models perform for algorithm selection,
demonstrating hybrid models to achieve performance superior to models
considering only regression or only ranking information. For a more
comprehensive overview of algorithm selection, we refer to [1].

###  2.2 Loss Functions

When considering algorithmic satisfaction problems, one is usually interested
in optimizing the runtime as a performance measure $m$ under the condition
that a given instance has to be solved. However, in the case of combinatorial
problems, some algorithms need extremely long to solve some instances [14],
resulting in a time-constrained evaluation. For this purpose, one usually
considers a so-called cutoff $C$, after which an algorithm is terminated and
the instance considered unsolved. The most common loss function in AS, taking
this into account, is the penalized average runtime (PAR10)

$\text{PAR10}(t)=\begin{cases}t&\text{ if }t\leq C\\\ 10\cdot C&\text{ else
}\end{cases}$ (3)

where $t$ is the runtime of the algorithm as chosen by the algorithm selector.
This score is averaged over all instances in order to rate a selector’s
performance on a set of instances, also called scenario. As PAR10 scores are
incomparable across scenarios, one often considers the normalized PAR10 score
defined as
$\text{nPAR10}(s)=(\text{PAR10}_{s}-\text{PAR10}_{\mathit{oracle}})/(\text{PAR10}_{\mathit{SBS}}-\text{PAR10}_{\mathit{oracle}})$,
where $\text{PAR10}_{s}$ is the PAR10 score of an algorithm selector $s$.

##  3 Per-Instance Meta-Algorithm-Selection

Per-instance meta-algorithm-selection (meta-AS) concerns the problem of
selecting an algorithm selector, for a given instance of an algorithmic
problem class. The selected algorithm selector then in turn selects the actual
algorithm solving the problem. Hence, instead of directly choosing an
algorithm to solve a problem instance, we take a detour by selecting an
algorithm selector.

Formally, in the meta-AS problem, we seek to find a mapping
$s_{\mathit{meta}}:\mathcal{I}\longrightarrow\mathcal{A}_{\mathit{meta}}$,
called algorithm selector selector (ASS), optimizing the original performance
measure $m$. While being in principle identical to the standard AS problem,
the set of algorithms $\mathcal{A}_{\mathit{meta}}$ to choose from does no
longer contain algorithms actually solving problem instances, but algorithm
selection approaches, which in turn choose one of the algorithms solving an
instance. Hence, the set of meta-algorithms to choose from, is a subset of all
possible algorithm selectors, i.e.,
$\mathcal{A}_{\mathit{meta}}\subseteq\\{s|s:\mathcal{I}\longrightarrow\mathcal{A}\\}$.

Similarly as in standard AS, we treat the algorithms, i.e., algorithm
selectors, from $\mathcal{A}_{\mathit{meta}}$ as black-boxes and ready to be
applied, although one could in principle slot a training phase for the
algorithm selectors in ahead. Hence, in the meta-algorithm selection problem
one only has to derive a meta-algorithm selector. The connection between
standard AS and meta-AS is depicted in Fig. 1.

Figure 1: Left: An algorithm selector
$s:\mathcal{I}\longrightarrow\mathcal{A}$ and its role as part of a meta-
algorithm selector. Right: A meta-algorithm selector $s_{\mathit{meta}}$ that
can choose from a set of algorithm selectors.

###  3.1 Meta-Algorithm-Selection Solutions

As the meta-AS problem is essentially a special case of the standard AS
problem with a very specific set of algorithms to choose from, standard AS
methods (see Sec. 2.1) can in principle be applied for solving the problem. It
is important to note that algorithm selection approaches that do not rely on
feature representations of instances do no longer necessarily have an
advantage in terms of runtime, if they select an algorithm selector which in
turn requires the feature representation. If the feature computation has to be
performed either on the meta or on the base-level, its time has to be
considered as well. However, there is no need to perform the computation twice
if both the algorithm selector selector and the algorithm selector require it,
as resulting features can be shared.

###  3.2 Implications for the Oracle and the Single-Best-Solver

It is important to note that the oracle on the meta-level, i.e., the
algorithm-selector-oracle (AS-oracle), is in general not identical to the
oracle on the base-level, as the set of algorithms to choose from changes. In
fact, the AS-oracle, is defined as

$s_{\mathit{meta}}^{*}(i)=\arg\min_{s\in\mathcal{A}_{\mathit{meta}}}\mathbb{E}\left[m(i,s(i))\right]\,\,\,.$
(4)

For a better understanding, consider the following example with two algorithms
$A_{1}$ and $A_{2}$ and two algorithm selectors $s_{1}$ and $s_{2}$, where
both always select algorithm $A_{1}$. Furthermore, assume that there exists an
instance, for which $A_{2}$ performs better than $A_{1}$, and hence the oracle
would select $A_{2}$. However, the AS-oracle can only select $s_{1}$ or
$s_{2}$, which in turn both select $A_{1}$.

Generally speaking, in order to preserve the original oracle, there has to be,
for each instance, at least one algorithm selector which selects the best
algorithm. By the time this is not the case for at least one instance, the AS-
oracle performance will degrade compared to the oracle. In practice, there
will be at least one such instance most of the time, and hence an important
question is how much the oracle performance degrades. As we show in our
experimental evaluation, the degradation strongly depends on the scenario at
hand, and ranges from less than $1\%$ to over $85\%$.

Similarly to the oracle, the SBS changes as well, since the single best
algorithm selector (SBAS), i.e., the SBS on the meta-level, is selecting from
algorithm selectors, making it a lot stronger baseline. Hence, while the
original oracle selects the actual problem solving algorithm best on average
and accordingly does not depend on instance features, the SBAS does in fact
depend on such features as long as it is not identical to the original SBS.
Observe that this results in a significant disadvantage for the SBAS in terms
of achievable PAR10 scores.

Obviously, these implications also influence the gains in performance, which
can be achieved by algorithm selector selectors in comparison to algorithm
selectors. As the oracle performance most likely degrades, while the SBS
performance most likely improves, the gap between the two also decreases,
offering less potential for algorithm selection approaches to close this gap.

##  4 Experimental Evaluation

In this section, we present the results of an experimental evaluation built
around the meta-algorithm selection problem based on the ASlib v4.0 benchmark
suite [15]. The study is organized into three parts: First, we relate the
performances of the SBS and the oracle for both levels to each other. Thereby,
we discuss the fact that the gaps between SBS and oracle become smaller
indeed, as already noted in Sec. 3.2. Second, we evaluate if and to what
extent the application of algorithm selector selectors does yield an
improvement in algorithm selection performance. Third, we investigate whether
algorithm selector selectors are able to improve over the SBAS.

We evaluate each approach for every ASlib scenario by performing a 10-fold
cross-validation over the instances of a scenario and report cropped means for
all measures we compute. For these cropped means, we remove the two best and
two worst splits. Often algorithm selection systems consist of more elements
than just a surrogate, such as pre-solvers or feature selectors. For our
experimental study, we consider pure algorithm selection models. All
experiments were run on machines featuring Intel Xeon<EMAIL_ADDRESS>CPUs
with 16 cores and 64GB RAM. In order to allow for full reproducibility of our
results, the code for all experiments and the generation of plots including
detailed documentation can be found on
Github111https://github.com/alexandertornede/meta_as.

###  4.1 Results

Fig. 2 shows the PAR10 scores of the oracle, AS-oracle, SBS and SBAS on a
subset of the ASlib v4.0 benchmark scenarios. As one can see, several of the
implications we noted in Sec. 3.2 can be validated empirically: Firstly and
most importantly, although the SBS/oracle gap is a lot larger than the
SBAS/AS-oracle gap, the SBAS/AS-oracle gaps are non-negligible, and hence
meta-algorithm selection does in principle make sense. For example, consider
scenarios SAT12-ALL, or QBF-2016 with large SBAS/AS-oracle gaps.

