Minimum Entropy Orientations

TL;DR

This paper investigates the problem of orienting graphs to minimize the entropy of in-degree sequences, proving NP-hardness and providing a simple approximation algorithm with a tight additive error bound.

Contribution

It establishes the NP-hardness of minimum entropy orientations even for planar graphs and introduces a linear-time approximation algorithm with a 1-bit additive error.

Findings

NP-hardness for planar graphs

Linear-time approximation algorithm

Additive error guarantee of 1 bit

Abstract

We study graph orientations that minimize the entropy of the in-degree sequence. The problem of finding such an orientation is an interesting special case of the minimum entropy set cover problem previously studied by Halperin and Karp [Theoret. Comput. Sci., 2005] and by the current authors [Algorithmica, to appear]. We prove that the minimum entropy orientation problem is NP-hard even if the graph is planar, and that there exists a simple linear-time algorithm that returns an approximate solution with an additive error guarantee of 1 bit. This improves on the only previously known algorithm which has an additive error guarantee of log_2 e bits (approx. 1.4427 bits).