Planar Feedback Vertex Set and Face Cover: Combinatorial Bounds and Subexponential Algorithms

The Planar Feedback Vertex Set problem asks whether an n-vertex planar graph contains at most k vertices meeting all its cycles. The Face Cover problem asks whether all vertices of a plane graph G lie on the boundary of at most k faces of G. Standard techniques from parameterized algorithm design indicate that both problems can be solved by sub-exponential parameterized algorithms (where k is the parameter). In this paper we improve the algorithmic analysis of both problems by proving a series of combinatorial results relating the branchwidth of planar graphs with their face cover. Combining this fact with duality properties of branchwidth, allows us to derive analogous results on feedback vertex set. As a consequence, it follows that Planar Feedback Vertex Set and Face Cover can be solved in \(O(2^{15.11\cdot\sqrt{k}}+n^{2})\) and \(O(2^{10.1\cdot\sqrt {k}}+n^{2})\) steps, respectively.     

Enumerating Minimal Subset Feedback Vertex Sets

The Subset Feedback Vertex Set problem takes as input a pair (G,S), where G=(V,E) is a graph with weights on its vertices, and S?V. The task is to find a set of vertices of total minimum weight to be removed from G, such that in the remaining graph no cycle contains a vertex of S. We show that this problem can be solved in time O(1.8638 ⁿ ), where n=|V|. This is a consequence of the main result of this paper, namely that all minimal subset feedback vertex sets of a graph can be enumerated in time O(1.8638 ⁿ ).     

Contracting Graphs to Paths and Trees

Vertex deletion and edge deletion problems play a central role in parameterized complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain a tree or a path, respectively, by a sequence of at most k edge contractions in G. For Tree Contraction, we present a randomized 4 ^k ? n ^O(1) time polynomial-space algorithm, as well as a deterministic 4.98 ^k ? n ^O(1) time algorithm, based on a variant of the color coding technique of Alon, Yuster and Zwick. We also present a deterministic 2 ^k+o(k)+n ^O(1) time algorithm for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ? coNP/poly. We find the latter result surprising because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k ² vertices.     

Applying Modular Decomposition to Parameterized Cluster Editing Problems

A graph G is said to be a bicluster graph if G is a disjoint union of bicliques (complete bipartite subgraphs), and a cluster graph if G is a disjoint union of cliques (complete subgraphs). In this work, we study the parameterized versions of the NP-hard Bicluster Graph Editing and Cluster Graph Editing problems. The former consists of obtaining a bicluster graph by making the minimum number of modifications in the edge set of an input bipartite graph. When at most k modifications are allowed (Bicluster(k) Graph Editing problem), this problem is FPT, and can be solved in O(4 ^k nm) time by a standard search tree algorithm. We develop an algorithm of time complexity O(4 ^k +n+m), which uses a strategy based on modular decomposition techniques; we slightly generalize the original problem as the input graph is not necessarily bipartite. The algorithm first builds a problem kernel with O(k ²) vertices in O(n+m) time, and then applies a bounded search tree. We also show how this strategy based on modular decomposition leads to a new way of obtaining a problem kernel with O(k ²) vertices for the Cluster(k) Graph Editing problem, in O(n+m) time. This problem consists of obtaining a cluster graph by modifying at most k edges in an input graph. A previous FPT algorithm of time O(1.92 ^k +n ³) for this problem was presented by Gramm et al. (Theory Comput. Syst. 38(4), 373–392, 2005, Algorithmica 39(4), 321–347, 2004). In their solution, a problem kernel with O(k ²) vertices is built in O(n ³) time.     

A Polynomial Kernel for Feedback Arc Set on Bipartite Tournaments

In the k-Feedback Arc/Vertex Set problem we are given a directed graph D and a positive integer k and the objective is to check whether it is possible to delete at most k arcs/vertices from D to make it acyclic. Dom et al. (J. Discrete Algorithm 8(1):76–86, 2010) initiated a study of the Feedback Arc Set problem on bipartite tournaments (k-FASBT) in the realm of parameterized complexity. They showed that k-FASBT can be solved in time O(3.373 ^k n ⁶) on bipartite tournaments having n vertices. However, until now there was no known polynomial sized problem kernel for k-FASBT. In this paper we obtain a cubic vertex kernel for k-FASBT. This completes the kernelization picture for the Feedback Arc/Vertex Set problem on tournaments and bipartite tournaments, as for all other problems polynomial kernels were known before. We obtain our kernel using a non-trivial application of “independent modules” which could be of independent interest.     

