Lazy clause generation (LCG) is an approach to building constraint programming (CP) solvers that uses ideas from Boolean Satisfiability (SAT) solvers. Propagation in the CP solver is extended to explain itself, lazily creating explanation clauses. This allows SAT style conflict analysis to be used to create clausal nogoods that record the reason for failure, and avoid the same failure in the future. LCG has proven to be a highly effective architecture, used in all most performant CP solvers. In this review we consider how LCG has been refined and expanded over the years since its invention.

This paper introduces a SAT-based technique that calculates a compact and complete symmetry-break for finite model finding, with the focus on structures with a single binary operation (magmas). Classes of algebraic structures are typically described as first-order logic formulas and the concrete algebras are models of these formulas. Such models include an enormous number of isomorphic, i.e. symmetric, algebras. A complete symmetry-break is a formula that has as models, exactly one canonical representative from each equivalence class of algebras. Thus, we enable answering questions about properties of the models so that computation and search are restricted to the set of canonical representations. For instance, we can answer the question: How many non-isomorphic semigroups are there of size n? Such questions can be answered by counting the satisfying assignments of a SAT formula, which already filters out non-isomorphic models. The introduced technique enables us calculating numbers of algebraic structures not present in the literature and going beyond the possibilities of pure enumeration approaches.

Symmetry breaking for graphs and other combinatorial objects is notoriously hard. On the one hand, complete symmetry breaks are exponential in size. On the other hand, current, state-of-the-art, partial symmetry breaks are often considered too weak to be of practical use. Recently, the concept of graph patterns has been introduced and provides a concise representation for (large) sets of non-canonical graphs, i.e. graphs that are not lex-leaders and can be excluded from search. In particular, four (specific) graph patterns apply to identify about 3/4 of the set of all non-canonical graphs. Taking this approach further, we discover that graph patterns that derive from permutations that are involutions play an important role in the construction of symmetry breaks for graphs. We take advantage of this to guide the construction of partial and complete symmetry-breaking constraints based on graph patterns. The resulting constraints are small in size and strong in the number of symmetries they break.

We formalize symmetry breaking as a set-covering problem. For the case of breaking symmetries on graphs, a permutation covers a graph if applying it to the graph yields a smaller graph in a given order. Canonical graphs are those that cannot be made smaller by any permutation. A complete symmetry break is then a set of permutations that covers all non-canonical graphs. A complete symmetry break with a minimal number of permutations can be obtained by solving an optimal set-covering problem. The challenge is in the sizes of the corresponding set-covering problems and in how these can be tamed. The set-covering perspective on symmetry breaking opens up a range of new opportunities deriving from decades of studies on both precise and approximate techniques for this problem. Application of our approach leads to optimal LexLeader symmetry breaks for graphs of order n≤ 10 as well as to partial symmetry breaks which improve on the state-of-the-art.

This paper proposes SAT-based techniques to calculate a specific normal form of a given finite mathematical structure (model). The normal form is obtained by permuting the domain elements so that the representation of the structure is lexicographically smallest possible. Such a normal form is of interest to mathematicians as it enables easy cataloging of algebraic structures. In particular, two structures are isomorphic precisely when their normal forms are the same. This form is also natural to inspect as mathematicians have been using it routinely for many decades. We develop a novel approach where a SAT solver is used in a black-box fashion to compute the smallest representative. The approach constructs the representative gradually and searches the space of possible isomorphisms, requiring a small number of variables. However, the approach may lead to a large number of SAT calls and therefore we devise propagation techniques to reduce this number. The paper focuses on finite structures with a single binary operation (encompassing groups, semigroups, etc.). However, the approach is generalizable to arbitrary finite structures. We provide an implementation of the proposed algorithm and evaluate it on a variety of algebraic structures.

This paper illustrates the application of graph invariants to break symmetries for graph search problems. The paper makes two contributions: (1) the use of higher dimensional graph invariants in symmetry breaking constraints; and (2) a novel technique to obtain symmetry breaking constraints by combining graph invariants. Experimentation demonstrates that the proposed approach applies to provide new results for the generation of a particular class of cubic graphs.