Figure 2: This figures shows the PAR10 scores of the oracle, AS-oracle, SBS
and SBAS on a subset of the ASlib v4.0 benchmark scenarios as bar charts.

As we noted earlier, the reason why these gaps become smaller is that oracle
performance degrades when moving to the meta-level for all scenarios whereas
the SBS performance improves when moving to the meta-level in general, since
the SBAS is essentially an algorithm selector. While the degradation in oracle
performance is moderate for the majority of scenarios (less than $10\%$), the
SBS improvement is non-negligible, as the more successful the algorithm
selectors considered by the algorithm selector selectors are, the larger this
performance gain is.

Table 1: Normalized PAR10 scores of all base- and meta-algorithm selectors
normalized wrt. the standard oracle and SBS. The result of the best approach
is marked in bold for each scenario. Moreover, for the meta-algorithm
selectors the values in brackets $(a/b)$ indicate that that the approach
achieves a performance better or equal to $a$ base-approaches and is worse
than $b$ base-approaches.

Level | Algorithm Selectors | Algorithm Selector Selectors (Meta)
---|---|---
Approach |  R2SExp |  R2SPAR10 |  ISAC |  MCC |  PAReg |  SATzilla’11 |  SUNNY |  R2SExp |  R2SPAR10 |  ISAC |  MCC |  PAReg |  SATzilla’11 |  SUNNY
Scenario |  |  |  |  |  |  |  |  |  |  |  |  |  |
ASP-POTASSCO | 0.30 | 0.32 | 0.60 | 0.64 | 0.34 | 0.47 | 0.17 | 0.23 (6/1) | 0.22 (6/1) | 0.23 (6/1) | 0.37 (3/4) | 0.30 (6/1) | 0.30 (6/1) | 0.26 (6/1)
BNSL-2016 | 0.18 | 0.21 | 0.84 | 0.31 | 0.18 | 0.18 | 0.25 | 0.22 (3/4) | 0.21 (4/3) | 0.19 (4/3) | 0.28 (2/5) | 0.22 (3/4) | 0.28 (2/5) | 0.27 (2/5)
CPMP-2015 | 0.76 | 0.69 | 0.90 | 0.85 | 0.78 | 0.70 | 0.94 | 0.78 (4/3) | 0.78 (4/3) | 0.89 (2/5) | 0.81 (3/4) | 0.77 (4/3) | 0.81 (3/4) | 0.89 (2/5)
CSP-2010 | 0.13 | 0.15 | 0.31 | 0.80 | 0.25 | 0.13 | 0.34 |  0.04 (7/0) |  0.04 (7/0) | 0.19 (4/3) | 0.13 (7/0) | 0.46 (1/6) | 0.21 (4/3) | 0.09 (7/0)
CSP-Minizinc-Time-2016 | 0.43 | 0.27 | 0.83 | 0.36 | 0.67 | 0.34 | 0.37 | 0.51 (2/5) | 0.51 (2/5) | 0.76 (1/6) | 0.60 (2/5) | 0.67 (2/5) | 0.35 (5/2) | 0.51 (2/5)
GRAPHS-2015 | 0.76 | 0.71 | 0.88 | 1.37 | 0.80 | 1.02 | 0.45 | 0.76 (5/2) | 0.73 (5/2) | 1.06 (1/6) | 1.05 (1/6) | 0.87 (3/4) | 1.22 (1/6) | 1.00 (2/5)
MAXSAT-PMS-2016 | 0.60 | 0.36 | 0.82 | 1.06 | 0.77 | 0.62 | 0.41 | 0.64 (3/4) | 0.65 (3/4) | 0.60 (5/2) | 0.71 (3/4) | 0.82 (2/5) | 0.98 (1/6) | 0.75 (3/4)
MAXSAT-WPMS-2016 | 0.44 | 0.37 | 0.76 | 0.85 | 0.52 | 0.31 | 0.16 | 0.37 (5/2) | 0.39 (4/3) | 0.60 (2/5) | 0.60 (2/5) | 0.54 (2/5) | 0.43 (4/3) | 0.44 (4/3)
MAXSAT12-PMS | 0.22 | 0.23 | 0.47 | 0.40 | 0.28 | 0.24 | 0.29 | 0.25 (4/3) | 0.25 (4/3) | 0.20 (7/0) |  0.19 (7/0) | 0.32 (2/5) | 0.20 (7/0) | 0.21 (7/0)
MAXSAT15-PMS-INDU | 0.34 | 0.44 | 0.89 | 1.06 | 0.55 | 0.39 | 0.24 | 0.36 (5/2) | 0.57 (2/5) | 0.33 (6/1) | 0.39 (5/2) | 0.40 (4/3) | 0.51 (3/4) | 0.26 (6/1)
MIP-2016 | 1.07 | 0.96 | 0.76 | 1.13 | 1.58 | 1.05 | 0.68 | 1.00 (4/3) | 0.98 (4/3) | 1.07 (3/4) | 1.25 (1/6) | 1.48 (1/6) | 1.29 (1/6) | 0.77 (5/2)
PROTEUS-2014 | 0.41 | 0.41 | 0.64 | 0.84 | 0.45 | 0.58 | 0.47 | 0.46 (4/3) | 0.46 (4/3) | 0.48 (3/4) | 0.49 (3/4) | 0.53 (3/4) | 0.62 (2/5) | 0.53 (3/4)
QBF-2011 | 0.21 | 0.20 | 0.36 | 0.35 | 0.18 | 0.15 | 0.22 | 0.21 (4/3) | 0.21 (4/3) | 0.21 (4/3) | 0.21 (4/3) | 0.29 (2/5) | 0.25 (2/5) | 0.25 (2/5)
QBF-2014 | 0.26 | 0.28 | 0.52 | 0.59 | 0.32 | 0.31 | 0.31 | 0.30 (5/2) | 0.32 (3/4) | 0.32 (3/4) | 0.35 (2/5) | 0.41 (2/5) | 0.39 (2/5) | 0.35 (2/5)
QBF-2016 | 0.52 | 0.51 | 0.65 | 0.69 | 0.61 | 0.61 | 0.49 | 0.55 (4/3) | 0.55 (4/3) | 0.52 (5/2) | 0.53 (4/3) | 0.62 (2/5) | 0.57 (4/3) | 0.58 (4/3)
SAT03-16_INDU | 0.71 | 0.76 | 0.98 | 0.99 | 0.77 | 0.82 | 0.82 | 0.92 (2/5) | 0.90 (2/5) | 0.78 (4/3) | 0.79 (4/3) | 0.81 (4/3) | 0.84 (2/5) | 0.86 (2/5)
SAT11-HAND | 0.34 | 0.34 | 0.65 | 0.57 | 0.46 | 0.44 | 0.60 | 0.42 (5/2) | 0.47 (3/4) | 0.42 (5/2) | 0.44 (5/2) | 0.50 (3/4) | 0.45 (4/3) | 0.56 (3/4)
SAT11-INDU | 0.69 | 0.69 | 1.08 | 0.71 | 0.63 | 0.79 | 0.76 | 0.78 (2/5) | 0.89 (1/6) | 0.84 (1/6) |  0.61 (7/0) | 0.79 (2/5) | 0.73 (3/4) | 0.85 (1/6)
SAT11-RAND | 0.13 | 0.06 | 0.58 | 0.17 | 0.09 | 0.39 | 0.12 | 0.13 (4/3) | 0.13 (4/3) | 0.19 (2/5) | 0.17 (3/4) | 0.19 (2/5) | 0.32 (2/5) | 0.20 (2/5)
SAT12-ALL | 0.36 | 0.36 | 0.67 | 0.38 | 0.37 | 0.44 | 0.38 | 0.37 (5/2) | 0.39 (2/5) | 0.39 (2/5) | 0.37 (5/2) | 0.40 (2/5) | 0.40 (2/5) | 0.43 (2/5)
SAT12-HAND | 0.34 | 0.34 | 0.65 | 0.41 | 0.37 | 0.27 | 0.43 | 0.34 (6/1) | 0.33 (6/1) | 0.31 (6/1) | 0.40 (3/4) | 0.40 (3/4) | 0.39 (3/4) | 0.38 (3/4)
SAT12-INDU | 0.70 | 0.73 | 1.02 | 0.94 | 0.79 | 0.59 | 0.78 | 0.62 (6/1) | 0.63 (6/1) | 0.75 (4/3) | 0.73 (5/2) | 0.65 (6/1) | 0.65 (6/1) | 0.66 (6/1)
SAT12-RAND | 0.96 | 0.86 | 0.84 | 5.20 | 1.17 | 0.93 | 1.14 | 1.03 (3/4) | 1.05 (3/4) | 0.95 (4/3) | 0.96 (4/3) | 1.19 (1/6) | 1.13 (3/4) | 1.05 (3/4)
SAT15-INDU | 0.95 | 0.83 | 0.76 | 0.91 | 0.74 | 0.75 | 1.00 | 0.68 (7/0) | 0.88 (3/4) | 1.00 (1/6) | 0.96 (1/6) |  0.65 (7/0) | 0.85 (3/4) | 0.81 (4/3)
TSP-LION2015 | 1.73 | 1.77 | 16.69 | 56.55 | 2.60 | 40.08 | 10.27 | 1.74 (6/1) |  1.73 (7/0) | 17.79 (2/5) | 36.21 (2/5) | 3.03 (4/3) | 44.3 (1/6) | 12.35 (3/4)

