Orienting Graphs to Optimize Reachability

TL;DR

This paper investigates how to orient edges in an undirected graph to either maximize or minimize reachability, providing a quadratic-time algorithm for maximization and proving NP-hardness for minimization.

Contribution

It introduces a quadratic-time algorithm for maximizing reachability and proves NP-hardness for minimizing reachability, establishing the complexity of these orientation problems.

Findings

Quadratic-time algorithm for maximizing reachability

NP-hardness of minimizing reachability

Equivalence to comparability graph completion

Abstract

The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to minimize the number of such pairs. The paper describes a quadratic-time algorithm for the first problem, and a proof that the second problem is NP-hard to approximate within some constant 1+epsilon > 1. The latter proof also shows that the second problem is equivalent to ``comparability graph completion''; neither problem was previously known to be NP-hard.