Symmetries are intrinsic to many combinatorial problems including Boolean Satisfiability (SAT) and Constraint Programming (CP). In SAT, the identification of symmetry breaking predicates (SBPs) is a well-known, often effective, technique for solving hard problems. The identification of SBPs in SAT has been the subject of significant improvements in recent years, resulting in more compact SBPs and more effective algorithms. The identification of SBPs has also been applied to pseudo-Boolean (PB) constraints, showing that symmetry breaking can also be an effective technique for PB constraints. This paper extends further the application of SBPs, and shows that SBPs can be identified and used in Maximum Satisfiability (MaxSAT), as well as in its most well-known variants, including partial MaxSAT, weighted MaxSAT and weighted partial MaxSAT. As with SAT and PB, symmetry breaking predicates for MaxSAT and variants are shown to be effective for a representative number of problem domains, allowing solving problem instances that current state of the art MaxSAT solvers could not otherwise solve.
Symmetries are a well-known research topic, that serve to tackle complexity in many combinatorial problems. The first ideas on symmetry breaking were developed in the 90s @cite_11 @cite_13 , by relating symmetries with the graph automorphism problem, and by proposing the first approach for generating symmetry breaking predicates. This work was later extended and optimized for propositional satisfiability @cite_10 .
@cite_10 @cite_13 @cite_11
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We investigate the role of symmetry detection and symmetry breaking in answer set programming to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We reduce symmetry detection to a graph automorphism problem which allows us to extract symmetries of a logic program from the symmetries of the constructed coloured graph. The correctness of our reduction is proven. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process to implicitly represent symmetries with exponential compression. These ideas are formulated as preprocessing and implemented in a completely automated flow that first detects symmetries from a given answer set program, adds symmetry-breaking constraints, and can be applied to any existing answer set solver. We demonstrate computational impact on benchmarks versus direct application of the solver. The presence of symmetry in Boolean satisfiability (SAT) problem instances often poses challenges to solvers. Currently, the most effective approach to handle symmetry is by static symmetry breaking, which generates asymmetric constraints to add to the instance. An alternative way is to handle symmetry dynamically during solving. As modern SAT solvers can be viewed as propositional proof generators, adding a symmetry rule in a solver’s proof system would be a straightforward technique to handle symmetry dynamically. However, none of these proposed symmetrical learning techniques are competitive to static symmetry breaking. In this paper, we present symmetric explanation learning, a form of symmetrical learning based on learning symmetric images of explanation clauses for unit propagations performed during search. A key idea is that these symmetric clauses are only learned when they would restrict the current search state, i.e., when they are unit or conflicting. We further provide a theoretical discussion on symmetric explanation learning and a working implementation in a state-of-the-art SAT solver. We also present extensive experimental results indicating that symmetric explanation learning is the first symmetrical learning scheme competitive with static symmetry breaking. We present a comprehensive study of the use of value precedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is both efficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreath value and partial interchangeability. We also show that value precedence is closely related to lexicographical ordering. Finally, we consider the interaction between value precedence and symmetry breaking constraints for variable symmetries. Symmetry-breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this article we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry-breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes the following: —An MIS algorithm running in O(log2D Δ L 2√log n, and comes close to the Ω(flog Δ log log Δ lower bound of Kuhn, Moscibroda, and Wattenhofer. —A maximal matching algorithm running in O(log Δ + log 4log n) time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on Δ is nearly optimal. —A (Δ + 1)-coloring algorithm requiring O(log Δ + 2o(√log log n) time, improving on an O(log Δ + √log n)-time algorithm of Schneider and Wattenhofer. —A method for reducing symmetry-breaking problems in low arboricity degeneracy graphs to low-degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an O(√log n)-time maximal matching algorithm for graphs with arboricity up to 2√log n and an O(log 2 3n)-time MIS algorithm for graphs with arboricity up to 2(log n)1 3. Each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry-breaking task to a corresponding deterministic one on a poly(log n)-size graph.