Table 1 shows the normalized PAR10 scores of all algorithm selectors and the
corresponding algorithm selector selectors, both normalized with respect to
the standard oracle and SBS. The result of the best approach is marked in bold
for each scenario. A value of 0 corresponds to oracle performance, a value of
1 to SBS performance, and values in-between indicate an improvement over the
SBS, whereas values above 1 indicate degradation compared to the SBS.
Unsurprisingly, most algorithm selector selectors are able to consistently
improve over the SBS. However, moving to the meta-level proves to be
beneficial for only 3 scenarios, which is disappointing. A possible cause of
this observation might be that the considered AS approaches are not able to
unleash their full potential on the meta-level, although considerable SBAS/AS-
oracle gaps exist, as we have seen previously. In order to further investigate
this, we evaluate the different meta-AS approaches w.r.t. their meta-algorithm
selection performance in the following.

Moreover, for the algorithm selector selectors, the values in brackets $(a/b)$
indicate that the approach achieves a performance better or equal to $a$ base
approaches and is worse than $b$ base approaches. These results indicate that
moving to the meta-level is beneficial in the sense that a more robust
performance across several scenarios can be achieved.

Table 2 shows a pairwise win/tie/loss between algorithm selectors and
algorithm selector selectors aggregated across scenarios, giving a fairer idea
of whether deciding for an algorithm selector selector is more beneficial than
deciding for an algorithm selector. Although some ASS are able to beat many
algorithm selectors on a majority of instances, ASS do not seem to be
advantageous.

Table 2: Win/Tie/Loss statistics, comparing ASS to algorithm selectors across
scenarios.

|  | Algorithm Selector Selectors (Meta)
---|---|---
|  | R2SExp | R2SPAR10 | ISAC | MCC | PAReg | Satzilla | Sunny
Alg. Selectors | R2SExp | 6/4/15 | 8/3/14 | 5/4/16 | 2/3/20 | 2/1/22 | 5/1/19 | 7/1/17
R2SPAR10 | 6/2/17 | 5/1/19 | 5/0/20 | 4/1/20 | 4/0/21 | 3/0/22 | 7/0/18
ISAC | 23/0/2 | 22/0/3 | 20/0/5 | 20/0/5 | 21/1/3 | 19/0/6 | 21/0/4
MCC | 23/0/2 | 22/0/3 | 19/0/6 | 21/1/3 | 20/0/5 | 21/0/4 | 20/0/5
PAReg | 17/2/6 | 13/2/10 | 11/1/13 | 11/1/13 | 6/1/18 | 11/0/14 | 11/0/14
Satzilla | 14/0/11 | 9/0/16 | 11/0/14 | 9/3/13 | 8/1/16 | 6/0/19 | 11/0/14
Sunny | 14/0/11 | 12/0/13 | 10/1/14 | 12/0/13 | 8/0/17 | 10/0/15 | 8/0/17

Table 3 shows the PAR10 scores normalized w.r.t. to the SBAS and AS-oracle of
all considered algorithm selector selectors, where improvements over the SBAS,
i.e., values below 1, are marked in bold. A first observation to be made is
that none of the meta-AS approaches is able to achieve an improvement over the
SBAS on the majority of instances, indicating weaknesses in performing
successful algorithm selection on the meta-level. This observation suggests
that very successful AS approaches seem to have trouble to perform well on a
meta-level. Note that these results have to be treated with care. Although the
SBAS is essentially an algorithm selector, a score below 1 does not
necessarily indicate that the corresponding ASS performs better than all
algorithm selectors. The SBAS is determined on the training data, and hence an
improvement over the SBAS on the test data is not equivalent to being better
than all algorithm selectors on the test data.

Table 3: PAR10 scores normalized wrt. to the SBAS and AS-oracle of all
considered meta-AS approaches. Values below 1, i.e., an improvement over the
SBAS, are marked in bold.

Approach | R2SExp | R2SPAR10 | ISAC | MCC | PAReg | SATzilla’11 | SUNNY
---|---|---|---|---|---|---|---
Scenario |  |  |  |  |  |  |
ASP-POTASSCO | 3.15 | 2.03 | 1.46 | 5.46 | 2.77 | 2.80 | 1.71
BNSL-2016 | 0.87 | 0.74 | 0.86 | 1.41 | 0.97 | 1.17 | 1.20
CPMP-2015 | 1.05 | 1.05 | 1.17 | 1.33 | 1.12 | 1.01 | 1.22
CSP-2010 | 1.07 | 0.87 | 2.69 | 0.91 | 272.51 | 1.13 | 0.79
CSP-Minizinc-Time-2016 | 1.64 | 1.64 | 2.04 | 14.69 | 3.11 | 1.03 | 0.94
GRAPHS-2015 | 0.95 | 0.93 | 1.64 | 1.31 | 1.31 | 1.74 | 1.50
MAXSAT-PMS-2016 | 2.72 | 1.27 | 1.05 | 3.18 | 5.09 | 8.18 | 3.05
MAXSAT-WPMS-2016 | 1.20 | 5.44 | 3.84 | 11.1 | 7.83 | 4.44 | 3.37
MAXSAT12-PMS | 0.70 | 0.77 | 0.74 | 0.82 | 1.09 | 0.70 | 0.66
MAXSAT15-PMS-INDU | 1.48 | 3.76 | 1.01 | 2.93 | 3.00 | 4.37 | 1.41
MIP-2016 | 1.01 | 1.00 | 1.13 | 0.99 | 1.00 | 0.95 | 0.76
PROTEUS-2014 | 1.18 | 1.20 | 1.30 | 1.34 | 1.49 | 1.76 | 1.50
QBF-2011 | 1.34 | 1.49 | 1.67 | 1.22 | 1.85 | 1.75 | 2.04
QBF-2014 | 1.16 | 1.22 | 1.51 | 1.53 | 1.55 | 1.66 | 1.54
QBF-2016 | 0.91 | 0.96 | 1.00 | 0.88 | 1.09 | 1.20 | 1.09
SAT03-16_INDU | 1.48 | 1.46 | 1.23 | 1.30 | 1.29 | 1.45 | 1.49
SAT11-HAND | 1.50 | 1.92 | 1.47 | 1.51 | 1.91 | 1.92 | 2.62
SAT11-INDU | 1.33 | 1.38 | 1.31 | 0.89 | 1.48 | 1.00 | 1.43
SAT11-RAND | 1.76 | 1.52 | 6.06 | 4.54 | 2.74 | 11.21 | 6.48
SAT12-ALL | 1.08 | 1.21 | 1.21 | 1.19 | 1.33 | 1.29 | 1.36
SAT12-HAND | 1.38 | 1.27 | 1.28 | 1.83 | 1.89 | 1.64 | 1.56
SAT12-INDU | 1.24 | 1.25 | 1.38 | 1.41 | 1.27 | 1.21 | 1.15
SAT12-RAND | 1.01 | 1.15 | 1.08 | 1.12 | 1.60 | 1.49 | 1.30
SAT15-INDU | 0.73 | 0.83 | 1.19 | 1.14 | 0.67 | 0.68 | 0.89
TSP-LION2015 | 1.14 | 1.05 | 10.86 | 22.55 | 2.00 | 27.62 | 8.00

