Slightly Superexponential Parameterized Problems

Abstract

A central problem in parameterized algorithms is to obtain algorithms with running time $f(k)\cdot n^{O(1)}$ such that $f$ is as slow growing a function of the parameter $k$ as possible. In particular, a large number of basic parameterized problems admit parameterized algorithms where $f(k)$ is single-exponential, that is, $c^k$ for some constant $c$, which makes aiming for such a running time a natural goal for other problems as well. However, there are still plenty of problems where the $f(k)$ appearing in the best-known running time is worse than single-exponential and it remained “slightly superexponential” even after serious attempts to bring it down. A natural question to ask is whether the $f(k)$ appearing in the running time of the best-known algorithms is optimal for any of these problems. In this paper, we examine parameterized problems where $f(k)$ is $k^{O(k)}=2^{O(k\log k)}$ in the best-known running time, and for a number of such problems we show that the dependence on $k$ in the running time cannot be improved to single-exponential. More precisely we prove the following tight lower bounds, for four natural problems, arising from three different domains: (1) In the Closest String problem, given strings $s_1$, $\dots$, $s_t$ over an alphabet $\Sigma$ of length $L$ each, and an integer $d$, the question is whether there exists a string $s$ over $\Sigma$ of length $L$, such that its hamming distance from each of the strings $s_i$, $1\leq i \leq t$, is at most $d$. The pattern matching problem Closest String is known to be solvable in times $2^{O(d\log d)}\cdot n^{O(1)}$ and $2^{O(d\log |\Sigma|)}\cdot n^{O(1)}$. We show that there are no $2^{o(d\log d)}\cdot n^{O(1)}$ or $2^{o(d\log |\Sigma|)}\cdot n^{O(1)}$ time algorithms, unless the Exponential Time Hypothesis (ETH) fails. (2) The graph embedding problem Distortion, that is, deciding whether a graph $G$ has a metric embedding into the integers with distortion at most $d$ can be solved in time $2^{O(d\log d)}\cdot n^{O(1)}$. We show that there is no $2^{o(d\log d)}\cdot n^{O(1)}$ time algorithm, unless the ETH fails. (3) The Disjoint Paths problem can be solved in time $2^{O(w\log w)}\cdot n^{O(1)}$ on graphs of treewidth at most $w$. We show that there is no $2^{o(w\log w)}\cdot n^{O(1)}$ time algorithm, unless the ETH fails. (4) The Chromatic Number problem can be solved in time $2^{O(w\log w)}\cdot n^{O(1)}$ on graphs of treewidth at most $w$. We show that there is no $2^{o(w\log w)}\cdot n^{O(1)}$ time algorithm, unless the ETH fails. To obtain our results, we first prove the lower bound for variants of basic problems: finding cliques, independent sets, and hitting sets. These artificially constrained variants form a good starting point for proving lower bounds on natural problems without any technical restrictions and could be of independent interest. Several follow-up works have already obtained tight lower bounds by using our framework, and we believe it will prove useful in obtaining even more lower bounds in the future.