An Investigation of the Recoverable Robust Assignment Problem

TL;DR

This paper explores the computational complexity of the recoverable robust assignment problem, revealing its hardness, special solvable cases, and connections to open problems in parallel algorithms.

Contribution

It establishes hardness results, provides a polynomial-time solution for specific cost structures, and analyzes a variant related to parallel complexity classes.

Findings

The problem is W[1]-hard with respect to parameters k and n-k.

A polynomial-time algorithm exists when one cost function is Monge and the other is Anti-Monge.

The second-stage matching problem is in RNC_2 and as hard as the open problem of Exact Matching in Red-Blue Bipartite Graphs.

Abstract

We investigate the so-called recoverable robust assignment problem on balanced bipartite graphs with $2n$ vertices, a mainstream problem in robust optimization: For two given linear cost functions $c_1$ and $c_2$ on the edges and a given integer $k$, the goal is to find two perfect matchings $M_1$ and $M_2$ that minimize the objective value $c_1(M_1)+c_2(M_2)$, subject to the constraint that $M_1$ and $M_2$ have at least $k$ edges in common. We derive a variety of results on this problem. First, we show that the problem is W[1]-hard with respect to the parameter $k$, and also with respect to the recoverability parameter $k'=n-k$. This hardness result holds even in the highly restricted special case where both cost functions $c_1$ and $c_2$ only take the values $0$ and $1$. (On the other hand, containment of the problem in XP is straightforward to see.) Next, as a positive result we construct a polynomial time algorithm for the special case where one cost function is Monge, whereas the other one is Anti-Monge. Finally, we study the variant where matching $M_1$ is frozen, and where the optimization goal is to compute the best corresponding matching $M_2$, the second stage recoverable assignment problem. We show that this problem variant is contained in the randomized parallel complexity class $\text{RNC}_2$, and that it is at least as hard as the infamous problem \probl{Exact Matching in Red-Blue Bipartite Graphs} whose computational complexity is a long-standing open problem