Rooting Out Entropy: Optimal Tree Extraction for Ultra-Succinct Graphs

TL;DR

This paper introduces a novel optimization problem for graph compression called MINETREX, analyzes its computational hardness, and proposes a greedy algorithm that enables ultra-succinct graph representations with practical query support.

Contribution

It formulates the MINETREX problem for entropy-based graph compression, proves its NP-hardness, and provides a simple greedy approximation algorithm with theoretical guarantees.

Findings

MINETREX is NP-hard to approximate within a certain additive error.

The greedy algorithm achieves an additive error of at most n / ln 2.

The resulting data structure enables space-efficient graph storage with logarithmic query time.

Abstract

We combine two methods for the lossless compression of unlabeled graphs - entropy compressing adjacency lists and computing canonical names for vertices - and solve an ensuing novel optimisation problem: Minimum-Entropy Tree-Extraction (MINETREX). MINETREX asks to determine a spanning forest $F$ to remove from a graph $G$ so that the remaining graph $G-F$ has minimal indegree entropy $H(d_1,\ldots,d_n) = \sum_{v\in V} d_v \log_2(m/d_v)$ among all choices for $F$. (Here $d_v$ is the indegree of vertex $v$ in $G-F$; $m$ is the number of edges.) We show that MINETREX is NP-hard to approximate with additive error better than $\delta n$ (for some constant $\delta>0$), and provide a simple greedy algorithm that achieves additive error at most $n / \ln 2$. By storing the extracted spanning forest and the remaining edges separately, we obtain a degree-entropy compressed ("ultrasuccinct") data structure for representing an arbitrary (static) unlabeled graph that supports navigational graph queries in logarithmic time. It serves as a drop-in replacement for adjacency-list representations using substantially less space for most graphs; we precisely quantify these savings in terms of the maximal subgraph density. Our inapproximability result uses an approximate variant of the hitting set problem on biregular instances whose hardness proof is contained implicitly in a reduction by Guruswami and Trevisan (APPROX/RANDOM 2005); we consider the unearthing of this reduction partner of independent interest with further likely uses in hardness of approximation.