IWOCA 2018: Combinatorial Algorithms pp 374-386

# How Far From a Worst Solution a Random Solution of a $$k\,$$CSP Instance Can Be?

• Jean-François Culus
• Sophie Toulouse
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)

## Abstract

Given an instance I of an optimization constraint satisfaction problem (CSP), finding solutions with value at least the expected value of a random solution is easy. We wonder how good such solutions can be. Namely, we initiate the study of ratio $$\rho _E(I) =(\mathrm {E}_X[v(I, X)] -\mathrm {wor}(I))/(\mathrm {opt}(I) -\mathrm {wor}(I))$$ where $$\mathrm {opt}(I)$$, $$\mathrm {wor}(I)$$ and $$\mathrm {E}_X[v(I, X)]$$ refer to respectively the optimal, the worst, and the average solution values on I. We here focus on the case when the variables have a domain of size $$q \ge 2$$ and the constraint arity is at most $$k \ge 2$$, where kq are two constant integers. Connecting this ratio to the highest frequency in orthogonal arrays with specified parameters, we prove that it is $$\varOmega (1/n^{k/2})$$ if $$q =2$$, $$\varOmega (1/n^{k -1 -\lfloor \log _{p^\kappa } (k -1)\rfloor })$$ where $$p^\kappa$$ is the smallest prime power such that $$p^\kappa \ge q$$ otherwise, and $$\varOmega (1/q^k)$$ in $$(\max \{q, k\} +1\})$$-partite instances.

## Keywords

Average differential ratio Optimization constraint satisfaction problems Orthogonal arrays

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