The Simultaneous Assignment Problem

TL;DR

This paper studies the complex simultaneous assignment problem involving weighted graphs, capacity constraints, and laminar system bounds, providing polynomial solutions for special cases, proving APX-hardness in general, and developing approximation algorithms.

Contribution

It introduces the simultaneous assignment problem, identifies solvable special cases, proves hardness results, and offers a framework for approximation algorithms applicable to related problems.

Findings

Polynomial-time solutions for special cases including trees.

APX-hardness of the general unweighted problem.

Constant-factor approximation algorithms for a fixed number of subgraphs.

Abstract

This paper introduces the Simultaneous assignment problem. Let us given a graph with a weight and a capacity function on its edges, and a set of its subgraphs along with a degree upper bound function for each of them. We are also given a laminar system on the node set with an upper bound on the degree-sum in each member of the system. Our goal is to assign each edge a non-negative integer below its capacity such that the total weight is maximized, the degrees in each subgraph are below the associated degree upper bound, and the degree-sum bound is respected in each member of the laminar system. We identify special cases when the problem is solvable in polynomial time. One of these cases is a common generalization of the hierarchical $b$-matching problem and the laminar matchoid problem. This implies that both problems can be solved efficiently in the weighted, capacitated case even if both lower and upper bounds are present -- generalizing the previous polynomial-time algorithms. The problem is solvable for trees provided that the laminar system is empty and a natural assumption holds for the subgraphs. The general problem, however, is shown to be APX-hard in the unweighted case, which implies that the $d$-distance matching problem is APX-hard. Furthermore, we prove that the approximation guarantee of any polynomial-time algorithm must increase linearly in the number of the given subgraphs unless P=NP. We give a generic framework for deriving approximation algorithms, which can be applied to a wide range of problems. As an application to our problem, a constant-approximation algorithm is derived when the number of the subgraphs is a constant. The approximation guarantee is the same as the integrality gap of a strengthened LP-relaxation when the number of the subgraphs is small. Improved approximation algorithms are given when the degree bounds are uniform or the graph is sparse.