This paper introduces, for the first time, a complete symmetry breaking constraint of polynomial size for a significant class of graphs: the class of uniquely Hamiltonian graphs. We introduce a canonical form for uniquely Hamiltonian graphs and prove that testing whether a given uniquely Hamiltonian graph is canonical can be performed efficiently. Based on this canonicity test, we construct a complete symmetry breaking constraint of polynomial size which is satisfied only by uniquely Hamiltonian graphs which are canonical. We apply the proposed symmetry breaking constraint to show new results regarding the class of uniquely Hamiltonian graphs. We also show that the proposed approach applies almost directly for the class of graphs which contain any cycle of known length where it shown to result in a partial symmetry breaking constraint. Given that it is unknown if there exist complete symmetry breaking constraints for graphs of polynomial size, this paper makes a first step in the direction of identifying specific classes of graphs for which such constraints do exist.

This paper introduces incremental symmetry breaking constraints for graph search problems which are complete and compact. We show that these constraints can be computed incrementally: A symmetry breaking constraint for order n graphs can be extended to one for order n + 1 graphs. Moreover, these constraints induce a special property on their canonical solutions: An order n canonical graph contains a canonical subgraph on the first k vertices for every 1 ≤ k ≤ n. This facilitates a “generate and extend” paradigm for parallel graph search problem solving: To solve a graph search problem φ on order n graphs, first generate the canonical graphs of some order k < n. Then, compute canonical solutions for φ by extending, in parallel, each canonical order k graph together with suitable symmetry breaking constraints. The contribution is that the proposed symmetry breaking constraints enable us to extend the order k canonical graphs to order n canonical solutions. We demonstrate our approach through its application on two hard graph search problems.

New properties of the front and back ends of sorting networks are studied, illustrating their utility when searching for bounds on optimal networks. Search focuses first on the “out-sides” of the network and then on the inner part. Previous works focused on properties of the front end to break symmetries in the search. The new, out-side-in, properties shed understanding on how sorting networks sort, and facilitate the computation of new bounds on optimality. We present new, faster, parallel sorting networks for 17–20 inputs. For 17 inputs, we show that no sorting network using less layers exists.

Many complex combinatorial problems arising from a range of scientific applications (such as computer networks, mathematical chemistry and bioinformatics) involve searching for an undirected graph satisfying a given property. Since for any possible solution there can be a large number of isomorphic representations, these problems can quickly become intractable. One way to mitigate this problem is to eliminate as many isomorphic copies as possible by breaking symmetry during search - i.e. by introducing constraints that ensure that at least one representative graph is generated for each equivalence class, but not the entire class. The goal is to generate as few members of each class as possible - ideally exactly one: the symmetry break is said to be complete in this case. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory. We also illustrate the application of our approach to graph edge coloring problems which exhibit additional symmetries due to the fact that the colors of the edges in any solution can be permuted.

Breaking symmetries is crucial when solving hard combinatorial problems. A common way to eliminate symmetries in CP/SAT is to add symmetry breaking constraints. Ideally, symmetry breaking constraints should be complete and compact. The aim of this paper is to find compact and complete symmetry breaks applicable when solving hard combinatorial problems using CP/SAT approach. In particular: graph search problems and matrix model problems where symmetry breaks are often specified in terms of lex constraints. We show that sets of lex constraints can be expressed with only a small portion of their inner lex implications which are a particular form of Horn clauses. We exploit this fact and compute a compact encoding of the row-wise LexLeader and state of the art partial symmetry breaking constraints. We illustrate the approach for graph search problems and matrix model problems.

This paper introduces a hybrid search method for optimization problems which combines techniques from Local Search methods and from SAT-based methods. At each iteration, the method performs a "big-step" move on a subset of variables of the current solution. This step is achieved by encoding the big-step itself as an optimization problem and solving it using a SAT (MaxSAT) solver such that the solution of the big-step results in a higher-quality solution to the entire problem. Experimentation illustrates a clear benefit of the approach over both methods: Local Search methods and SAT-based methods.

This note describes a SAT encoding for the $n$-fractions puzzle which is problem 041 of the CSPLib. Using a SAT solver we obtain a solution for two of the six remaining open instances of this problem.

