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

Symmetry-Based Search Space Reduction For Grid Maps

In this paper we explore a symmetry-based search space reduction technique which can speed up optimal pathfinding on undirected uniform-cost grid maps by up to 38 times. Our technique decomposes grid maps into a set of empty rectangles, removing from each rectangle all interior nodes and possibly some from along the perimeter. We then add a series of macro-edges between selected pairs of remaining perimeter nodes to facilitate provably optimal traversal through each rectangle. We also develop a novel online pruning technique to further speed up search. Our algorithm is fast, memory efficient and retains the same optimality and completeness guarantees as searching on an unmodified grid map

Symmetry reduction and heuristic search for error detection in model checking

The state explosion problem is the main limitation of model checking. Symmetries in the system being verified can be exploited in order to avoid this problem by defining an equivalence (symmetry) relation on the states of the system, which induces a semantically equivalent quotient system of smaller size. On the other hand, heuristic search algorithms can be applied to improve the bug finding capabilities of model checking. Such algorithms use heuristic functions to guide the exploration. Bestfirst is used for accelerating the search, while A* guarantees optimal error trails if combined with admissible estimates. We analyze some aspects of combining both approaches, concentrating on the problem of finding the optimal path to the equivalence class of a given error state. Experimental results evaluate our approach

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

Symmetry-reinforced Nogood Recording from Restarts

dans le cadre de CP'11International audienceNogood recording from restarts is a form of lightweight learn- ing that combines nogood recording with a restart strategy. At the end of each run, nogoods are extracted from the current (rightmost) branch of the search tree. These nogoods can be used to prevent parts of the search space from being explored more than once. In this paper, we propose to reinforce nogood recording (from restarts) by exploiting symmetries: every time the solver has to be restarted, not only the nogoods that are extracted from the current branch are recorded, but also some additional nogoods that can be computed by means of the previously identi ed problem symmetries. This mechanism of computing symmetric nogoods can be iterated until a xed-point is reached, and controlled (if necessary) by limiting the number and/or the size of recorded nogoods

Symmetry Breaking for Answer Set Programming

In the context of answer set programming, this work investigates symmetry detection and symmetry breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We contribute a reduction of symmetry detection to a graph automorphism problem which allows to extract symmetries of a logic program from the symmetries of the constructed coloured graph. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process which implicitly represent symmetries and always 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. Furthermore, we explore symmetry breaking for answer set programming in two domains: first, constraint answer set programming as a novel approach to represent and solve constraint satisfaction problems, and second, distributed nonmonotonic multi-context systems. In particular, we formulate a translation-based approach to constraint answer set solving which allows for the application of our symmetry detection and symmetry breaking methods. To compare their performance with a-priori symmetry breaking techniques, we also contribute a decomposition of the global value precedence constraint that enforces domain consistency on the original constraint via the unit-propagation of an answer set solver. We evaluate both options in an empirical analysis. In the context of distributed nonmonotonic multi-context system, we develop an algorithm for distributed symmetry detection and also carry over symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201

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