How Far From a Worst Solution a Random Solution of a $$k\,$$ CSP Instance Can Be? - Culus, Jean-François; Toulouse, Sophie (Combinatorial Algorithms, 2018)
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 $$ρ _E(I) =(E_X[v(I, X)] -wor(I))/(opt(I) -wor(I))$$ where $$opt(I)$$ , $$wor(I)$$ and $$E_X[v(I, X)]$$ 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 $$q =2$$ , $$\varOmega (1/n^{k -1 -\lfloor \log _{p^κ } (k -1)\rfloor })$$ where $$p^κ $$ is the smallest prime power such that $$p^κ \ge q$$ otherwise, and $$\varOmega (1/q^k)$$ in $$(\max \{q, k\} +1\})$$ -partite instances.