Applications of Random Algebraic Constructions to Hardness of Approximation

TL;DR

This paper constructs new extremal combinatorial objects, panchromatic and threshold graphs, to establish stronger conditional lower bounds for the parameterized set intersection problem under ETH and SETH.

Contribution

It introduces efficient constructions of previously conjectural extremal graphs and applies them to derive improved time lower bounds for set intersection approximation.

Findings

Constructed panchromatic graphs confirming their existence

Constructed threshold graphs confirming their existence

Established stronger ETH and SETH-based lower bounds for set intersection approximation

Abstract

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a balanced bipartite graph with one partition class equipartitioned into k colour classes in which the common neighbourhoods of panchromatic k-sets of vertices are much larger than those of k-sets that repeat a colour. The question of their existence was raised by Karthik and Manurangsi [Combinatorica 2020]. (*) Threshold Graphs: For fixed integer k, a k-threshold graph is, roughly speaking, a balanced bipartite graph in which the common neighbourhoods of k-sets of vertices on one side are much larger than those of (k+1)-sets. The question of their existence was raised by Lin [JACM 2018]. As applications of our constructions, we show the following conditional time lower bounds on the parameterized set intersection problem where, given a collection of n sets over universe [n] and a parameter k, the goal is to find k sets with the largest intersection. (*) Assuming ETH, for any computable function F, no $n^{o(k)}$-time algorithm can approximate the parameterized set intersection problem up to factor F(k). This improves considerably on the previously best-known result under ETH due to Lin [JACM 2018], who ruled out any $n^{o(\sqrt{k})}$ time approximation algorithm for this problem. (*) Assuming SETH, for every $\varepsilon>0$ and any computable function F, no $n^{k-\varepsilon}$-time algorithm can approximate the parameterized set intersection problem up to factor F(k). No result of comparable strength was previously known under SETH, even for solving this problem exactly.