Generalized Cuts and Grothendieck Covers: a Primal-Dual Approximation Framework Extending the Goemans--Williamson Algorithm

TL;DR

This paper introduces a primal-dual approximation framework using semidefinite programming that extends existing algorithms to new convex optimization problems, achieving tight approximation factors for a broad class of combinatorial problems.

Contribution

It develops a unified primal-dual framework based on Grothendieck covers that extends the Goemans--Williamson algorithm to new APX-complete problems with tight approximation guarantees.

Findings

Framework applies to MaxCut, Max2Sat, MaxDicut, and MaxQ.

Achieves reciprocal approximation factors for primal and dual problems.

Provides feasible solutions and certificates for both primal and dual problems.

Abstract

We provide a primal-dual framework for randomized approximation algorithms utilizing semidefinite programming (SDP) relaxations. Our framework pairs a continuum of APX-complete problems including MaxCut, Max2Sat, MaxDicut, and more generally, Max-Boolean Constraint Satisfaction and MaxQ (maximization of a positive semidefinite quadratic form over the hypercube) with new APX-complete problems which are stated as convex optimization problems with exponentially many variables. These new dual counterparts, based on what we call Grothendieck covers, range from fractional cut covering problems (for MaxCut) to tensor sign covering problems (for MaxQ). For each of these problem pairs, our framework transforms the randomized approximation algorithms with the best known approximation factors for the primal problems to randomized approximation algorithms for their dual counterparts with reciprocal approximation factors which are tight with respect to the Unique Games Conjecture. For each APX-complete pair, our algorithms solve a single SDP relaxation and generate feasible solutions for both problems which also provide approximate optimality certificates for each other. Our work utilizes techniques from areas of randomized approximation algorithms, convex optimization, spectral sparsification, as well as Chernoff-type concentration results for random matrices.