Improved FPT Algorithms Research Articles

Improved FPT Algorithms for Deletion to Forest-Like Structures

Improved FPT Algorithms for Deletion to Forest-Like Structures

New Results on Directed Edge Dominating Set

We study a family of generalizations of Edge Dominating Set on directed graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc $(u,v)$ is said to dominate itself, as well as all arcs which are at distance at most $q$ from $v$, or at distance at most $p$ to $u$. First, we give significantly improved FPT algorithms for the two most important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by $p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is added as a second parameter), where $tw$ is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of $p,q$, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

Fixed parameterized algorithms for generalized feedback vertex set problems

A graph is an r-pseudoforest if every connected component of it has a feedback edge set of size at most r. A graph is a d-quasi-forest if every connected component of it has a feedback vertex set of size at most d. The r-Pseudoforest Deletion problem (d-Quasi-Forest Deletion problem) asks to delete a minimum number of vertices to get an r-pseudoforest (a d-quasi-forest, respectively). The well-studied feedback vertex set problem is the special case of r-Pseudoforest Deletion (d-Quasi-Forest Deletion, respectively) in which r=0 (d=0, respectively).We provide an improved FPT algorithm and a smaller kernel for r-Pseudoforest Deletion when parameterized by the solution size (when r is fixed). For d-Quasi-Forest Deletion, we show that it is FPT as well when parameterized by d and the solution size.

An improved FPT algorithm for the flip distance problem

Given a set P of points in the Euclidean plane and two triangulations of P, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other. The Parameterized Flip Distance problem is to decide if the flip distance between two given triangulations is equal to a given integer k. The previous best FPT algorithm runs in time O⁎(k⋅ck) (c≤2×1411), where each step has fourteen possible choices, and the length of the action sequence is bounded by 11k. By analyzing the underlying properties of the flip sequence, each step of our algorithm has only five possible choices. Based on an auxiliary graph G, we prove that the length of the action sequence for our algorithm is bounded by 2|G|. As a result, we present an FPT algorithm running in time O⁎(k⋅32k).

An Improved FPT Algorithm for Independent Feedback Vertex Set

We study the Independent Feedback Vertex Set problem — a variant of the classic Feedback Vertex Set problem where, given a graph G and an integer k, the problem is to decide whether there exists a vertex set Ssubseteq V(G) such that G ∖ S is a forest and S is an independent set of size at most k. We present an mathcal {O}^{ast }((1+varphi ^{2})^{k})-time FPT algorithm for this problem, where φ < 1.619 is the golden ratio, improving the previous fastest mathcal {O}^{ast }(4.1481^{k})-time algorithm given by Agrawal et al. (2016). The exponential factor in our time complexity bound matches the fastest deterministic FPT algorithm for the classic Feedback Vertex Set problem. On the technical side, the main novelty is a refined measure of an input instance in a branching process, that allows for a simpler and more concise description and analysis of the algorithm.

Simultaneous consecutive ones submatrix and editing problems: Classical complexity and fixed-parameter tractable results

A binary matrix M has the consecutive ones property (C1P) for rows (resp. columns) if there exists a permutation of its columns (resp. rows) that arranges the ones consecutively in all the rows (resp. columns). If M has the C1P for rows and the C1P for columns, then M is said to have the simultaneous consecutive ones property (SC1P). In this article, we consider the classical complexity and fixed-parameter tractability of a few variants of decision problems related to the SC1P. Given a binary matrix M and a positive integer d, we focus on problems that decide whether there exists a set of rows, columns, and rows as well as columns, respectively, of size at most d in M, whose deletion results in a matrix with the SC1P. We also consider problems that decide whether there exists a set of 0-entries, 1-entries and 0-entries as well as 1-entries, respectively, of size at most d in M, whose flipping results in a matrix with the SC1P. In this paper, we show that all the above mentioned problems are NP-complete. We could also prove that all these problems are fixed-parameter tractable with respect to solution size as the parameter, except for two variants (flipping 1-entries and flipping 0/1-entries). We also give improved FPT algorithms for certain problems on restricted binary matrices.

