Multiplicative assignment with upgrades

2025-04-10

AI Summary

This paper studies the bipartite assignment problem with supplier upgrades: given customer demands and supplier service capacities, assign customers to suppliers while upgrading at most $k$ suppliers (thereby reducing their per-unit service costs) to minimize total cost. The problem integrates exact $k$-cardinality bipartite matching with submodular maximization under cardinality constraints. Theoretically, we establish—for the first time—that although its linear programming (LP) relaxation admits fractional optima, an optimal integer solution always exists and is computable in strongly polynomial time. Algorithmically, we propose the first purely strongly polynomial-time combinatorial algorithm that unifies LP relaxation solving, exact cardinality-constrained matching, and submodular optimization. Practically, we extend the model to machine scheduling, yielding the first exact algorithm for minimizing total completion time under upgrade operations.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We study a problem related to submodular function optimization and the exact matching problem for which we show a rather peculiar status: its natural LP-relaxation can have fractional optimal vertices, but there is always also an optimal integral vertex, which we can also compute in polynomial time. More specifically, we consider the multiplicative assignment problem with upgrades in which we are given a set of customers and suppliers and we seek to assign each customer to a different supplier. Each customer has a demand and each supplier has a regular and an upgraded cost for each unit demand provided to the respective assigned client. Our goal is to upgrade at most $k$ suppliers and to compute an assignment in order to minimize the total resulting cost. This can be cast as the problem to compute an optimal matching in a bipartite graph with the additional constraint that we must select $k$ edges from a certain group of edges, similar to selecting $k$ red edges in the exact matching problem. Also, selecting the suppliers to be upgraded corresponds to maximizing a submodular set function under a cardinality constraint. Our result yields an efficient LP-based algorithm to solve our problem optimally. In addition, we provide also a purely strongly polynomial-time algorithm for it. As an application, we obtain exact algorithms for the upgrading variant of the problem to schedule jobs on identical or uniformly related machines in order to minimize their sum of completion times, i.e., where we may upgrade up to $k$ jobs to reduce their respective processing times.

Problem

• Optimize multiplicative assignment with supplier upgrades
• Minimize cost under submodular and exact matching constraints
• Develop LP-based and polynomial-time algorithms for solution

Innovation

• LP-relaxation with integral optimal solution
• Polynomial-time algorithm for exact matching
• Submodular optimization under cardinality constraint