Multiplicative assignment with upgrades

TL;DR

The paper studies upgrading up to $k$ suppliers in a bipartite assignment between $I$ and $J$, where upgraded and non-upgraded costs are $b_i$ and $c_i$ with demands $d_j$. It proves that the natural LP relaxation may be fractional, yet there always exists an optimal integral vertex that can be computed in polynomial time, and it develops a strongly polynomial combinatorial algorithm. A key technical contribution is a redistribution lemma that enables iterative refinement of upgrade sets and a convexity property for the upgrade-cost function $h(k')$, leading to both LP-based and purely combinatorial solution methods. The results extend to scheduling with upgrades on identical or uniformly related machines and connect to exact matching problems, while also outlining limitations via counterexamples for non-complete graphs and partition matroids, highlighting practical significance for scheduling and optimization problems with upgrade decisions.

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.

Figures (12)

• Figure 1: Instance of the multiplicative assignment problem with upgrades with $|I|=|J|=3$ and $k=1$, viewed as a perfect matching problem in a bipartite graph with red and blue edges. The right matching corresponds to upgrading supplier $3$ and assigning customer $1$ to supplier $3$, customer $2$ to supplier $1$, and customer $3$ to supplier $2$, resulting in a cost $b_{3}d_{1}+c_{1}d_{2}+c_{2}d_{3}$.
• Figure 2: Instance with an optimal integral vertex (middle) and an optimal fractional vertex (right).
• Figure 3: Illustration for the redistribution. The figure shows only the suppliers in $A\, \Delta \, B$. The dotted and solid rectangles are suppliers in $A$ and $B$, respectively. The green and blue rectangles are suppliers in $A'$ and $B'$ after redistribution.
• Figure 4: Illustration of the fractional vertices without (i).
• Figure 5: Illustration of Contracting Operation. The left part is the graph $M_{\bar{A}}\Delta M_{\bar{B}}$, which consists of two cycles. The purple and green nodes/edges are the set $\bar{A}$ and $\bar{B}$, respectively. The middle part shows the contracting operation, in which we shall remove some edges and contract two nodes. The edges labeled by "$\times$" are edges that will be deleted. The right part is the new graph after the contracting operation. The first cycle still includes more than $4$ edges, so one more contracting operation will be performed.

Theorems & Definitions (38)

• Lemma 2.0
• Theorem 3.1
• Theorem 3.2
• Lemma 3.2
• Definition 3.3
• Lemma 3.4
• proof
• Lemma 3.4
• proof
• Definition 3.5