匹配三角形和基于硬度的一个非常流行的猜想

Amir Abboud, V. V. Williams, Huacheng Yu
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引用次数: 119

摘要

由于缺乏无条件多项式下界,现在流行证明条件下界,以提高我们对p类的理解。这些下界中的绝大多数是基于三个著名的假设之一:3-SUM猜想,APSP猜想和强指数时间假设。只有已知的间接证据支持这些假设,它们之间没有正式的关系。为了得到“条件更少”从而更可靠的下界,我们考虑上述三个假设中至少有一个为真的猜想。我们从3-SUM, APSP和CNF-SAT设计了新的约简,并得出了这个非常合理的猜想的有趣结果,包括:关于无加权图中三角形的纯组合问题的紧密n3-o(1)下界。单源可达性、强连接组件和Max-Flow的动态算法的平摊更新和查询次数的新n1-o(1)下界。新的n1.5-o(1)下界,用于计算n个节点~O(n)条边的有向图的n个st-maximum-flow值集。对于c∈(2,3),存在n个节点上的自然图问题的层次结构,其复杂度为nc。在我们工作之前,人们只知道这个猜想的一些不重要的结果。在此过程中,我们还得到了单源最大流问题的新的条件下界。
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Matching Triangles and Basing Hardness on an Extremely Popular Conjecture
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 ~O(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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