Symmetry within Solutions

arXiv:1004.2624v1 [cs.AI] 15 Apr 2010 Symmetry within Solutions Marijn Heule TU Delft The Netherlands marijn@heule.nl Toby Walsh NICTA and UNSW Sydney, Australia toby.walsh@nicta.com.au Abstract We define the concept of an internal symmetry. This is a symmety within a solution of a constraint satisfaction prob- lem. We compare this to solution symmetry, which is a map- ping between different solutions of the same problem. We argue that we may be able to exploit both types of symmetry when finding solutions. We illustrate the potential of exploit- ing internal symmetries on two benchmark domains: Van der Waerden numbers and graceful graphs. By identifying inter- nal symmetries we are able to extend the state of the art in both cases. Introduction Symmetry is an important feature of many combinato- rial search problems. To be able to solve such prob- lems, we often need to take account of symmetry. For ex- ample, when finding magic squares (prob019 in CSPLib (Gent and Walsh 1999)), we have the symmetries that rotate and reflect the square. Factoring such symmetry out of the search space is often critical when trying to solve large in- stances of a problem. Up till now, research on symmetry has mostly focused on symmetries between different solutions of the same problem. In this paper, we propose considering in addition the internal symmetries (that is, symmetries within each solution). Whilst it appears to be challenging to iden- tify useful internal symmetries, such symmetries are easy to exploit. We simply add constraints that restrict search to those solutions with the required internal symmetry and limit branching to the subset of decisions that generate a complete solution. We will demonstrate the value of exploit- ing internal symmetries within solutions with experimental results on two benchmark domains: Van der Waerden num- bers and graceful graphs. Symmetry between solutions A symmetry σ is a bijection on assignments. Given a set of assignments A and a symmetry σ, we write σ(A) for {σ(a) | a ∈A}. Similarly, given a set of symmetries Σ, we write Σ(A) for {σ(a) | a ∈A, σ ∈Σ}. A special type of symmetry, called solution symmetry is a symmetry be- tween the solutions of a problem. Such a symmetry maps Copyright c⃝2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. solutions onto solutions. A solution is simply a set of as- signments that satisfy every constraint in the problem. More formally, we say that a problem has the solution symmetry σ iff σ of any solution is itself a solution (Cohen et al. 2006). As such mappings are associativity, and the inverse of a so- lution symmetry and the identity mapping are solution sym- metries, the set of solution symmetries Σ of a problem forms a group under composition. We say that two sets of assign- ments A and B are in the same symmetry class of Σ iff there exists σ ∈Σ such that σ(A) = B. Running example. The magic squares problem is to label a n by n square so that the sum of every row, column and diag- onal are equal (prob019 in CSPLib (Gent and Walsh 1999)). A normal magic square contains the integers 1 to n2. We model this with n2 variables Xi,j where Xi,j = k iff the ith column and jth row is labelled with the integer k. “Lo Shu”, the smallest non-trivial normal magic square has been known for over four thousand years and is an im- portant object in ancient Chinese mathematics: 4 9 2 3 5 7 8 1 6 (1) The magic squares problem has a number of solution sym- metries. For example, consider the symmetry σd that reflects a solution in the leading diagonal. This map “Lo Shu” onto a symmetric solution: 6 7 2 1 5 9 8 3 4 (2) Any other rotation or reflection of the square maps one so- lution onto another. The 8 symmetries of the square are thus all solution symmetries of this problem. In fact, there are only 8 different magic square of order 3, and all are in the same symmetry class. One way to factor solution symmetry out of the search space is to post symmetry break- ing constraints. See, for instance, (Puget 1993; Crawford et al. 1996; Flener et al. 2002; Frisch et al. 2002; Walsh 2006a; Walsh 2006b; Law et al. 2007; Walsh 2007). For example, we can eliminate σd by posting a constraint which ensures that the top left corner is smaller than its symmetry, the bottom right corner. This selects (1) and eliminates (2). Symmetry within a solution Symmetries can also be found within individual solutions of a constraint satisfaction problem. We say that a solution A contains the internal symmetry σ (or equivalently σ is a internal symmetry within this solution) iff σ(A) = A. Running example. Consider again “Lo Shu”. This con- tains an internal symmetry. To see this, consider the solution symmetry σinv that inverts labels, mapping k onto n2+1−k. This solution symmetry maps “Lo Shu” onto a different (but symmetric) solution. However, if we now apply the solution symmetry σ180 that rotates the square 180◦, we map back onto the origi

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