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

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 the most popular such metrics of importance 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 p3{2 ́ δq-approximation of the diameter in sparse and unweighted graphs faster that n2 ́op1q 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 and very active 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 n2 ́op1q unless some popular conjecture is false. Furthermore they ruled out a p2 ́ δq-approximation for RC, a p3{2 ́ δq-approximation for Radius and a p5{3 ́ δq-approximation for computing all eccentricities of a graph for any δ ą 0. In this paper 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 graphs with maximum degree 3. Specifically we show that no p3{2 ́δq approximation of Radius or Diameter, p2 ́δqapproximation of RC, p5{3 ́ δq-approximation of all eccentricities or exact algorithm for BC exists in time n2 ́op1q for such graphs and any δ ą 0. For BC, this strengthens the result of Abboud et al. [SODA’16] by showing a hardness result for unweighted graphs. Our results follow in the footsteps of Abboud et al. [SODA’16] and Abboud and Dahlgaard [FOCS’16] by showing conditional lower bounds for restricted but realistic graph classes.