An improved FPT algorithm for Almost Forest Deletion problem

Almost Forest Deletion problem (AFD) is a generalization of the Feedback Vertex Set problem, which decides whether there exist at most k vertices in a given graph G whose removal from G results in an l-forest, where k and l are two given non-negative integers, and an l-forest is a graph which can be transformed into a forest by deleting at most l edges. Based on the iterative compression technique, we study the iterative version of the AFD problem, called Almost Forest Deletion Disjoint Compression problem (AFDDC), which asks for a new l-forest deletion set X′ of size at most k for a given graph G that is disjoint with a given l-forest deletion set X of graph G for two given non-negative integers k and l. For the AFDDC problem, we develop some reduction rules to simplify a given instance, and give a new branching algorithm for the problem. A new branching measure is presented to evaluate the efficiency of the algorithm, which results in an algorithm of running time O⁎(4k+l). Based on the proposed algorithm for the AFDDC problem, a parameterized algorithm for the AFD problem with running time O⁎(5k4l) is presented, improving the previous result O⁎(5.0024(k+l)).

Improved FPT algorithms for weighted independent set in bull-free graphs

Very recently, Thomassé et al. (2017) have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time 2O(k5)⋅n9. In this article we improve this running time to 2O(k2)⋅n7. As a byproduct, we also improve the previous Turing-kernel for this problem from O(k5) to O(k2). Furthermore, for the subclass of bull-free graphs without holes of length at most 2p−1 for p≥3, we speed up the running time to 2O(k⋅k1p−1)⋅n7. As p grows, this running time is asymptotically tight in terms of k, since we prove that for each integer p≥3, Weighted Independent Set cannot be solved in time 2o(k)⋅nO(1) in the class of {bull,C4,…,C2p−1}-free graphs unless the ETH fails.

The Graph Motif problem parameterized by the structure of the input graph

The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. In the many utilizations of Graph Motif, however, the input graph originates from real-life applications and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. For the FPT cases, we also give some kernelization lower bounds as well as some ETH-based lower bounds on the worst case running time. Interestingly, we establish that Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way.

Foreword: Special Issue on IPEC 2014

We are pleased to present this special issue of Algorithmica, which contains the extended journal versions of selected papers previously presented at the 9th International Symposium on Parameterized and Exact Computation, IPEC 2014, September 10–12, Wroclaw, Poland. The symposium is an established annual meeting of the multivariate and exact algorithms communities. This issue consists of nine papers, reviewed thoroughly according to the usual, high standard of the journal. In The Parameterized Complexity of Geometric Graph Isomorphism, V. Arvind and Gaurav Rattan present improved FPT algorithm for Geometric Graph Isomorphism problem, where one is to decide whether there is a distance preserving bijection between two sets of points in k-dimensional euclidian space. Igor Razgon in On the read-once property of branching programs and CNFs of bounded treewidthproves a space lower bound for non-deterministic read-oncebranching programs on functions expressible as CNFswith treewidth at most k of their primal graphs. In Finding Shortest Paths between Graph Colourings, Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, and Daniel Paulusma give a complete picture of the parameterized complexity of the k-colouring reconfiguration problem, where the goal is to modify one proper colouring into another one, by changing the colour of one vertex at a time. Given a system of linear equations Ax = b over the binary field one can ask whether there is a solution of weight at most t , exactly t or at least t . In Solving Linear

An Improved FPT Algorithm and a Quadratic Kernel for Pathwidth One Vertex Deletion

The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph G and an integer k, one can delete at most k vertices from G so that the remaining graph has pathwidth at most 1. The question can be considered as a natural variation of the extensively studied Feedback Vertex Set (FVS) problem, where the deletion of at most k vertices has to result in the remaining graph having treewidth at most 1 (i.e., being a forest). Recently Philip et al. (WG, Lecture Notes in Computer Science, vol. 6410, pp. 196–207, 2010) initiated the study of the parameterized complexity of POVD, showing a quartic kernel and an algorithm which runs in time 7k n O(1). In this article we improve these results by showing a quadratic kernel and an algorithm with time complexity 4.65k n O(1), thus obtaining almost tight kernelization bounds when compared to the general result of Dell and van Melkebeek (STOC, pp. 251–260, ACM, New York, 2010). Techniques used in the kernelization are based on the quadratic kernel for FVS, due to Thomasse (ACM Trans. Algorithms 6(2), 2010).

Improved FPT algorithm for feedback vertex set problem in bipartite tournament

We present an O ( 3 k n 2 + n 3 ) time FPT algorithm for the feedback vertex set problem in a bipartite tournament on n vertices with integral weights. This improves the previously best known O ( 3.12 k n 4 ) time FPT algorithm for the problem.