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

• Published in: Combinatorial Algorithms pp. 374 - 386
• Main Authors: Culus, Jean-François, Toulouse, Sophie
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing 04.07.2018
• Series: Lecture Notes in Computer Science
• Subjects: Average differential ratio; Optimization constraint satisfaction problems; Orthogonal arrays
• ISBN: 9783319946665; 3319946668
• ISSN: 0302-9743; 1611-3349; 1611-3349
• DOI: 10.1007/978-3-319-94667-2_31

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.