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Parameterized Complexity of Fair Many-to-One Matchings

Ramin Javadi, Hossein Shokouhi

TL;DR

This paper investigates parameterized complexity of fair left-perfect many-to-one matching problem with respect to the structural parameters of the input graph and proves that the problem is W[1]-hard with respect to the feedback vertex number, tree-depth and the maximum degree of $U, combined.

Abstract

Given a bipartite graph $G=(U\cup V,E)$, a left-perfect many-to-one matching is a subset $M \subseteq E$ such that each vertex in $U$ is incident with exactly one edge in $M$. If $U$ is partitioned into some groups, the matching is called fair if for every $v\in V$, the difference between the number of vertices matched with $v$ in any two groups does not exceed a given threshold. In this paper, we investigate parameterized complexity of fair left-perfect many-to-one matching problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the feedback vertex number, tree-depth and the maximum degree of $U$, combined. Also, it is W[1]-hard with respect to the path-width, the number of groups and the maximum degree of $U$, combined. In the positive side, we prove that the problem is FPT with respect to the treewidth and the maximum degree of $V$. Also, it is FPT with respect to the neighborhood diversity of the input graph (which implies being FPT with respect to vertex cover and modular-width). Finally, we prove that the problem is FPT with respect to the tree-depth and the number of groups.

Parameterized Complexity of Fair Many-to-One Matchings

TL;DR

This paper investigates parameterized complexity of fair left-perfect many-to-one matching problem with respect to the structural parameters of the input graph and proves that the problem is W[1]-hard with respect to the feedback vertex number, tree-depth and the maximum degree of $U, combined.

Abstract

Given a bipartite graph , a left-perfect many-to-one matching is a subset such that each vertex in is incident with exactly one edge in . If is partitioned into some groups, the matching is called fair if for every , the difference between the number of vertices matched with in any two groups does not exceed a given threshold. In this paper, we investigate parameterized complexity of fair left-perfect many-to-one matching problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the feedback vertex number, tree-depth and the maximum degree of , combined. Also, it is W[1]-hard with respect to the path-width, the number of groups and the maximum degree of , combined. In the positive side, we prove that the problem is FPT with respect to the treewidth and the maximum degree of . Also, it is FPT with respect to the neighborhood diversity of the input graph (which implies being FPT with respect to vertex cover and modular-width). Finally, we prove that the problem is FPT with respect to the tree-depth and the number of groups.

Paper Structure

This paper contains 5 sections, 8 theorems, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. W[1]-hard Results
  4. FPT Results
  5. Concluding Remark

Key Result

Theorem 1

Let $G=(U\cup V, E)$ be the input graph and $|V|=k$. Then, Generalized Fair Matching can be solved in time $O^*(k^{O(k)})$.

Figures (2)

  • Figure 1: Example of a left-perfect many-to-one fair matching
  • Figure 2: Summary of our main results regarding Generalized Fair Matching problem. An arrow from $f$ to $g$ means that $g$ is bounded by a function of $f$ and so W[1]-hardness result with respect to $f$ implies W[1]-hardness with respect to $g$. Parameters marked by green are proved to be FPT. Parameters marked by red are proved to be W[1]-hard and the situation of the parameter marked by yellow is unknown.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8