Maximum Reachability Orientation of Mixed Graphs

TL;DR

This paper investigates the problem of orienting mixed graphs to maximize reachability, proving NP-hardness and APX-hardness, and explores parameterized algorithms based on the number of pre-oriented arcs.

Contribution

It establishes the NP-hardness and APX-hardness of the maximum reachability orientation problem in mixed graphs and analyzes its parameterized complexity with respect to pre-oriented arcs.

Findings

The problem is NP-hard in mixed graphs.

The problem is APX-hard in mixed graphs.

Algorithms with runtime depending on the number of pre-oriented arcs are developed.

Abstract

We aim to find orientations of mixed graphs optimizing the total reachability, a problem that has applications in causality and biology. For given a digraph $D$, we use $P(D)$ for the set of ordered pairs of distinct vertices in $V(D)$ and we define $\kappa_D:P(D)\rightarrow \{0,1\}$ by $\kappa_D(u,v)=1$ if $v$ is reachable from $u$ in $D$, and $\kappa_D(u,v)=0$, otherwise. We use $R(D)=\sum_{(u,v)\in P(D)}\kappa_D(u,v)$. Now, given a mixed graph $G$, we aim to find an orientation $\vec{G}$ of $G$ that maximizes $R(\vec{G})$. Hakimi, Schmeichel, and Young proved that the problem can be solved in polynomial time when restricted to undirected inputs. They inquired about the complexity in mixed graphs. We answer this question by showing that this problem is NP-hard, and, moreover, APX-hard. We then develop a finer understanding of how quickly the problem becomes difficult when going from undirected to mixed graphs. To this end, we consider the parameterized complexity of the problem with respect to the number $k$ of preoriented arcs of $G$, a poorly understood form of parameterization. We show that the problem can be solved in time $n^{O(k)}$ and that a $(1-\epsilon)$-approximation can be computed in time $f(k,\epsilon)n^{O(1)}$ for any $\epsilon > 0$.