Approximation Algorithms for the $b$-Matching and List-Restricted Variants of MaxQAP

TL;DR

The paper studies approximation algorithms for two natural generalizations of MaxQAP: list-restricted MaxQAP and MaxQbAP (b-matching). It extends the LP-relaxation and randomized rounding framework of Makarychev et al. to handle list constraints and multiple-matchings, achieving an $O(\sqrt{n})$-approximation when lists are size $n- O(\sqrt{n})$ and an $O(\sqrt{bn})$-approximation for MaxQbAP, with constant $b$ matching the best-known ratio for standard MaxQAP. A key technical idea is to decompose the LP objective into heavy and light parts and to construct auxiliary matchings (M'_{rand}, M'_{star}) to guarantee a constant fraction of the LP optimum in expectation. The results also connect to dup-MaxQbAP, showing that a $\alpha$-approximation for the dup version yields a $2\alpha$-approximation for the original MaxQbAP, broadening the applicability of the approach and setting a foundation for further improvements. Overall, the work initiates a principled approximation-theoretic study of these variants and provides near-optimal guarantees under natural constraints.

Abstract

We study approximation algorithms for two natural generalizations of the Maximum Quadratic Assignment Problem (MaxQAP). In the Maximum List-Restricted Quadratic Assignment Problem, each node in one partite set may only be matched to nodes from a prescribed list. For instances on $n$ nodes where every list has size at least $n - O(\sqrt{n})$, we design a randomized $O(\sqrt{n})$-approximation algorithm based on the linear-programming relaxation and randomized rounding framework of Makarychev, Manokaran, and Sviridenko. In the Maximum Quadratic $b$-Matching Assignment Problem, we seek a $b$-matching that maximizes the MaxQAP objective. We refine the standard MaxQAP relaxation and combine randomized rounding over $b$ independent iterations with a polynomial-time algorithm for maximum-weight $b$-matching problem to obtain an $O(\sqrt{bn})$-approximation. When $b$ is constant and all lists have size $n - O(\sqrt{n})$, our guarantees asymptotically match the best known approximation factor for MaxQAP, yielding the first approximation algorithms for these two variants.

Theorems & Definitions (24)

• lemma 1
• lemma 2
• theorem 1
• proof
• lemma 3
• lemma 4
• theorem 2
• proof
• lemma F1
• proof