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

2025-12-08

AI Summary

This paper studies two natural extensions of the Maximum Quadratic Assignment Problem (MaxQAP): (1) the Maximum List-Restricted MaxQAP, where each node on one side can only be assigned to a predefined candidate list; and (2) the Maximum Quadratic b-Matching Assignment Problem, requiring the solution to be a b-matching on a graph. We design the first LP-based approximation algorithms with randomized rounding for both problems, leveraging a maximum-weight b-matching solver in b independent sampling rounds. When list sizes are at least $ n - O(sqrt{n}) $ and $ b $ is constant, our algorithms achieve $ O(sqrt{n}) $ and $ O(sqrt{bn}) $ approximation ratios, respectively—matching the best-known asymptotic lower bounds. Our key contribution is the development of the first unified approximation framework applicable to both list-restricted and degree-constrained variants of MaxQAP, thereby overcoming long-standing algorithmic barriers in handling these combinatorial constraints.

Technology Category

• Constraint Satisfaction and Optimization: Applications
• Machine Learning: Quantum Machine Learning
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Security and privacy of machine learning and AI applications

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.

Problem

• Develops approximation algorithm for list-restricted MaxQAP with large lists
• Provides approximation algorithm for b-matching variant of MaxQAP
• Matches best known MaxQAP approximation for constant b and large lists

Innovation

• Randomized rounding for list-restricted MaxQAP
• Refined relaxation for b-matching MaxQAP
• Independent iterations with maximum-weight b-matching