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Breaking Symmetries with Involutions

Michael Codish, Mikoláš Janota

Abstract

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.

Breaking Symmetries with Involutions

Abstract

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.

Paper Structure

This paper contains 5 sections, 1 theorem, 4 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Preliminaries and Notation
  2. Breaking Symmetries with Graph Patterns: A Greedy Approach
  3. Tweaking CEGAR for Better Partial and Complete Symmetry Breaks
  4. Symmetry Breaks for a Specific Graph Search Problem
  5. Conclusions and Future Work

Key Result

Proposition 12

For $n\geq 5$, the four top ranking graph patterns always take the following particular form where $m=\binom{n}{2}-2$. Below, we adopt the cycle notation for permutations. The permutation $(j,k)$ is the permutation which swaps elements $j$ and $k$ and leaves all other elements fixed. The first two graph patterns are orthogonal and each covers $2^{m-2}$ graphs. The second two are also orthogonal a

Figures (4)

  • Figure 1: Comparison of CEGAR and Layered CEGAR (log scale)
  • Figure 2: Computation time for partial symmetry breaks with layered CEGAR (log scale)
  • Figure 3: Comparison of CEGAR's Performance on Ramsey graphs (log scale)
  • Figure 4: The impact of various partial symmetry breaks on Ramsey graphs (log scale)

Theorems & Definitions (12)

  • Example 1
  • Example 2
  • Definition 3: cpaior2025
  • Example 4
  • Definition 5: cover
  • Definition 6: dominate
  • Example 7
  • Definition 8: orthogonal
  • Example 9
  • Definition 10
  • ...and 2 more