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

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


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.


Average differential ratio Optimization constraint satisfaction problems Orthogonal arrays 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CEREGMIAUniversité des AntillesPointe-à-PitreFrance
  2. 2.LIPN (UMR CNRS 7030), Institut GaliléeVilletaneuseFrance

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