Local Search is Better than Random Assignment for Bounded Occurrence Ordering k-CSPs

TL;DR

This paper demonstrates that a simple local search algorithm outperforms random assignment for bounded occurrence ordering k-CSPs, showing these problems are not approximation resistant and providing concrete performance guarantees.

Contribution

The paper introduces a straightforward local search method that guarantees better solutions than random assignment for bounded occurrence ordering k-CSPs, addressing an open question in approximation resistance.

Findings

Local search outperforms random assignment in bounded occurrence ordering k-CSPs.

Bounded occurrence ordering k-CSPs are not approximation resistant.

The algorithm's expected value exceeds the average by a quantifiable amount.

Abstract

We prove that the Bounded Occurrence Ordering k-CSP Problem is not approximation resistant. We give a very simple local search algorithm that always performs better than the random assignment algorithm. Specifically, the expected value of the solution returned by the algorithm is at least Alg > Avg + a(B,k) (Opt - Avg), where "Opt" is the value of the optimal solution; "Avg" is the expected value of the random solution; and a(B,k)=Omega_k(B^{-(k+O(1))} is a parameter depending only on "k" (the arity of the CSP) and "B" (the maximum number of times each variable is used in constraints). The question whether bounded occurrence ordering k-CSPs are approximation resistant was raised by Guruswami and Zhou (APPROX 2012) who recently showed that bounded occurrence 3-CSPs and "monotone" k-CSPs admit a non-trivial approximation.