##  5 Conclusion

We proposed the problem of meta-algorithm selection, where the task is to
select the right algorithm selector, and investigated how standard AS
approaches perform on the meta-level. Our analysis reveals interesting
insights about the meta-AS problem concerning changes of the oracle and the
SBS. As suggested by a considerable gap between the SBAS and the AS-oracle, a
potential improvement through meta-AS definitely exists. This is confirmed by
several cases, in which moving to the meta-level proves beneficial in terms of
algorithm selection performance. In general, however, standard AS methods,
when applied on the meta-level, do not seem to fully exploit this potential.

One may wonder, therefore, how AS on the meta-level could be improved. One
idea is to consider features of the algorithm selectors that would allow the
application of methods from extreme algorithm selection [16, 17]. Another idea
worth investigation is to move to even higher meta-levels. Moreover, the
problem of oracle degradation could be tackled by including the original
algorithms (which one can think of as static algorithm selectors always
choosing the same algorithm) in the set of algorithm selectors to choose from.

## Acknowledgments and Disclosure of Funding

This work was partially supported by the German Research Foundation (DFG)
within the Collaborative Research Center “On-The-Fly Computing” (SFB 901/3
project no. 160364472). The authors gratefully acknowledge support of this
project through computing time provided by the Paderborn Center for Parallel
Computing (PC2).

## References

  * [1] P. Kerschke, H. H. Hoos, F. Neumann, and H. Trautmann.  Automated algorithm selection: Survey and perspectives.  Evol. Comput., 27(1), 2019.
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  * [6] M. Feurer and F. Hutter.  Towards further automation in AutoML.  In AutoML workshop @ ICML, 2018.
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# Magnetized hot neutron matter: lowest order constrained variational
calculations

G.H. Bordbar 1,2 111Corresponding author. E-mail<EMAIL_ADDRESS>Z. Rezaei1 1Department of Physics, Shiraz University, Shiraz 71454,
Iran222Permanent address,
and
2Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha 55177-36698, Iran

###### Abstract

We have studied the spin polarized hot neutron matter in the presence of
strong magnetic field. In this work, using the lowest order constrained
variational method at finite temperature and employing $AV_{18}$ nuclear
potential, some thermodynamic properties of spin polarized neutron matter such
as spin polarization parameter, free energy, equation of state and effective
mass have been calculated. It has been shown that the strong magnetic field
breaks the symmetry of the free energy, leading to a magnetized equilibrium
state. We have found that the equation of state becomes stiffer by increasing
both magnetic field and temperature. The magnetic field dependence of
effective mass for the spin-up and spin-down neutrons has been investigated.

Neutron matter - strong magnetic field - equation of state

##  I INTRODUCTION

Based on the supernova models, after the gravitational collapse of a
degenerate stellar core and the explosive ejection of outer layers material, a
protoneutron star would be born Haensel . Through the formation of the
protoneutron star, the system arrives at a temperature of about $20-50\ MeV$
Camenzind . This protoneutron star is hot, opaque to neutrinos, and larger
than an ordinary neutron star Haensel . After the formation of the
protoneutron star, neutrino emission is the dominant process in cooling of the
neutron star (mainly by URCA process and neutrino Bremsstrahlung) Camenzind .

Woltjer has predicted a magnetic field strength of order $10^{15}\ G$ for
neutron stars as a result of the magnetic flux conservation from the
progenitor star Woltjer . This in agreement with the experimental indication
that the surface magnetic field strength of magnetars can be of the order
$B_{magnetar}\approx 10^{14}-10^{15}\ G$ Thompson ; Lazzati . The magnetic
field can be distorted or amplified by some mixture of convection,
differential rotation, and magnetic instabilities Tayler ; Spruit . The
relative importance of these ingredients depends on the initial field strength
and the rotation rate of the star. For both convection and differential
rotation, the field and its supporting currents are not likely to be confined
to the solid crust of the star but instead distributed in most of the stellar
interior, which is mostly a fluid mixture of neutrons, protons, electrons, and
other more exotic particles Reisen . Thompson et al. argued that newborn
neutron stars probably combine vigorous convection and differential rotation,
making it likely that a dynamo process might operate in them Thompson2 . They
expected fields up to $10^{15}-10^{16}\ G$ in neutron stars with few-
millisecond initial periods. In the core of high density inhomogeneous
gravitationally bound neutron stars, the magnetic field strength can be as
large as $10^{20}\ G$ Ferrer . In addition, considering the formation of a
quark core in the high density interior of a neutron star, the maximum field
reaches up to about $10^{20}\ G$ Ferrer ; Tatsumi . According to the scalar
virial theorem based on Newtonian gravity, the magnetic field strength is
allowed to be up to $10^{18}\ G$ in the interior of a magnetar Lai . Moreover,
general relativity predicts the allowed maximum value of the neutron star
magnetic field to be $10^{18}-10^{20}\ G$ shapiro . By comparing the cooling
curves of neutron stars with the observational data, Yuan et al. obtained the
magnetic field strength of order $10^{19}\ G$ for many not so old neutron
stars Yuan .

The finite temperature and strong magnetic field in the interior of a
protoneutron star can influence different astrophysical quantities. Therefore,
to have a better understanding of different astrophysical phenomena such as
supernova explosion, thermal evolution and cooling of protoneutron stars,
gravitational wave emission spectrum from neutron star mergers, and to get the
more precise astrophysical quantities such as properties of very young hot
neutron stars and composition of neutron stars, one should consider the
neutron star matter at finite temperatures (excited neutron star matter) and
strong magnetic fields.

