OV Graphs Are (Probably) Hard Instances

A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . . , vn ∈ {0, 1} such that nodes i and j are adjacent in G if and only if 〈vi, vj〉 = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: ˆ Determining whether G contains a triangle. ˆ More generally, determining whether G contains a directed k-cycle for any k ≥ 3. ˆ Computing the square of the adjacency matrix of G over Z or F2. ˆ Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication. Harvard University, jalman@seas.harvard.edu. Supported by a Michael O. Rabin Postdoctoral Fellowship. Supported by an NSF CAREER Award, NSF Grants CCF-1528078, CCF-1514339 and CCF-1909429, a BSF Grant BSF:2012338, a Google Research Fellowship and a Sloan Research Fellowship.

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