Slightly Superexponential Parameterized Problems

A central problem in parameterized algorithms is to obtain algorithms with running time f(k) center dot 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,..., s(t) over an alphabet Sigma of length L each, and an integer d, the question is whether there exists a string s over E of length L, such that its hamming distance from each of the strings s,, 1 <= i <= t, is at most d. The pattern matching problem CLOSEST STRING is known to be solvable in times 2(O(d log d)) center dot n(O(1)) and 2(O(d log vertical bar Sigma vertical bar)) center dot n(O(1)). We show that there are no 2(O(d log d)) center dot n(O(1)) or 2(O(d log vertical bar Sigma vertical bar)) 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)) center dot n(O(1)). We show that there is no 2(O(w log w)) center dot n(O(1)) time algorithm, unless the ETH fails. (3) The DISJOINT PATHS problem can be solved in time 2(O(w log w)) center dot n(O(1)) on graphs of treewidth at most w. We show that there is no 2(O(w log w)) center dot n(O(1)) time algorithm, unless the ETH fails. (4) The CHROMATIC NUMBER problem can be solved in time 2(O(w log w)) center dot n(O(1)) on graphs of treewidth at most w. We show that there is no 2(O(w log w)) center dot 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

A Complexity Dichotomy for Hitting Small Planar Minors Parameterized by Treewidth

For a fixed graph H, we are interested in the parameterized complexity of the following problem, called {H}-M-Deletion, parameterized by the treewidth tw of the input graph: given an n-vertex graph G and an integer k, decide whether there exists S subseteq V(G) with |S| <= k such that G setminus S does not contain H as a minor. In previous work [IPEC, 2017] we proved that if H is planar and connected, then the problem cannot be solved in time 2^{o(tw)} * n^{O(1)} under the ETH, and can be solved in time 2^{O(tw * log tw)} * n^{O(1)}. In this article we manage to classify the optimal asymptotic complexity of {H}-M-Deletion when H is a connected planar graph on at most 5 vertices. Out of the 29 possibilities (discarding the trivial case H = K_1), we prove that 9 of them are solvable in time 2^{Theta (tw)} * n^{O(1)}, and that the other 20 ones are solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}. Namely, we prove that K_4 and the diamond are the only graphs on at most 4 vertices for which the problem is solvable in time 2^{Theta (tw * log tw)} * n^{O(1)}, and that the chair and the banner are the only graphs on 5 vertices for which the problem is solvable in time 2^{Theta (tw)} * n^{O(1)}. For the version of the problem where H is forbidden as a topological minor, the case H = K_{1,4} can be solved in time 2^{Theta (tw)} * n^{O(1)}. This exhibits, to the best of our knowledge, the first difference between the computational complexity of both problems

Optimal Algorithms for Hitting (Topological) Minors on Graphs of Bounded Treewidth

For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a subset S of V(G) of size at most k such that G-S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-DELETION when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F}, the smallest function f_F such that F-M-DELETION can be solved in time f_F(tw)n^{O(1)} on n-vertex graphs. Using and enhancing the machinery of boundaried graphs and small sets of representatives introduced by Bodlaender et al. [J ACM, 2016], we prove that when all the graphs in F are connected and at least one of them is planar, then f_F(w) = 2^{O(wlog w)}. When F is a singleton containing a clique, a cycle, or a path on i vertices, we prove the following asymptotically tight bounds: - f_{K_4}(w) = 2^{Theta(wlog w)}. - f_{C_i}(w) = 2^{Theta(w)} for every i4. - f_{P_i}(w) = 2^{Theta(w)} for every i5. The lower bounds hold unless the Exponential Time Hypothesis fails, and the superexponential ones are inspired by a reduction of Marcin Pilipczuk [Discrete Appl Math, 2016]. The single-exponential algorithms use, in particular, the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove essentially the same set of results holds

Lower bounds for approximation schemes for Closest String

In the Closest String problem one is given a family $\mathcal S$ of equal-length strings over some fixed alphabet, and the task is to find a string $y$ that minimizes the maximum Hamming distance between $y$ and a string from $\mathcal S$. While polynomial-time approximation schemes (PTASes) for this problem are known for a long time [Li et al., J. ACM'02], no efficient polynomial-time approximation scheme (EPTAS) has been proposed so far. In this paper, we prove that the existence of an EPTAS for Closest String is in fact unlikely, as it would imply that $\mathrm{FPT}=\mathrm{W}[1]$, a highly unexpected collapse in the hierarchy of parameterized complexity classes. Our proof also shows that the existence of a PTAS for Closest String with running time $f(\varepsilon)\cdot n^{o(1/\varepsilon)}$, for any computable function $f$, would contradict the Exponential Time Hypothesis