Since $\beta$-equilibrium leads to an increase in the number of neutrons in
neutron star matter, it is possible to approximate the neutron star matter by
the pure neutron matter. Many works have dealt with the study of dense neutron
matter at finite temperature Alonso2 ; Mukherjee ; Panda ; Rios2 ; Bombaci ;
Rios3 . Alonso et al. have used a field theoretical model for the analysis of
relativistic neutron matter Alonso2 . By solving the model in the renormalized
Hartree approximation, they have investigated the effect of central
temperature on the maxima of mass for stable configurations, the radii of the
configurations, and the gravitational red shift at the surface of a neutron
star. Panda et al. used a mean-field description of nonoverlaping nucleon bags
bound by the self-consistent exchange of $\sigma$, $\omega$, and $\rho$ mesons
to investigate the properties of neutron matter at finite temperature Panda .
They showed that by increasing the temperature, the equation of state becomes
stiffer. Within the framework of the Brueckner-Hartree-Fock formalism and
using the $AV_{18}$ nucleon-nucleon interaction for the spin polarized neutron
matter, Bombaci et al. found that an increase in the temperature moderately
affects the single-particle potentials Bombaci . In the Hartree-Fock
approximation using Skyrme type interactions for spin polarized neutron
matter, Rios et al. showed that the critical density at which the
ferromagnetism takes place, decreases by temperature Rios2 . Within the self-
consistent Green’s-function approach applying the CD Bonn and the $AV_{18}$
potential for neutron matter, Rios et al. found that the effect of dynamical
correlations on the macroscopic properties is rather insensitive to the
thermal effects Rios3 . The variational theory for fermions at finite
temperature and high density has been applied by Mukherjee to neutron matter
Mukherjee . It has been found that the temperature dependence of the
correlation operator is weak, but it is not negligible. Besides, it has been
shown that the first order phase transition due to neutral pion condensation
has a critical temperature of about $22MeV$ for neutron matter. The effect of
strong magnetic field on the properties of dense neutron matter has also been
considered. Perez-Garcia showed that for the neutron matter in the presence of
strong magnetic fields, in the Skyrme model, there is a ferromagnetic phase
transition at $\rho\sim 4\rho_{0}$, whereas it is forbidden in the $D1P$ model
Perez-Garcia . The results indicate that the effects of temperature on the
neutron magnetization remain moderate at temperatures up to about $T=40\ MeV$.
In the context of the Landau theory of normal Fermi liquids, using Skyrme and
Gogny effective interactions, some thermodynamical quantities such as
isothermal compressibility and spin susceptibility of pure neutron matter have
also been studied Perez-Garcia2 .

In our previous works, we have investigated the spin polarized neutron matter
Bordbar75 , symmetric nuclear matter Bordbar76 , asymmetric nuclear matter as
well as neutron star matter Bordbar77  and magnetized neutron matter Bordbar83
at zero temperature using lowest order constrained variational (LOCV) method
with the realistic strong interactions. We have also investigated the
thermodynamic properties of spin polarized neutron matter Bordbar78 ,
symmetric nuclear matter Bordbar80 , and asymmetric nuclear matter Bordbar81
at finite temperature with no magnetic field. In the present work, we
calculate different thermodynamic properties of spin polarized neutron matter
at finite temperature in the presence of strong magnetic field using LOCV
technique employing $AV_{18}$ potential.

##  II LOCV formalism for the spin polarized hot neutron matter in the
presence of magnetic field

We consider a homogeneous system of $N$ interacting particles with $N^{(+)}$
spin-up and $N^{(-)}$ spin-down neutrons under the influence of a uniform
magnetic field, i.e. $\mathbf{B}=B\widehat{k}$. The number densities of spin-
up and spin-down neutrons are presented by $\rho^{(+)}$ and $\rho^{(-)}$
respectively. We introduce the spin polarization parameter, $\delta$, by

$\displaystyle\delta=\frac{\rho^{(+)}-\rho^{(-)}}{\rho},$ (1)

where $-1\leq\delta\leq 1$, and $\rho$ is the total number density of system.
The magnetization density of neutron matter is defined as

$\displaystyle m=\mu_{n}\delta\rho,$ (2)

where $\mu_{n}$ is the neutron magnetic moment. The total magnetization of a
given volume is also as follows

$\displaystyle M=\int mdV.$ (3)

In order to calculate the energy of this system, we use LOCV method as
follows: we consider a trial many-body wave function of the form

$\displaystyle\psi=\mathcal{F}\phi,$ (4)

where $\phi$ is the uncorrelated ground-state wave function of $N$ independent
neutrons, and $\mathcal{F}$ is a proper $N$-body correlation function. Using
Jastrow approximation Jastrow , $\mathcal{F}$ can be replaced by

$\displaystyle\mathcal{F}=S\prod_{i>j}f(ij),$ (5)

where $S$ is a symmetrizing operator. We consider a cluster expansion of the
energy functional up to the two-body term,

$\displaystyle
E([f])=\frac{1}{N}\frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}=E_{1}+E_{2}\cdot$
(6)

The one-body term, $E_{1}$, is given by

$\displaystyle E_{1}=-\frac{MH}{N}+\sum_{i=+,-}\overline{\varepsilon}_{i},$
(7)

where the first term of Eq. (7), $H$ shows the external magnetic field. It
should be noted that we have used $B$ to present the magnetic field strength;
while the total magnetic field is the sum of the external magnetic field and
the induced magnetization, $B=H+4\pi M$. Because of the tiny value of the
neutron magnetic moment, we assume that the induced magnetization has a small
contribution to the total magnetic field. Consequently,

$\displaystyle B\sim H.$ (8)

Using Eqs. (2), (3) and (8), the one-body term can be written as

$\displaystyle E_{1}=-\mu_{n}B\delta+\sum_{i=+,-}\overline{\varepsilon}_{i}.$
(9)

The second term in Eq. (7), $\overline{\varepsilon}_{i}$ is as follows

$\displaystyle\overline{\varepsilon}_{i}=\sum_{k}\frac{\hbar^{2}k^{2}}{2m}\overline{n}_{i}(k,T,B,\rho^{(i)}),$
(10)

where $\overline{n}_{i}(k,T,B,\rho^{(i)})$ is the Fermi-Dirac distribution
function in the presence of magnetic field,

$\displaystyle\overline{n}_{i}(k,T,B,\rho^{(i)})=\frac{1}{e^{\beta[\overline{\epsilon}_{i}(k,T,B,\rho^{(i)})-\overline{\mu}_{i}(T,B,\rho^{(i)})]}+1}.$
(11)

In Eq. (11), $\overline{\epsilon}_{i}$ and $\overline{\mu}_{i}$ are the
single-particle energy of a neutron and the neutron chemical potential
respectively. The single-particle energy, $\overline{\epsilon}_{i}$, of a
neutron with momentum $k$ and spin projection $i$ in the presence of magnetic
field is approximately written in terms of the effective mass as follows Rios2

$\displaystyle\overline{\epsilon}_{i}(k,T,B,\rho^{(i)})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\hbar^{2}k^{2}}{2m_{+}^{*}(T,\rho)}-\mu_{n}B+U_{+}(T,\rho^{(+)})&;~{}~{}i=+,\\\
\\\
\frac{\hbar^{2}k^{2}}{2m_{-}^{*}(T,\rho)}+\mu_{n}B+U_{-}(T,\rho^{(-)})&;~{}~{}i=-.\end{array}\right.$
(15)

In fact, we use a quadratic approximation for the single particle potential
incorporated in the single particle energy as a momentum independent effective
mass. $U_{i}(T,\rho^{(i)})$ is the momentum independent single particle
potential. The effective mass, $m_{i}^{*}$, is determined variationally
Modarres3 ; Modarres5 ; Modarres7 ; Modarres8 ; Friedman . The chemical
potential, $\overline{\mu}_{i}$, is also obtained by applying the constraint

$\displaystyle\sum_{k}\overline{n}_{i}(k,T,B,\rho^{(i)})=N^{(i)}.$ (16)

The two-body energy, $E_{2}$, is

$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum_{ij}\langle ij\left|\nu(12)\right|ij-
ji\rangle,$ (17)

where

$\nu(12)=-\frac{\hbar^{2}}{2m}[f(12),[\nabla_{12}^{2},f(12)]]+f(12)V(12)f(12).$

$f(12)$ and $V(12)$ are the two-body correlation function and nuclear
potential respectively. In the LOCV formalism, the two-body correlation
function, $f(12)$, is considered as follows Owen ,

