Multitasking Capacity: Hardness Results and Improved Constructions

Abstract

We consider the problem of determining the maximal α ∈ (0, 1] such that every matching M of size k (or at most k) in a bipartite graph G contains an induced matching of size at least α|M |. This measure was recently introduced in [ARS + 17] and is motivated by connectionist models of cognition as well as modeling interference in wireless and communication networks.We prove various hardness results for computing α either exactly or approximately. En route to our results, we also consider the maximum connected matching problem: determining the largest matching N in a graph G such that every two edges in N are connected by an edge. We prove a nearly optimal n 1−ε hardness of approximation result (under randomized reductions) for connected matching in bipartite graphs (with both sides of cardinality n). Towards this end we define bipartite half-covers: A new combinatorial object that may be of independent interest. To the best of our knowledge, the best previous hardness result for the connected matching problem was some constant β > 1.Finally, we demonstrate the existence of bipartite graphs with n vertices on each side of average degree d, that achieve α = 1/2 − ε for matchings of size sufficiently smaller than n/poly(d). This nearly matches the trivial upper bound of 1/2 on α which holds for any graph containing a path of length 3.