Vertex Cover Kernelization Revisited

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G′,k′) in polynomial time with the guarantee that G′ has at most 2k′ vertices (and thus $\mathcal{O}((k')^{2})$ edges) with k′≤k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Θ(k ²) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number $\mathop{\mathrm{\mbox {\textsc{vc}}}}(G)$ since $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\leq\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)$ and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ : an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G′,X′,k′) such that |V(G′)|≤2k and $|V(G')| \in\mathcal{O}(|X'|^{3})$ . A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP ? coNP/poly and the polynomial hierarchy collapses to the third level.     

Kernels for feedback arc set in tournaments

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T^′ on O(k) vertices. In fact, given any fixed ?>0, the kernelized instance has at most (2+?)k vertices. Our result improves the previous known bound of O(k²) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.     

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph. A preliminary version of this paper appeared in the Proceedings of 10th Scandinavian Workshop on Algorithm Theory (SWAT), Lecture Notes in Computer Science, vol. 4059, pp. 304–315, 2006.     

Quadratic Kernelization for Convex Recoloring of Trees

The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting a so-called “perfect phylogeny”. For an input consisting of a vertex-colored tree T, the problem is to determine whether recoloring at most k vertices can achieve a convex coloring, meaning by this a coloring where each color class induces a subtree. The problem was introduced by Moran and Snir (J. Comput. Syst. Sci. 73:1078–1089, 2007; J. Comput. Syst. Sci. 74:850–869, 2008) who showed that CR is NP-hard, and described a search-tree based FPT algorithm with a running time of O(k(k/log k) ^k n ⁴). The Moran and Snir result did not provide any nontrivial kernelization. In this paper, we show that CR has a kernel of size O(k ²).     

The super spanning connectivity and super spanning laceability of the enhanced hypercubes

A k -container C(u,v) of a graph G is a set of k disjoint paths between u and v. A k-container C(u,v) of G is a k ^* -container if it contains all vertices of G. A graph G is k ^* -connected if there exists a k ^*-container between any two distinct vertices of G. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is Hamiltonian connected (respectively, Hamiltonian). A graph G is super spanning connected if there exists a k ^*-container between any two distinct vertices of G for every k with 1≤k≤κ(G) where κ(G) is the connectivity of G. A bipartite graph G is k ^* -laceable if there exists a k ^*-container between any two vertices from different partite set of G. A bipartite graph G is super spanning laceable if there exists a k ^*-container between any two vertices from different partite set of G for every k with 1≤k≤κ(G). In this paper, we prove that the enhanced hypercube Q _n,m is super spanning laceable if m is an odd integer and super spanning connected if otherwise.

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?C207, 2010) initiated the study of the parameterized complexity of POVD, showing a quartic kernel and an algorithm which runs in time 7 ^k n ^O(1). In this article we improve these results by showing a quadratic kernel and an algorithm with time complexity 4.65 ^k n ^O(1), thus obtaining almost tight kernelization bounds when compared to the general result of Dell and van Melkebeek (STOC, pp.?251?C260, ACM, New York, 2010). Techniques used in the kernelization are based on the quadratic kernel for FVS, due to Thomassé (ACM Trans. Algorithms 6(2), 2010).     

A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R ⁺, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k _u . A solution consists of a function x:V→ℕ₀ and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x _u k _u . Our objective is to find a cover that minimizes ∑ _v∈V x _v w _v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time. Research supported by NSF Awards CCR 0113192 and CCF 0430650, and the University of Maryland Dean’s Dissertation Fellowship.     

Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization

We show that the NP-complete Feedback Vertex Set problem, which asks for the smallest set of vertices to remove from a graph to destroy all cycles, is deterministically solvable in O(ck⋅m) time. Here, m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. We extend this to an algorithm enumerating all solutions in O(dk⋅m) time for a (larger) constant d. As a further result, we present a fixed-parameter algorithm with runtime O(k²⋅m²) for the NP-complete Edge Bipartization problem, which asks for at most k edges to remove from a graph to make it bipartite.     