$\displaystyle f(12)$ $\displaystyle=$
$\displaystyle\sum^{3}_{k=1}f^{(k)}(r_{12})P^{(k)}_{12},$ (18)

where

$\displaystyle P^{(k=1-3)}_{12}$ $\displaystyle=$
$\displaystyle(\frac{1}{4}-\frac{1}{4}\sigma_{1}.\sigma_{2}),\
(\frac{1}{2}+\frac{1}{6}\sigma_{1}.\sigma_{2}+\frac{1}{6}S_{12}),\
(\frac{1}{4}+\frac{1}{12}\sigma_{1}.\sigma_{2}-\frac{1}{6}S_{12}).$ (19)

In Eq. (19), $S_{12}$ and $\sigma_{1}.\sigma_{2}$ are the tensor and Pauli
operators respectively. Using the above two-body correlation function and the
$AV_{18}$ two-body potential Wiringa , after doing some algebra, we find the
following equation for the two-body energy:

$\displaystyle E_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{4}\rho}\left(\frac{\hbar^{2}}{2m}\right)\sum_{JLSS_{z}}\frac{(2J+1)}{2(2S+1)}[1-(-1)^{L+S+1}]\left|\left\langle\frac{1}{2}\sigma_{z1}\frac{1}{2}\sigma_{z2}\mid
SS_{z}\right\rangle\right|^{2}\times$ (20) $\displaystyle\times\int
dr\left\\{\left[{f_{\alpha}^{(1)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(1)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+\frac{2m}{\hbar^{2}}(\\{V_{c}-3V_{\sigma}+V_{\tau}-3V_{\sigma\tau}+2(V_{T}-3V_{\sigma
T})-2V_{\tau z}\\}{a_{\alpha}^{(1)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}-3V_{l2\sigma}+V_{l2\tau}-3V_{l2\sigma\tau}]{c_{\alpha}^{(1)}}^{2}(r,\rho,T))(f_{\alpha}^{(1)})^{2}\right]+\sum_{k=2,3}\left[{f_{\alpha}^{(k)^{{}^{\prime}}}}^{2}{a_{\alpha}^{(k)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+\frac{2m}{\hbar^{2}}(\\{V_{c}+V_{\sigma}+V_{\tau}+V_{\sigma\tau}+(-6k+14)(V_{t\tau}+V_{t})-(k-1)(V_{ls\tau}+V_{ls})\right.\right.$
$\displaystyle\left.\left.+2[V_{T}+V_{\sigma T}+(-6k+14)V_{tT}-V_{\tau
z}]\\}{a_{\alpha}^{(k)}}^{2}(r,\rho,T)\right.\right.$
$\displaystyle\left.\left.+[V_{l2}+V_{l2\sigma}+V_{l2\tau}+V_{l2\sigma\tau}]{c_{\alpha}^{(k)}}^{2}(r,\rho,T)+[V_{ls2}+V_{ls2\tau}]{d_{\alpha}^{(k)}}^{2}(r,\rho,T)){f_{\alpha}^{(k)}}^{2}\right]\right.$
$\displaystyle\left.+\frac{2m}{\hbar^{2}}\\{V_{ls}+V_{ls\tau}-2(V_{l2}+V_{l2\sigma}+V_{l2\sigma\tau}+V_{l2\tau})-3(V_{ls2}+V_{ls2\tau})\\}b_{\alpha}^{2}(r,\rho,T)f_{\alpha}^{(2)}f_{\alpha}^{(3)}\right.$
$\displaystyle\left.+\frac{1}{r^{2}}(f_{\alpha}^{(2)}-f_{\alpha}^{(3)})^{2}b_{\alpha}^{2}(r,\rho,T)\right\\},$

where $\alpha=\\{J,L,S,S_{z}\\}$ and the coefficients
${a_{\alpha}^{(i)}}^{2}$, $b_{\alpha}^{2}$, ${c_{\alpha}^{(i)}}^{2}$, and
${d_{\alpha}^{(i)}}^{2}$ are defined as

$\displaystyle{a_{\alpha}^{(1)}}^{2}(r,\rho,T)=r^{2}I_{L,S_{z}}(r,\rho,T),$
(21) $\displaystyle{a_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\beta
I_{J-1,S_{z}}(r,\rho,T)+\gamma I_{J+1,S_{z}}(r,\rho,T)],$ (22)
$\displaystyle{a_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\gamma
I_{J-1,S_{z}}(r,\rho,T)+\beta I_{J+1,S_{z}}(r,\rho,T)],$ (23) $\displaystyle
b_{\alpha}^{2}(r,\rho,T)=r^{2}[\beta_{23}I_{J-1,S_{z}}(r,\rho,T)-\beta_{23}I_{J+1,S_{z}}(r,\rho,T)],$
(24)
$\displaystyle{c_{\alpha}^{(1)}}^{2}(r,\rho,T)=r^{2}\nu_{1}I_{L,S_{z}}(r,\rho,T),$
(25)
$\displaystyle{c_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\eta_{2}I_{J-1,S_{z}}(r,\rho,T)+\nu_{2}I_{J+1,S_{z}}(r,\rho,T)],$
(26)
$\displaystyle{c_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\eta_{3}I_{J-1,S_{z}}(r,\rho,T)+\nu_{3}I_{J+1,S_{z}}(r,\rho,T)],$
(27)
$\displaystyle{d_{\alpha}^{(2)}}^{2}(r,\rho,T)=r^{2}[\xi_{2}I_{J-1,S_{z}}(r,\rho,T)+\lambda_{2}I_{J+1,S_{z}}(r,\rho,T)],$
(28)
$\displaystyle{d_{\alpha}^{(3)}}^{2}(r,\rho,T)=r^{2}[\xi_{3}I_{J-1,S_{z}}(r,\rho,T)+\lambda_{3}I_{J+1,S_{z}}(r,\rho,T)],$
(29)

with

$\displaystyle\beta=\frac{J+1}{2J+1},\ \gamma=\frac{J}{2J+1},\
\beta_{23}=\frac{2J(J+1)}{2J+1},$ (30) $\displaystyle\nu_{1}=L(L+1),\
\nu_{2}=\frac{J^{2}(J+1)}{2J+1},\ \nu_{3}=\frac{J^{3}+2J^{2}+3J+2}{2J+1},$
(31) $\displaystyle\eta_{2}=\frac{J(J^{2}+2J+1)}{2J+1},\
\eta_{3}=\frac{J(J^{2}+J+2)}{2J+1},$ (32)
$\displaystyle\xi_{2}=\frac{J^{3}+2J^{2}+2J+1}{2J+1},\
\xi_{3}=\frac{J(J^{2}+J+4)}{2J+1},$ (33)
$\displaystyle\lambda_{2}=\frac{J(J^{2}+J+1)}{2J+1},\
\lambda_{3}=\frac{J^{3}+2J^{2}+5J+4}{2J+1},$ (34)

and

$\displaystyle I_{J,S_{z}}(r,\rho,T)=\frac{1}{2\pi^{6}\rho^{2}}\int dk_{1}\
dk_{2}\
\overline{n}_{i}(k_{m},T,B,\rho^{(i)})\overline{n}_{j}(k_{m},T,B,\rho^{(j)})J_{J}^{2}(|k_{2}-k_{1}|r)\cdot$
(35)

In the above equation, $J_{J}(x)$ is the Bessel function.

