799 research outputs found
Breaking Instance-Independent Symmetries In Exact Graph Coloring
Code optimization and high level synthesis can be posed as constraint
satisfaction and optimization problems, such as graph coloring used in register
allocation. Graph coloring is also used to model more traditional CSPs relevant
to AI, such as planning, time-tabling and scheduling. Provably optimal
solutions may be desirable for commercial and defense applications.
Additionally, for applications such as register allocation and code
optimization, naturally-occurring instances of graph coloring are often small
and can be solved optimally. A recent wave of improvements in algorithms for
Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests
generic problem-reduction methods, rather than problem-specific heuristics,
because (1) heuristics may be upset by new constraints, (2) heuristics tend to
ignore structure, and (3) many relevant problems are provably inapproximable.
Problem reductions often lead to highly symmetric SAT instances, and
symmetries are known to slow down SAT solvers. In this work, we compare several
avenues for symmetry breaking, in particular when certain kinds of symmetry are
present in all generated instances. Our focus on reducing CSPs to SAT allows us
to leverage recent dramatic improvement in SAT solvers and automatically
benefit from future progress. We can use a variety of black-box SAT solvers
without modifying their source code because our symmetry-breaking techniques
are static, i.e., we detect symmetries and add symmetry breaking predicates
(SBPs) during pre-processing.
An important result of our work is that among the types of
instance-independent SBPs we studied and their combinations, the simplest and
least complete constructions are the most effective. Our experiments also
clearly indicate that instance-independent symmetries should mostly be
processed together with instance-specific symmetries rather than at the
specification level, contrary to what has been suggested in the literature
Colorings of odd or even chirality on hexagonal lattices
We define two classes of colorings that have odd or even chirality on
hexagonal lattices. This parity is an invariant in the dynamics of all loops,
and explains why standard Monte-Carlo algorithms are nonergodic. We argue that
adding the motion of "stranded" loops allows for parity changes. By
implementing this algorithm, we show that the even and odd classes have the
same entropy. In general, they do not have the same number of states, except
for the special geometry of long strips, where a Z symmetry between even
and odd states occurs in the thermodynamic limit.Comment: 18 pages, 13 figure
Random Graph Coloring - a Statistical Physics Approach
The problem of vertex coloring in random graphs is studied using methods of
statistical physics and probability. Our analytical results are compared to
those obtained by exact enumeration and Monte-Carlo simulations. We critically
discuss the merits and shortcomings of the various methods, and interpret the
results obtained. We present an exact analytical expression for the 2-coloring
problem as well as general replica symmetric approximated solutions for the
thermodynamics of the graph coloring problem with p colors and K-body edges.Comment: 17 pages, 9 figure
Algorithms Transcending the SAT-Symmetry Interface
Dedicated treatment of symmetries in satisfiability problems (SAT) is
indispensable for solving various classes of instances arising in practice.
However, the exploitation of symmetries usually takes a black box approach.
Typically, off-the-shelf external, general-purpose symmetry detection tools are
invoked to compute symmetry groups of a formula. The groups thus generated are
a set of permutations passed to a separate tool to perform further analyzes to
understand the structure of the groups. The result of this second computation
is in turn used for tasks such as static symmetry breaking or dynamic pruning
of the search space. Within this pipeline of tools, the detection and analysis
of symmetries typically incurs the majority of the time overhead for symmetry
exploitation.
In this paper we advocate for a more holistic view of what we call the
SAT-symmetry interface. We formulate a computational setting, centered around a
new concept of joint graph/group pairs, to analyze and improve the detection
and analysis of symmetries. Using our methods, no information is lost
performing computational tasks lying on the SAT-symmetry interface. Having
access to the entire input allows for simpler, yet efficient algorithms.
Specifically, we devise algorithms and heuristics for computing finest direct
disjoint decompositions, finding equivalent orbits, and finding natural
symmetric group actions. Our algorithms run in what we call
instance-quasi-linear time, i.e., almost linear time in terms of the input size
of the original formula and the description length of the symmetry group
returned by symmetry detection tools. Our algorithms improve over both
heuristics used in state-of-the-art symmetry exploitation tools, as well as
theoretical general-purpose algorithms
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