Exact Algorithms for <Emphasis Type="Italic">L</Emphasis>(2,1)-Labeling of Graphs

The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,…,k} is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k≥4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive O ^*((k+1) ⁿ ) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of k=4, where the running time of our algorithm is O(1.3006 ⁿ ). Furthermore we show that dynamic programming can be used to establish an O(3.8730 ⁿ ) algorithm to compute an optimal L(2,1)-labeling.     

An O(2<Superscript>O(k)</Superscript>n<Superscript>3</Superscript>) FPT Algorithm for the Undirected Feedback Vertex Set Problem

We describe an algorithm for the Feedback Vertex Set problem on undirected graphs, parameterized by the size k of the feedback vertex set, that runs in time O(c^kn³) where c = 10.567 and n is the number of vertices in the graph. The best previous algorithms were based on the method of bounded search trees, branching on short cycles. The best previous running time of an FPT algorithm for this problem, due to Raman, Saurabh and Subramanian, has a parameter function of the form 2^{O(k log k /log log k)}. Whether an exponentially linear in k FPT algorithm for this problem is possible has been previously noted as a significant challenge. Our algorithm is based on the new FPT technique of iterative compression. Our result holds for a more general form of the problem, where a subset of the vertices may be marked as forbidden to belong to the feedback set. We also establish "exponential optimality" for our algorithm by proving that no FPT algorithm with a parameter function of the form O(2^o(k)) is possible, unless there is an unlikely collapse of parameterized complexity classes, namely FPT = M[1].     

A linear time algorithm for the minimum Weighted Feedback Vertex Set on diamonds

Given an undirected and vertex weighted graph G, the Weighted Feedback Vertex Problem (WFVP) consists in finding a subset F⊆V of vertices of minimum weight such that each cycle in G contains at least one vertex in F. The WFVP on general graphs is known to be NP-hard. In this paper we introduce a new class of graphs, namely the diamond graphs, and give a linear time algorithm to solve WFVP on it.     

On Clark and Suen bound-type results for k-domination and Roman domination of Cartesian product graphs

A set S of vertices of a graph G is a dominating set for G if every vertex of G is adjacent to at least one vertex of S. The domination number γ(G), of G, is the minimum cardinality of a dominating set in G. Moreover, if the maximum degree of G is Δ, then for every positive integer k≤Δ, the set S is a k-dominating set in G if every vertex outside of S is adjacent to at least k vertices of S. The k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a k-dominating set in G. A map f: V→<texlscub>0, 1, 2</texlscub>is a Roman dominating function for G if for every vertex v with f(v)=0, there exists a vertex u∈N(v) such that f(u)=2. The weight of a Roman dominating function is f(V)=∑ _u∈V f(u). The Roman domination number γ_R(G), of G, is the minimum weight of a Roman dominating function on G. In this paper, we obtain that for any two graphs G and H, the k-domination number of the Cartesian product of G and H is bounded below by γ(G)γ _k (H)/2. Also, we obtain that the domination number of Cartesian product of G and H is bounded below by γ(G)γ_R(H)/3.     

Multicolorings of Series-Parallel Graphs

Let G be a graph, and let each vertex v of G have a positive integer weight ω(v). A multicoloring of G is to assign each vertex v a set of ω(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given series-parallel graph G with the minimum number of colors in time O(n W), where n is the number of vertices and W is the maximum weight of vertices in G.     

Parameterized algorithm for eternal vertex cover

In this paper we initiate the study of a “dynamic” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem introduced by Klostermeyer and Mynhardt, from the perspective of parameterized algorithms. This problem consists in placing a minimum number of guards on the vertices of a graph such that these guards can protect the graph from any sequence of attacks on its edges. In response to an attack, each guard is allowed either to stay in his vertex, or to move to a neighboring vertex. However, at least one guard has to fix the attacked edge by moving along it. The other guards may move to reconfigure and prepare for the next attack. Thus at every step the vertices occupied by guards form a vertex cover. We show that the problem admits a kernel of size k⁴(k+1)+2k, which shows that the problem is fixed parameter tractable when parameterized by the number of available guards k. Finally, we also provide an algorithm with running time O(²O(k²)+nm) for Eternal Vertex Cover, where n is the number of vertices and m the number of edges of the input graph. In passing we also observe that Eternal Vertex Cover is NP-hard, yet it has a polynomial time 2-approximation algorithm.