Now, we minimize the two-body energy with respect to the variations in the
function $f_{\alpha}^{(i)}$ subject to the normalization constraint,

$\displaystyle\frac{1}{N}\sum_{ij}\langle
ij\left|h_{S_{z}}^{2}-f^{2}(12)\right|ij\rangle_{a}=0.$ (36)

The minimization subject to the above normalization constraint leads to the
normalization of the two body wave function to unity Owen . For the spin
polarized hot neutron matter, the function $h_{S_{z}}(r)$ is defined as
follows,

$\displaystyle h_{S_{z}}(r)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\left[1-\left(\frac{\gamma_{i}(r)}{\rho}\right)^{2}\right]^{-1/2}&;~{}~{}S_{z}=\pm
1,\\\ \\\ 1&;~{}~{}S_{z}=0,\end{array}\right.$ (40)

where

$\displaystyle\gamma_{i}(r)=\frac{1}{2\pi^{2}}\int dk\
\overline{n}_{i}(k_{m},T,B,\rho^{(i)})J_{0}(kr)k^{2}.$ (41)

From the minimization of the two-body cluster energy, we get a set of coupled
and uncoupled differential equations BordMod98 . By solving these equations,
we can obtain the correlation functions to compute the two-body energy term,
$E_{2}$. As the final step, we calculate the free energy per particle, $F$, to
get different thermodynamic properties of spin polarized hot neutron matter,

$\displaystyle F(\rho,T,B)=E(\rho,T,B)-TS(\rho,T,B),$ (42)

where S is the entropy per particle,

$\displaystyle S(\rho,T,B)$ $\displaystyle=$
$\displaystyle-\frac{1}{N}\sum_{i=+,-}\sum_{k}\\{{[1-\overline{n}_{i}(k,T,B,\rho^{(i)})]\textrm{ln}[1-\overline{n}_{i}(k,T,B,\rho^{(i)})]}$
(43) $\displaystyle+$
$\displaystyle{\overline{n}_{i}(k,T,B,\rho^{(i)})\textrm{ln}[\overline{n}_{i}(k,T,B,\rho^{(i)})]}\\}.$

It should be noted that in our calculations, we introduce the effective
masses, $m_{i}^{*}$, as variational parameters Modarres3 ; Modarres5 ;
Modarres7 ; Modarres8 ; Friedman . We minimize the free energy with respect to
the variations in the effective masses, and then we obtain the chemical
potentials and the effective masses of spin-up and spin-down neutrons at the
equilibrium state. In our approach, the effective mass depends on both density
and temperature but it is independent of the momentum. The effective mass of a
quasiparticle near the Fermi surface for the spin polarized neutron matter at
low temperatures is also the static physical quantity of interest in the
context of Landau Fermi liquid theory Landau .

##  III RESULTS and DISCUSSION

In Fig. 1, we present the free energy per particle of spin polarized neutron
matter versus the spin polarization parameter $\delta$. Fig. 1a shows that in
the presence of magnetic field, the free energy is not a symmetric function of
spin polarization parameter and the equilibrium configuration would experience
a net magnetization. Clearly, the effects of magnetic fields below $B\sim
10^{18}\ G$ are almost insignificant, but by increasing the magnetic field,
the equilibrium value of the spin polarization parameter (i.e. the
polarization that minimizes the free energy) and the free energy decrease,
leading to a more stable system. Fig. 1b indicates that the effect of
temperature on the fully spin polarized neutron matter is less than that of
the unpolarized one.

Fig. 2 presents the equilibrium value of the spin polarization parameter
versus density $\rho$. Fig. 2a shows that at low densities ($\rho\leq 0.2\
fm^{-3}$), the magnitude of the spin polarization parameter decreases by
increasing the temperature. However, at higher densities, the related values
of the spin polarization parameter at different finite temperatures are almost
identical to the one for zero temperature. This is due to smaller values of
$T/\varepsilon_{f}^{*}$ at high densities (Fig. 2b), in which
$\varepsilon_{f}^{*}$ is defined as follows,

$\displaystyle\varepsilon_{f}^{*}=\sum_{i=+,-}\frac{\rho^{(i)}}{\rho}\varepsilon_{fi}^{*},$
(44)

with

$\displaystyle\varepsilon_{f+}^{*}=\frac{\hbar^{2}k_{F}^{(+)^{2}}}{2m}-\mu_{n}B,$
(45)

and

$\displaystyle\varepsilon_{f-}^{*}=\frac{\hbar^{2}k_{F}^{(-)^{2}}}{2m}+\mu_{n}B.$
(46)

In the above equations, $\varepsilon_{fi}^{*}$ and
$k_{F}^{(i)}=(6\pi^{2}\rho^{(i)})^{\frac{1}{3}}$ are the Fermi energy and
Fermi momentum of neutrons with spin projection $i$ in the presence of the
magnetic field. It is evident from Eq. (44) that $\varepsilon_{f}^{*}$ gives
an average of the Fermi energy of magnetized neutron matter. Therefore, the
ratio $T/\varepsilon_{f}^{*}$ is a criterion for the fraction of particles
which are thermally excited Pathria , and how much the system is disordered.
In Fig. 3, we show the spin polarization parameter at the equilibrium state as
a function of the magnetic field $B$. At each temperature, the magnitude of
spin polarization parameter grows by increasing the magnetic field (Fig. 3a).
We have found that at strong magnetic fields, the effect of finite temperature
is more significant because the ratio $T/\varepsilon_{f}^{*}$ rises with the
increase in the magnetic field (Fig. 3b).

The free energy per particle at the equilibrium value of the spin polarization
parameter is presented in Fig. 4. It can be seen that at finite temperature,
the free energy is an increasing function of density (Fig. 4a). At low
densities, the rate of increase in the free energy varies by increasing the
density, but at high densities, this rate of increase is nearly constant.
Moreover, the effect of temperature on the free energy is more pronounced at
low densities. Fig. 4b shows that the free energy decreases by growing the
temperature nearly at the same rate for different magnetic fields. The free
energy decreases by increasing the magnetic field (Fig. 4c). We can see that
by increasing the magnetic field up to a value of about $B\simeq 10^{18}\ G$,
the free energy per particle slowly decreases, and then it rapidly decreases
for the magnetic fields greater than this value. This indicates that, above
$B\simeq 10^{18}\ G$, the effect of magnetic field on the free energy of the
spin polarized neutron matter becomes more important.

From the free energy per particle of magnetized neutron matter, $F$, we can
obtain the corresponding pressure of neutron matter using the following
relation,

$\displaystyle P(\rho,T,B)=\rho^{2}{\left(\frac{\partial
F(\rho,T,B)}{\partial\rho}\right)_{T,B}}.$ (47)

This equation of state (EoS) is plotted in Fig. 5. For each value of the
density, pressure increases by growing the magnetic field (Fig. 5a). This
stiffening of the equation of state is due to the inclusion of neutron
anomalous magnetic moments. From the astrophysical point of view, it should be
noted that this stiffening of the EoS leads to the larger value for the
maximum mass of neutron star Bordbar2006 ; Bordbar2009 ; Bordbar2011 . At each
density, the pressure at finite temperature is larger than that of zero
temperature (Fig. 5b). It means that the equation of state of neutron matter
becomes stiffer by increasing the temperature. Fig. 5c also shows that by
increasing the temperature, the pressure increases nearly at the same rate for
different magnetic fields.

Fig. 6 shows the effective mass corresponding with the equilibrium of the
system for the spin-up and spin-down neutrons as a function of the magnetic
field $B$. At low magnetic fields, the effective masses of spin-up and spin-
down neutrons are nearly identical because the effective masses of spin-up and
spin-down neutrons have the same values at $\delta\simeq 0$. Fig. 6 indicates
that the effective mass of spin-up (spin-down) neutrons decreases (increases)
by increasing the magnetic field in agreement with the results obtained in
Ref. Perez-Garcia2 . From the comparison of Fig. 3a and 6, we can see that the
shift in mass is due to the polarization of neutron matter. For the maximum
value of the magnetic field considered in this work, i.e. $5\times 10^{18}\
G$, and at $T=10\ MeV$ and $\rho=0.3\ fm^{-3}$, the equilibrium value of the
spin polarization parameter is about $\delta_{B_{max}}=-0.23$ and that
corresponds to a mass shift of an amount of $\bigtriangleup(m^{*}/m)\approx
0.02$ with respect to the unpolarized case.

