The Strongish Planted Clique Hypothesis and Its Consequences

Authors Pasin Manurangsi, Aviad Rubinstein, Tselil Schramm

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Pasin Manurangsi
  • Google Research, Mountain View, CA, USA
Aviad Rubinstein
  • Stanford University, CA, USA
Tselil Schramm
  • Stanford University, CA, USA


Pasin would like to thank Michal Pilipczuk and Daniel Lokshtanov for posing the approximability of Densest k-Subhypergraph as an open problem at Dagstuhl Seminar on New Horizons in Parameterized Complexity, and for helpful discussions.

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Pasin Manurangsi, Aviad Rubinstein, and Tselil Schramm. The Strongish Planted Clique Hypothesis and Its Consequences. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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^Ω(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, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Fixed parameter tractability
  • Planted Clique
  • Densest k-Subgraph
  • Hardness of Approximation


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