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

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 k, q 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.