New hardness results for planar graph problems in p and an algorithm for sparsest cut

Abstract

The Sparsest Cut is a fundamental optimization problem that has been extensively studied. For planar inputs the problem is in P and can be solved inÕ(n 3) time if all vertex weights are 1. Despite a significant amount of effort, the best algorithms date back to the early 90's and can only achieve O(log n)-approximation inÕ(n) time or a constant factor approximation inÕ(n 2) time [Rao, STOC92]. Our main result is an Ω(n 2−ε) lower bound for Sparsest Cut even in planar graphs with unit vertex weights, under the (min, +)-Convolution conjecture, showing that approximations are inevitable in the near-linear time regime. To complement the lower bound, we provide a constant factor approximation in near-linear time, improving upon the 25-year old result of Rao in both time and accuracy. Our lower bound accomplishes a repeatedly raised challenge by being the first fine-grained lower bound for a natural planar graph problem in P. Moreover, we prove near-quadratic lower bounds under SETH for variants of the closest pair problem in planar graphs, and use them to show that the popular Average-Linkage procedure for Hierarchical Clustering cannot be simulated in truly subquadratic time. At the core of our constructions is a diamond-like gadget that also settles the complexity of Diameter in distributed planar networks. We prove an Ω(n/ log n) lower bound on the number of communication rounds required to compute the weighted diameter of a network in the CONGEST model, even when the underlying graph is planar and all nodes are D = 4 hops away from each other. This is the first poly(n) + ω(D) lower bound in the planar-distributed setting, and it complements the recent poly(D, log n) upper bounds of Li and Parter [STOC 2019] for (exact) unweighted diameter and for (1 + ε) approximate weighted diameter.

On the Fine-Grained Complexity of Parity Problems

“…Recently, a few other problems have been shown to be harder, e.g. [2], or subquadratic-equivalent to it, e.g. Maximum consecutive subsums [17,31], 0/1-Knapsack [17,30], and a (1 + ε)-approximation for Subset Sum [14].…”

On Dasgupta's hierarchical clustering objective and its relation to other graph parameters

“…The results by Charikar and Chatziafratis [3] already indicate that the balanced cut algorithm gives an 8-approximation of the DC-value of trees (and other graph classes for which the sparsest cut can be found in polynomial time, like planar graphs, see [1] for more information). In the following, we prove that for trees, we can guarantee a 2-approximation.…”

A Quasipolynomial $(2+\varepsilon)$-Approximation for Planar Sparsest Cut

“…For the uniform version of sparsest cut, O(1)-approximations exist for classes of graphs that exclude non-trivial minors, via low-diameter decompositions [KPR93]. Park and Philips [PP93] solve the uniform problem on planar graphs in O(n 3 ) time; see Abboud et al [ACAK20] for a speedup. Abboud et al [ACAK20] give an O(1)-approximation in near-linear time, improving upon the near-linear time O(log n)-approximation of Rao [Rao92].…”

“…Park and Philips [PP93] solve the uniform problem on planar graphs in O(n 3 ) time; see Abboud et al [ACAK20] for a speedup. Abboud et al [ACAK20] give an O(1)-approximation in near-linear time, improving upon the near-linear time O(log n)-approximation of Rao [Rao92]. Finally, Patel [Pat13] showed that the uniform sparsest-cut problem can be exactly solved in time n O(g) for graphs that embed into a surface of genus at most g. The approximation factor for the uniform problem on general graphs has been improved from O(log n) [LR10] by rounding LP relaxations, to O( √ log n) [ARV09] by rounding SDP relaxations.…”