4,083 research outputs found
Reinforcement learning based local search for grouping problems: A case study on graph coloring
Grouping problems aim to partition a set of items into multiple mutually
disjoint subsets according to some specific criterion and constraints. Grouping
problems cover a large class of important combinatorial optimization problems
that are generally computationally difficult. In this paper, we propose a
general solution approach for grouping problems, i.e., reinforcement learning
based local search (RLS), which combines reinforcement learning techniques with
descent-based local search. The viability of the proposed approach is verified
on a well-known representative grouping problem (graph coloring) where a very
simple descent-based coloring algorithm is applied. Experimental studies on
popular DIMACS and COLOR02 benchmark graphs indicate that RLS achieves
competitive performances compared to a number of well-known coloring
algorithms
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
Complete Multipartite Graphs and the Relaxed Coloring Game
Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted χ gd (G). It is known that there exist graphs such that χ g0 (G) = 3, but χ g1 (G) \u3e 3. We will show that for all positive integers m, there exists a complete multipartite graph G such that m ≤ χ g0 (G) \u3c χ g1 (G)
The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs
The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k
A combinatorial approach to knot recognition
This is a report on our ongoing research on a combinatorial approach to knot
recognition, using coloring of knots by certain algebraic objects called
quandles. The aim of the paper is to summarize the mathematical theory of knot
coloring in a compact, accessible manner, and to show how to use it for
computational purposes. In particular, we address how to determine colorability
of a knot, and propose to use SAT solving to search for colorings. The
computational complexity of the problem, both in theory and in our
implementation, is discussed. In the last part, we explain how coloring can be
utilized in knot recognition
The Relaxed Edge-Coloring Game and \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs
The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in [10]. More broadly, we generalize the main result in [10] by showing that if G is k-degenerate with ∆(G) = ∆ and j ∈ [∆ + k − 1], then there exists a function h(k, j) such that Alice has a winning strategy for this game with r = ∆ + k − j and d ≥ h(k, j)
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