Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A 3-path vertex cover in a graph is a vertex subset $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201

A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set

An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of \Omega(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.Comment: Full version. A preliminary version appeared in the proceedings of WG 200

Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, notably Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for a more precise and improved running time analysi

Improved Analysis of Highest-Degree Branching for Feedback Vertex Set

Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning heuristic. In this paper, we prove that this empirically fast algorithm runs in O(3.460^k n) time, where k is the solution size. This improves the previous best O(3.619^k n)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf. Process. Lett., 2014)

Minimizing and Computing the Inverse Geodesic Length on Trees

For any fixed measure H that maps graphs to real numbers, the MinH problem is defined as follows: given a graph G, an integer k, and a target tau, is there a set S of k vertices that can be deleted, so that H(G - S) is at most tau? In this paper, we consider the MinH problem on trees. We call H balanced on trees if, whenever G is a tree, there is an optimal choice of S such that the components of G - S have sizes bounded by a polynomial in n / k. We show that MinH on trees is Fixed-Parameter Tractable (FPT) for parameter n / k, and furthermore, can be solved in subexponential time, and polynomial space, whenever H is additive, balanced on trees, and computable in polynomial time. A particular measure of interest is the Inverse Geodesic Length (IGL), which is used to gauge the efficiency and connectedness of a graph. It is defined as the sum of inverse distances between every two vertices: IGL(G) = sum_{{u,v} subseteq V} 1/d_G(u,v). While MinIGL is W[1]-hard for parameter treewidth, and cannot be solved in 2^{o(k + n + m)} time, even on bipartite graphs with n vertices and m edges, the complexity status of the problem remains open in the case where G is a tree. We show that IGL is balanced on trees, to give a 2^O((n log n)^(5/6)) time, polynomial space algorithm. The distance distribution of G is the sequence {a_i} describing the number of vertex pairs distance i apart in G: a_i = |{{u, v}: d_G(u, v) = i}|. Given only the distance distribution, one can easily determine graph parameters such as diameter, Wiener index, and particularly, the IGL. We show that the distance distribution of a tree can be computed in O(n log^2 n) time by reduction to polynomial multiplication. We also extend the result to graphs with small treewidth by showing that the first p values of the distance distribution can be computed in 2^(O(tw(G))) n^(1 + epsilon) sqrt(p) time, and the entire distance distribution can be computed in 2^(O(tw(G))) n^{1 + epsilon} time, when the diameter of G is O(n^epsilon\u27) for every epsilon\u27 > 0

An O∗(1.0821n)O^*(1.0821^n)-Time Algorithm for Computing Maximum Independent Set in Graphs with Bounded Degree 3

We give an $O^*(1.0821^n)$-time, polynomial space algorithm for computing Maximum Independent Set in graphs with bounded degree 3. This improves all the previous running time bounds known for the problem