Michael Codish

Michael Codish, Mikoláš Janota

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 $n\leq 10$ as well as to partial symmetry breaks which improve on the state-of-the-art.

Michael Codish, Mikoláš Janota

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.

Avraham Itzhakov, Michael Codish

There are many complex combinatorial problems which involve searching for an undirected graph satisfying given constraints. Such problems are often highly challenging because of the large number of isomorphic representations of their solutions. This paper introduces effective and compact, complete symmetry breaking constraints for small graph search. Enumerating with these symmetry breaks generates all and only non-isomorphic solutions. For small search problems, with up to $10$ vertices, we compute instance independent symmetry breaking constraints. For small search problems with a larger number of vertices we demonstrate the computation of instance dependent constraints which are complete. We illustrate the application of complete symmetry breaking constraints to extend two known sequences from the OEIS related to graph enumeration. We also demonstrate the application of a generalization of our approach to fully-interchangeable matrix search problems.

Michael Codish, Michael Frank, Avraham Itzhakov et al.

The number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found "soon". Yet, its precise value has remained unknown for almost 50 years. This paper presents a methodology based on \emph{abstraction} and \emph{symmetry breaking} that applies to solve hard graph edge-coloring problems. The utility of this methodology is demonstrated by using it to compute the value $R(4,3,3)=30$. Along the way it is required to first compute the previously unknown set ${\cal R}(3,3,3;13)$ consisting of 78{,}892 Ramsey colorings.

Michael Codish, Michael Frank, Avraham Itzhakov et al.

This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number $R(4,3,3)=30$. The number $R(4,3,3)$ is often presented as the unknown Ramsey number with the best chances of being found "soon". Yet, its precise value has remained unknown for more than 50 years. We illustrate our approach by showing that: (1) there are precisely 78{,}892 $(3,3,3;13)$ Ramsey colorings; and (2) if there exists a $(4,3,3;30)$ Ramsey coloring then it is (13,8,8) regular. Specifically each node has 13 edges in the first color, 8 in the second, and 8 in the third. We conjecture that these two results will help provide a proof that no $(4,3,3;30)$ Ramsey coloring exists implying that $R(4,3,3)=30$.

Amit Metodi, Michael Codish, Peter James Stuckey

We present an approach to propagation-based SAT encoding of combinatorial problems, Boolean equi-propagation, where constraints are modeled as Boolean functions which propagate information about equalities between Boolean literals. This information is then applied to simplify the CNF encoding of the constraints. A key factor is that considering only a small fragment of a constraint model at one time enables us to apply stronger, and even complete, reasoning to detect equivalent literals in that fragment. Once detected, equivalences apply to simplify the entire constraint model and facilitate further reasoning on other fragments. Equi-propagation in combination with partial evaluation and constraint simplification provide the foundation for a powerful approach to SAT-based finite domain constraint solving. We introduce a tool called BEE (Ben-Gurion Equi-propagation Encoder) based on these ideas and demonstrate for a variety of benchmarks that our approach leads to a considerable reduction in the size of CNF encodings and subsequent speed-ups in SAT solving times.