Tight Hardness Results for Distance and Centrality Problems in Constant Degree Graphs

TL;DR

This paper establishes tight hardness bounds for approximating centrality and distance metrics in constant degree graphs, extending previous results from sparse graphs to bounded-degree unweighted graphs, indicating no significantly faster algorithms exist under SETH.

Contribution

The authors extend hardness results for distance and centrality problems from sparse graphs to unweighted graphs with constant degree, using new constructions to match previous bounds.

Findings

No (3/2 - δ)-approximation for Radius or Diameter in time n^{2-o(1)}.

No (2 - δ)-approximation for Reach Centrality in time n^{2-o(1)}.

No exact algorithm for Betweenness Centrality in time n^{2-o(1)}.

Abstract

Finding important nodes in a graph and measuring their importance is a fundamental problem in the analysis of social networks, transportation networks, biological systems, etc. Among popular such metrics are graph centrality, betweenness centrality (BC), and reach centrality (RC). These measures are also very related to classic notions like diameter and radius. Roditty and Vassilevska Williams~[STOC'13] showed that no algorithm can compute a (3/2-\delta)-approximation of the diameter in sparse and unweighted graphs faster that n^{2-o(1)} time unless the widely believed strong exponential time hypothesis (SETH) is false. Abboud et al.~[SODA'15] and [SODA'16] further analyzed these problems under the recent line of research on hardness in P. They showed that in sparse and unweighted graphs (weighted for BC) none of these problems can be solved faster than n^{2-o(1)} unless some popular conjecture is false. Furthermore they ruled out a (2-\delta)-approximation for RC, a (3/2-\delta)-approximation for Radius and a (5/3-\delta)-approximation for computing all eccentricities of a graph for any \delta > 0. We extend these results to the case of unweighted graphs with constant maximum degree. Through new graph constructions we are able to obtain the same approximation and time bounds as for sparse graphs even in unweighted bounded-degree graphs. We show that no (3/2-\delta) approximation of Radius or Diameter, (2-\delta)-approximation of RC, (5/3-\delta)-approximation of all eccentricities or exact algorithm for BC exists in time n^{2-o(1)} for such graphs and any \delta > 0. This strengthens the result for BC of Abboud et al.~[SODA'16] by showing a hardness result for unweighted graphs, and follows in the footsteps of Abboud et al.~[SODA'16] and Abboud and Dahlgaard~[FOCS'16] in showing conditional lower bounds for restricted but realistic graph classes.