##  IV Summary and Concluding Remarks

We have investigated the effect of strong magnetic fields on the thermodynamic
properties of spin polarized hot neutron matter applying LOCV method and using
$AV_{18}$ potential. We have found that in the presence of a strong magnetic
field, the free energy is not a symmetric function of the spin polarization
parameter and the system is macroscopically magnetized. By increasing both
density and temperature, the magnitude of the equilibrium value of the spin
polarization parameter decreases. At low magnetic fields, the free energy
decreases very slowly by increasing the magnetic field, but at stronger
magnetic fields, the free energy decreases rapidly with the increase in the
magnetic field. It has been found that the equation of state becomes stiffer
by increasing the magnetic field. This stiffening of the EoS leads to the
larger value for the maximum mass of neutron star.

###### Acknowledgements.

We wish to thank the Research Institute for Astronomy and Astrophysics of
Maragha and Shiraz University Research Council.

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Figure 1:  Free energy per particle as a function of the spin polarization
parameter $\delta$: (a) for the cases $B=0\ G$ (solid curve), $B=10^{18}\ G$
(dashed curve) and $B=5\times 10^{18}\ G$ (dashdot curve) at the fixed values
of the temperature, $T=10\ MeV$, and the density, $\rho=0.2\ fm^{-3}$, (b) for
the cases $T=0\ MeV$ (solid curve), $T=10\ MeV$ (dashed curve) and $T=20\ MeV$
(dashdot curve) at the fixed values of the magnetic field, $B=5\times 10^{18}\
G$, and the density, $\rho=0.2\ fm^{-3}$.

Figure 2:  (a) Spin polarization parameter at the equilibrium state as a
function of the density $\rho$ for the cases $T=0\ MeV$ (solid curve), $T=10\
MeV$ (dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value of the
magnetic field, $B=5\times 10^{18}\ G$. (b) Same as in the top panel but for
the ratio $T/\varepsilon_{f}^{*}$.

Figure 3:  (a) Magnetic field dependence of the spin polarization parameter
$\delta$ at the equilibrium state for the cases $T=0\ MeV$ (solid curve),
$T=10\ MeV$ (dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value
of the density, $\rho=0.2\ fm^{-3}$. (b) Same as in the top panel but for the
ratio $T/\varepsilon_{f}^{*}$.

Figure 4:  Free energy per particle at the equilibrium state as a function of:
(a) the density $\rho$ for the cases $T=0\ MeV$ (solid curve), $T=10\ MeV$
(dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value of the
magnetic field, $B=5\times 10^{18}\ G$, (b) the temperature $T$ for the cases
$B=0\ G$ (solid curve), $B=10^{18}\ G$ (dashed curve) and $B=5\times 10^{18}\
G$ (dashdot curve), and a fixed value of the density, $\rho=0.2\ fm^{-3}$, (c)
the magnetic field $B$ for the cases $T=0\ MeV$ (solid curve), $T=10\ MeV$
(dashed curve) and $T=20\ MeV$ (dashdot curve), and a fixed value of the
density, $\rho=0.2\ fm^{-3}$.

Figure 5:  Pressure of spin polarized neutron matter as a function of: (a) the
density $\rho$ for the cases $B=0\ G$ (solid curve), $B=10^{18}\ G$ (dashed
curve) and $B=5\times 10^{18}\ G$ (dashdot curve), and a fixed value of the
temperature, $T=20\ MeV$, (b) the density $\rho$ for the cases $T=0\ MeV$
(solid curve), $T=10\ MeV$ (dashed curve) and $T=20\ MeV$ (dashdot curve), and
a fixed value of the magnetic field, $B=5\times 10^{18}\ G$, (c) the
temperature $T$ for the cases $B=0\ G$ (solid curve), $B=10^{18}\ G$ (dashed
curve) and $B=5\times 10^{18}\ G$ (dashdot curve), and a fixed value of the
density, $\rho=0.2\ fm^{-3}$.

Figure 6:  Magnetic field dependence of the effective masses of spin-up (solid
curve) and spin-down (dashed curve) neutrons corresponding with the
equilibrium state at the fixed values of the temperature, $T=10\ MeV$, and the
density, $\rho=0.3\ fm^{-3}$.


11institutetext: D. Moździerski 22institutetext: Instytut Astronomiczny,
Uniwersytet Wrocławski, Kopernika 11, 51-622, Wrocław, Poland, 22email:
<EMAIL_ADDRESS>
# Variability survey in the young open cluster NGC 457

Dawid Moździerski  Andrzej Pigulski  Grzegorz Kopacki  Marek Stȩślicki and
Zbigniew Kołaczkowski

###### Abstract

We present preliminary results of the photometric variability search in the
field of view of the young open cluster NGC 457. We find over 60 variable
stars in the field, including 25 pulsating or candidate pulsating stars.

##  1 The cluster

NGC 457 is a young open cluster in Cassiopeia, nearby $\phi$ Cas. Its age is
estimated for 10–20 Myr PheJan1994 ; Loktin2001 , distance, for about 2.5–3.0
kpc. It is located in the Perseus arm of the Galaxy. Six variable or suspected
variable stars were known in the observed cluster field prior to our study
(see, e.g., Maciejewski2008 ). We present preliminary results of the
photometric variability survey in this cluster aimed at discovery of B-type
pulsating stars and bright eclipsing binaries.

##  2 Observations and results

The photometric observations of NGC 457 were obtained during three runs: the
run consisting of 4 nights in 1993 carried out in Ostrowik station, University
of Warsaw, by one of us (GK), the second run of 31 nights in the years
1999–2002 (Białków station, University of Wrocław) and the third one
consisting of 24 nights made again in Białków between December 2010 and March
2011. Here, we present results based on the 2010–2011 observations only. Of
the three runs, the last one is of the best quality. During this run we used
60-cm reflecting telescope equipped with the Andor Tech. DW 432-BV CCD camera
covering 13${}^{\prime}\times$ 12′ field of view. All frames were calibrated
in a standard way and reduced with the Daophot II package Stetson1987 .

We have found 64 variable stars in the observed field, including the six
already known. The most interesting is the discovery of pulsating stars,
likely members of the cluster. The sample of pulsating stars includes: a
single $\beta$ Cephei star NGC 457-8 (see Fig. 1), 12 (candidate) SPB stars
and 12 $\delta$ Scuti stars. In addition, nine eclipsing and ellipsoidal
variables were found. The sample includes also stars showing irregular or
(quasi)periodic variations of unknown origin both in the cluster main sequence
and among the reddest stars in the field. Some of B-type stars showing this
type of variability are known as Be stars.

[scale=.65]Mozdzierski_Fig1.eps

Figure 1: Fourier amplitude spectrum for the $V$-filter data of $\beta$ Cephei
star NGC 457-8. The inset shows the light curve phased with the main period.

###### Acknowledgements.

We thank Dominik Drobek, Piotr Śródka and Ewa Zahajkiewicz for making some
observations of NGC 457. The work was supported by the MNiSzW grant No. N N203
302635.

## References

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  * (2) Maciejewski, G., Bukowiecki, L., Brożek, T., Georgiev, Ts., Boeva, S., Kacharov, N., Mihov, B., Latev, G., Ovcharov, E., Valcheva, A.: Variable stars in the field of the open cluster NGC 457. Information Bulletin on Variable Stars. 5864, 1–7 (2008)
  * (3) Phelps, R.L., Janes, K.A.: Young open clusters as probes of the star formation process. I. An atlas of open cluster photometry. ApJS. 90, 31–82 (1994)
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