TY - GEN

T1 - Matching triangles and basing hardness on an extremely popular conjecture

AU - Abboud, Amir

AU - Williams, Virginia Vassilevska

AU - Yu, Huacheng

N1 - Publisher Copyright:
© Copyright 2015 ACM.

PY - 2015/6/14

Y1 - 2015/6/14

N2 - Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the APSP conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining "less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including: • Tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs. • New n1-o(1) lower bounds for the amortized update and query times of dynamic algorithms for single-source reachability, strongly connected components, and Max-Flow. • New n1.5-o(1) lower bound for computing a set of n st-maximum-flow values in a directed graph with n nodes and Õ(n) edges. • There is a hierarchy of natural graph problems on n nodes with complexity nc for c ∈ (2,3). Only slightly non-trivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source-Max-Flow problem.

AB - Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the APSP conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining "less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including: • Tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs. • New n1-o(1) lower bounds for the amortized update and query times of dynamic algorithms for single-source reachability, strongly connected components, and Max-Flow. • New n1.5-o(1) lower bound for computing a set of n st-maximum-flow values in a directed graph with n nodes and Õ(n) edges. • There is a hierarchy of natural graph problems on n nodes with complexity nc for c ∈ (2,3). Only slightly non-trivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source-Max-Flow problem.

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UR - http://www.scopus.com/inward/citedby.url?scp=84958281695&partnerID=8YFLogxK

U2 - 10.1145/2746539.2746594

DO - 10.1145/2746539.2746594

M3 - Conference contribution

AN - SCOPUS:84958281695

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 41

EP - 50

BT - STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

T2 - 47th Annual ACM Symposium on Theory of Computing, STOC 2015

Y2 - 14 June 2015 through 17 June 2015

ER -