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A Study of 0/1 Encodings Prosser & Selensky

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A Study of 0/1 Encodings Prosser & Selensky

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You can encode a constraint satisfaction problem in a number of different, but logically equivalent ways. That is, each encoding explores the same search space, but they take different times. If you encode in a different constraint programming language you again get different relative performance for those encodings This is bad news Paper in a nutshell

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You can encode a constraint satisfaction problem (with n variables) as a problem of finding an independent set of size n That independent set is also maximal. Will the maximality constraint improve performance? Conclusion: maybe, maybe not. It depends on your toolkit! Presentation in a nutshell

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1 2 3 4 5 7 9 8 6 A HypergraphEncode CSP as independent set of size n

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Independent set of a hypergraph G = (V,E) - a set I of vertices such that no edge in E is totally subsumed by I

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1 2 3 4 5 7 9 8 6 1 2 3 4 5 7 9 8 6 An Independent Set You could add vertex 3 or vertex 8! Encode CSP as independent set of size n

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1 2 3 4 5 7 9 8 6 The Largest Independent Set Just so you know. There is only one for this graph

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Represent a CSP with n variables as an independent set as follows each vertex of the graph corresponds to an assignment of a value to a variable if n variables each of domain size m, we have n.m vertices there is an m-clique for each variable therefore a variable can only take one value constraints are explicitly represented as nogoods a nogood is a hyper edge select n vertices of the graph corresponds to instantiation of all n variables

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X + Y + Z = 8 where X, Y and Z are in {2,3} An Example We have n.m 0/1 vertices A hyper edge for each nogood An m-clique for each variables domain Give me an independent set of size n

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X2X2 X3X3 Y2Y2 Y3Y3 Z2Z2 Z3Z3

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1 2 3 4 5 7 9 8 6 Encode CSP as independent set of size n Constraints 3 and 4 can be implemented in two ways sum of variables equals some value (r or k) the number of occurrences of 1 equals some value (r or k) Could this make a difference?

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A Maximal Independent Set An independent set (as before), but we cannot add an element to the set without loss of the independence property. Note: the largest independent set is maximal … obviously

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1 2 3 4 5 7 9 8 6 A Maximal Independent Set There are 11 maximal independent sets of size 6 Remember, there is one largest independent set, size 7

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CP encoding 1 2 3 4 5 7 9 8 6 Encoding Maximality That is, we state when a variable MUST be selected and when it MUST NOT be selected An example, vertex 2

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CP encoding 1 2 3 4 5 7 9 8 6 Example, vertices 1,2, and 3

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More Generally

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Actual Encoding of Maximality Example, vertice 3 We have a sum implement using sumVars or implement using occurs We have the biconditional. This can be implemented in (at least) 3 ways Therefore 6 ways to implement maximality!

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CSP Independent Set of Hypergraph Add redundant maximality constraint Solve How it might go So?

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So, does maximality help? This IS exciting.

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An experiment Encode independence using sumVars (rather than occurs) Encode biconditional using ifOnlyIf Given a hypergraph find an independent set of size k using just independence constraint using redundant maximality constraint Carry out experiments using Choco 1.07 ILOG Solver 5.0 Is run time reduced when we use maximality? Does maximality help?

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Choco 1.07 Conclusion: Use maximality! Run time in milliseconds Note: As largest indSet is size 14 Bs largest indSet is size 15 both encodings look for same thing About 3 times faster A & B are regular degree hypergraphs

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Solver 5.0 Conclusion: Avoid maximality! About 8 times slower

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The paper has other confounding results such as summation is faster than occurrence in Solver occurrence is faster than summation in Choco ((p & q) or (¬p & ¬q)) is fastest implementation of in Solver ifOnlyIf is the fastest implementation of in Choco maximality helps in Choco, and does not help in Solver Confused?

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So? Why the differences, between Solver and Choco? Read the paper What does this mean? What lesson can I learn? Be paranoid

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… and maximality is interesting!

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Any questions?

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