Multitasking Capacity: Hardness Results and Improved Constructions

TL;DR

This paper investigates the computational hardness of determining the maximal induced matching ratio in bipartite graphs, introduces bipartite half-covers, and constructs graphs with near-optimal multitasking capacity, advancing understanding of these complex problems.

Contribution

It establishes new hardness results for computing multitasking capacity and connected matchings, introduces bipartite half-covers, and constructs graphs with near-optimal induced matching ratios.

Findings

Proves nearly optimal hardness of approximation for connected matching in bipartite graphs.

Introduces bipartite half-covers as a new combinatorial object.

Constructs bipartite graphs with high multitasking capacity close to the theoretical maximum.

Abstract

We consider the problem of determining the maximal $\alpha \in (0,1]$ 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 $\alpha |M|$. This measure was recently introduced in Alon et al. (NIPS 2018) 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 $\alpha$ 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 $n^{1-\epsilon}$ 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 $\beta>1$. Finally, we demonstrate the existence of bipartite graphs with $n$ vertices on each side of average degree $d$, that achieve $\alpha=1/2-\epsilon$ for matchings of size sufficiently smaller than $n/poly(d)$. This nearly matches the trivial upper bound of $1/2$ on $\alpha$ which holds for any graph containing a path of length 3.