The Strongish Planted Clique Hypothesis and Its Consequences

TL;DR

This paper introduces the Strongish Planted Clique Hypothesis (SPCH), a new computational hardness assumption, and demonstrates its implications for inapproximability results and graph pattern detection problems, establishing tighter lower bounds.

Contribution

The paper formulates the SPCH, a novel hardness assumption, and applies it to derive nearly tight inapproximability and lower bounds for various graph problems.

Findings

SPCH implies no polynomial time algorithm can approximate Densest k-Subgraph within o(k).

SPCH leads to tighter lower bounds for graph pattern detection problems.

The results improve upon previous bounds based on the Exponential Time Hypothesis.

Abstract

We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time $n^{\Omega(\log{n})}$ (so that the state-of-the-art running time of $n^{O(\log n)}$ is optimal up to a constant in the exponent). We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter $k$: Densest $k$-Subgraph, Smallest $k$-Edge Subgraph, Densest $k$-Subhypergraph, Steiner $k$-Forest, and Directed Steiner Network with $k$ terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves $o(k)$-approximation for Densest $k$-Subgraph. This inapproximability ratio improves upon the previous best $k^{o(1)}$ factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter $k$. Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, which improves the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis.