We study the classic set cover problem from the perspective of sublinear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sublinear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sublinear query model, that returns an αapproximate cover using Õ(m(n/k)^1/(α–1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is , even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1 + ε)approximation algorithm with Õ(mn/kε^2) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing amore »
This content will become publicly available on June 9, 2023
Hardness of approximation in p via short cycle removal: cycle detection, distance oracles, and beyond
We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost kcycle free graphs, for any constant k≥ 4.
Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many 4 or 5cycles in a worstcase instance had been the obstacle towards resolving major open questions.
Hardness of approximation: Are there distance oracles with m1+o(1) preprocessing time and mo(1) query time that achieve a constant approximation? Existing algorithms with such desirable time bounds only achieve superconstant approximation factors, while only 3− factors were conditionally ruled out (Pătraşcu, Roditty, and Thorup; FOCS 2012). We prove that no O(1) approximations are possible, assuming the 3SUM or APSP conjectures. In particular, we prove that kapproximations require Ω(m1+1/ck) time, which is tight up to the constant c. The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths. The 4Cycle problem: An infamous open question in finegrained complexity is to establish any surprising consequences from a subquadratic or more »
 Award ID(s):
 1900460
 Publication Date:
 NSFPAR ID:
 10338450
 Journal Name:
 STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
 Page Range or eLocationID:
 1487 to 1500
 Sponsoring Org:
 National Science Foundation
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