This paper presents pl-cliquer, a Prolog interface to the cliquer tool for the maximum clique problem. Using pl-cliquer facilitates a programming style that allows logic programs to integrate with other tools such as: Boolean satisfiability solvers, finite domain constraint solvers, and graph isomorphism tools. We illustrate this programming style to solve the Graph Coloring problem, applying a symmetry break that derives from finding a maximum clique in the input graph. We present an experimentation of the resulting Graph Coloring solver on two benchmarks, one from the graph coloring community and the other from the examination timetabling community. The implementation of pl-cliquer consists of two components: A lightweight C interface, connecting cliquer’s C library and Prolog, and a Prolog module which loads the library. The complete tool is available as a SWI-Prolog module. 1998 ACM Subject Classification D.1.6 Logic Programming

We prove properties of extremal graphs of girth 5 and order 20 <=v <= 32. In each case we identify the possible minimum and maximum degrees, and in some cases prove the existence of (non-trivial) embedded stars. These proofs allow for tractable search for and identification of all non isomorphic cases.

In this paper, we show how the theory of sorting networks can be applied to synthesize optimized general-purpose sorting libraries. Standard sorting libraries are often based on combinations of the classic Quicksort algorithm, with insertion sort applied as base case for small, fixed, numbers of inputs. Unrolling the code for the base case by ignoring loop conditions eliminates branching, resulting in code equivalent to a sorting network. By replacing it with faster sorting networks, we can improve the performance of these algorithms. We show that by considering the number of comparisons and swaps alone we are not able to predict any real advantage of this approach. However, significant speed-ups are obtained when taking advantage of instruction level parallelism and non-branching conditional assignment instructions, both of which are common in modern CPU architectures. Furthermore, a close control of how often registers have to be spilled to memory gives us a complete explanation of the performance of different sorting networks, allowing us to choose an optimal one for each particular architecture. Our experimental results show that using code synthesized from these efficient sorting networks as the base case for Quicksort libraries results in significant real-world speed-ups.

We solve a 40-year-old open problem on the depth optimality of sorting networks. In 1973, Donald E. Knuth detailed, in Volume 3 of "The Art of Computer Programming", sorting networks of the smallest depth known at the time for n =< 16 inputs, quoting optimality for n =< 8. In 1989, Parberry proved the optimality of the networks with 9 =< n =< 10 inputs. In this article, we present a general technique for obtaining such optimality results, and use it to prove the optimality of the remaining open cases of 11 =< n =< 16 inputs. We show how to exploit symmetry to construct a small set of two-layer networks on n inputs such that if there is a sorting network on n inputs of a given depth, then there is one whose first layers are in this set. For each network in the resulting set, we construct a propositional formula whose satisfiability is necessary for the existence of a sorting network of a given depth. Using an off-the-shelf SAT solver we show that the sorting networks listed by Knuth are optimal. For n =< 10 inputs, our algorithm is orders of magnitude faster than the prior ones.

A fundamental problem in coding theory concerns the computation of the maximum cardinality of a set S of length n code words over an alphabet of size q, such that every pair of code words has Hamming distance at least d, and the set of additional constraints U on S is satisfied. This problem has application in several areas, one of which is the design of DNA codes where q=4 and the alphabet is {A,C,G,T}. We describe a new constraint model for this problem and demonstrate that it improves on previous solutions (computes better lower bounds) for various instances of the problem. Our approach is based on a clustering of DNA words into small sets of words. Solutions are then obtained as the union of such clusters. Our approach is SAT based: we specify constraints on clusters of DNA words and solve these using a Boolean satisfiability solver.

© Springer International Publishing Switzerland 2016.Symmetry breaking is an essential component when solving graph search problems as it restricts the search space to that of canonical representations. There are an abundance of powerful tools, such as nauty, which apply to find the canonical representation of a given graph and to test for isomorphisms given a set of graphs. In contrast, for graph search problems, current symmetry breaking techniques are partial and solvers unnecessarily explore an abundance of isomorphic parts of the search space. This paper is novel in that it introduces complete symmetry breaking for graph search problems by modeling, in terms of constraints, the same ideas underlying the algorithm applied in tools like nauty. Whereas nauty tests given graphs, symmetry breaks restrict the search space and apply during generation.

Symmetry breaking is an essential component when solving graph search problems as it restricts the search space to that of canonical representations. There are an abundance of powerful tools, such as nauty, which apply to find the canonical representation of a given graph and to test for isomorphisms given a set of graphs. In contrast, for graph search problems, current symmetry breaking techniques are partial and solvers unnecessarily explore an abundance of isomorphic parts of the search space. This paper is novel in that it introduces complete symmetry breaking for graph search problems by modeling, in terms of constraints, the same ideas underlying the algorithm applied in tools like nauty. Whereas nauty tests given graphs, symmetry breaks restrict the search space and apply during generation.