The Simultaneous Assignment Problem

This paper introduces the \emph{Simultaneous Assignment Problem}. Here, we are given an assignment problem on some of the subgraphs of a given graph, and we are looking for a heaviest assignment which is feasible when restricted to any of the assignment problems. More precisely, we are given a graph with a weight- and a capacity function on its edges and a set of its subgraphs $H_1,\dots,H_k$ along with a degree upper bound function for each of them. In addition, we are also given a laminar system on the node set with an upper bound on the degree-sum of the nodes in each set in the system. We want 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 degree upper bound associated with the subgraph, and the degree-sum bound is respected in each set of the laminar system. The problem is shown to be APX-hard in the unweighted case even if the graph is a forest and $k=2$. This also implies that the Distance matching problem is APX-hard in the weighted case and that the Cyclic distance matching problem is APX-hard in the unweighted case. We identify multiple special cases when the problem can be solved in strongly polynomial time. One of these cases, the so-called locally laminar case, is a common generalization of the Hierarchical b-matching problem and the Laminar matchoid problem, and it implies that both of these problems can be solved efficiently in the weighted, capacitated case -- improving upon the most general polynomial-time algorithms for these problems. The problem can be constant approximated when $k$ is a constant, and we show that the approximation factor matches the integrality gap of a strengthened LP-relaxation for small $k$. We give improved approximation algorithms for special cases, for example, when the degree bounds are uniform or the graph is sparse.