Lower Bounds Based on the Exponential Time Hypothesis: Edge Clique Cover

Years and Authors of Summarized Original Work

• 2008; Gramm, Guo, Hüffner, Niedermeier

• 2013; Cygan, Pilipczuk, Pilipczuk

The Exponential Time Hypothesis and Its Consequences

In 2001, Impagliazzo, Paturi, and Zane [5, 6] introduced the Exponential Time Hypothesis (ETH): a complexity assumption saying that there exists a constant c > 0 such that no algorithm for 3-SATcan achieve the running time of $\mathcal{O}(2^{cn})$, where n is the number of variables of the input formula. In particular, this implies that there is no subexponential-time algorithm for 3-SAT, that is, one with running time 2^o(n). The key result of Impagliazzo, Paturi, and Zane is the Sparsification Lemma, proved in [6]. Without going into technical details, the Sparsification Lemma provides a reduction that allows us to assume that the input instance of 3-SATis sparse in the following sense: the number of clauses is linear in the number of variables. Thus, a direct consequence is that, assuming ETH, there is a...