Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP

TL;DR

This paper develops optimal constant-time approximation algorithms for bounded-degree CSPs and establishes unconditional inapproximability results, extending the understanding of CSP complexity in the bounded-degree model.

Contribution

It introduces a universal approach to constant-time approximation and hardness for bounded-degree CSPs, unconditionally matching integrality gaps with approximation ratios.

Findings

Constructed an oracle for nearly optimal LP solutions in constant time.

Provided a constant-time rounding scheme matching the LP integrality gap.

Established a generic conversion from LP gaps to hardness results.

Abstract

Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it. In this paper, we show that similar results hold for constant-time approximation algorithms in the bounded-degree model. Specifically, we present the followings: (i) For every CSP, we construct an oracle that serves an access, in constant time, to a nearly optimal solution to a basic LP relaxation of the CSP. (ii) Using the oracle, we give a constant-time rounding scheme that achieves an approximation ratio coincident with the integrality gap of the basic LP. (iii) Finally, we give a generic conversion from integrality gaps of basic LPs to hardness results. All of those results are \textit{unconditional}. Therefore, for every bounded-degree CSP, we give the best constant-time approximation algorithm among all. A CSP instance is called $\epsilon$-far from satisfiability if we must remove at least an $\epsilon$-fraction of constraints to make it satisfiable. A CSP is called testable if there is a constant-time algorithm that distinguishes satisfiable instances from $\epsilon$-far instances with probability at least $2/3$. Using the results above, we also derive, under a technical assumption, an equivalent condition under which a CSP is testable in the bounded-degree model.