Improved Algorithms and Complexity Results for Power Domination in Graphs

The NP-complete Power Dominating Set problem is an “electric power networks variant” of the classical domination problem in graphs: Given an undirected graph G=(V,E), find a minimum-size set P?V such that all vertices in V are “observed” by the vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “bounded-treewidth dynamic programs.” For treewidth being upper-bounded by a constant, we achieve a linear-time algorithm. In particular, we present a simplified linear-time algorithm for Power Dominating Set in trees. Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by |P| is W[2]-hard and it cannot be better approximated than Dominating Set.     

On the (Non-)Existence of Polynomial Kernels for P l -Free Edge Modification Problems

Given a graph G=(V,E) and a positive integer k, an edge modification problem for a graph property Π consists in deciding whether there exists a set F of pairs of V of size at most k such that the graph $H=(V,E\vartriangle F)$ satisfies the property Π. In the Π edge-completion problem, the set F is constrained to be disjoint from E; in the Π edge-deletion problem, F is a subset of E; no constraint is imposed on F in the Π edge-editing problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity (Cai in Inf. Process. Lett. 58:171–176, 1996; Fellows et al. in FCT, pp. 312–321, 2007; Heggernes et al. in STOC, pp. 374–381, 2007). When parameterized by the size k of the set F, it has been proved that if Π is a hereditary property characterized by a finite set of forbidden induced subgraphs, then the three Π edge-modification problems are FPT (Cai in Inf. Process. Lett. 58:171–176, 1996). It was then natural to ask (Bodlaender et al. in IWPEC, 2006) whether these problems also admit a polynomial kernel. in polynomial time to an equivalent instance (G′,k′) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlström answered this question negatively (Kratsch and Wahlström in IWPEC, pp. 264–275, 2009). However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. question to characterize for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlström asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P ₄-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that Parameterized cograph edge-modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the P _l -free edge-deletion and the C _l -free edge-deletion problems for l?7 and l≥4 respectively. Indeed, if they exist, then NP?coNP/poly.     

The k-in-a-Path Problem for Claw-free Graphs

The k-in-a-Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP-complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw-free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k-Induced Disjoint Paths and the k-in-a-Cycle problem are solvable in polynomial time on claw-free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s _i ,t _i ) for i=1,…,k and is to test whether G contain k mutually induced paths P _i such that P _i connects s _i and t _i for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP-complete, even for the class of line graphs, which form a subclass of the class of claw-free graphs.     

On Cutwidth Parameterized by Vertex Cover

We study the Cutwidth problem, where the input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for Cutwidth with running time O(2 ^k n ^O(1)). Here k is the size of a minimum vertex cover of the input graph G, and n is the number of vertices in G. Our algorithm gives an O(2 ^n/2 n ^O(1)) time algorithm for Cutwidth on bipartite graphs as a corollary. This is the first non-trivial exact exponential time algorithm for Cutwidth on a graph class where the problem remains NP-complete. Additionally, we show that Cutwidth parameterized by the size of the minimum vertex cover of the input graph does not admit a polynomial kernel unless NP?coNP/poly. Our kernelization lower bound contrasts with the recent results of Bodlaender et al. (ICALP, Springer, Berlin, 2011; SWAT, Springer, Berlin, 2012) that both Treewidth and Pathwidth parameterized by vertex cover do admit polynomial kernels.     

The Kernelization Complexity of Connected Domination in Graphs with (no) Small Cycles

In the Parameterized Connected Dominating Set problem the input consists of a graph G and a positive integer k, and the question is whether there is a set S of at most k vertices in G—a connected dominating set of G—such that (i) S is a dominating set of G, and (ii) the subgraph G[S] induced by S is connected; the parameter is k. The underlying decision problem is a basic connectivity problem which is long known to be NP-complete, and it has been extensively studied using several algorithmic approaches. Parameterized Connected Dominating Set is W[2]-hard, and therefore it is unlikely (Downey and Fellows, Parameterized Complexity, Springer, 1999) that the problem has fixed-parameter tractable (FPT) algorithms or polynomial kernels in graphs in general. We investigate the effect of excluding short cycles, as subgraphs, on the kernelization complexity of Parameterized Connected Dominating Set. The girth of a graph G is the length of a shortest cycle in G. It turns out that the Parameterized Connected Dominating Set problem is hard on graphs with small cycles, and becomes progressively easier as the girth increases. More precisely, we obtain the following kernelization landscape: Parameterized Connected Dominating Set

• does not have a kernel of any size on graphs of girth three or four (since the problem is W[2]-hard);
• admits a kernel of size 2 ^k k ^3k on graphs of girth at least five;
• has no polynomial kernel (unless the Polynomial Hierarchy collapses to the third level) on graphs of girth at most six, and,
• has a cubic ( $\mathcal {O}(k^{3})$ ) vertex kernel on graphs of girth at least seven.

While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs.     

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.     

Detecting Fixed Patterns in Chordal Graphs in Polynomial Time

The Contractibility problem takes as input two graphs G and H, and the task is to decide whether H can be obtained from G by a sequence of edge contractions. The Induced Minor and Induced Topological Minor problems are similar, but the first allows both edge contractions and vertex deletions, whereas the latter allows only vertex deletions and vertex dissolutions. All three problems are NP-complete, even for certain fixed graphs H. We show that these problems can be solved in polynomial time for every fixed H when the input graph G is chordal. Our results can be considered tight, since these problems are known to be W[1]-hard on chordal graphs when parameterized by the size of H. To solve Contractibility and Induced Minor, we define and use a generalization of the classic Disjoint Paths problem, where we require the vertices of each of the k paths to be chosen from a specified set. We prove that this variant is NP-complete even when k=2, but that it is polynomial-time solvable on chordal graphs for every fixed k. Our algorithm for Induced Topological Minor is based on another generalization of Disjoint Paths called Induced Disjoint Paths, where the vertices from different paths may no longer be adjacent. We show that this problem, which is known to be NP-complete when k=2, can be solved in polynomial time on chordal graphs even when k is part of the input. Our results fit into the general framework of graph containment problems, where the aim is to decide whether a graph can be modified into another graph by a sequence of specified graph operations. Allowing combinations of the four well-known operations edge deletion, edge contraction, vertex deletion, and vertex dissolution results in the following ten containment relations: (induced) minor, (induced) topological minor, (induced) subgraph, (induced) spanning subgraph, dissolution, and contraction. Our results, combined with existing results, settle the complexity of each of the ten corresponding containment problems on chordal graphs.     

Property Testing on k-Vertex-Connectivity of?Graphs

We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called ε-far from k-vertex-connectivity when at least $\frac{\epsilon dn}{2}$ edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is ε-far from k-vertex-connectivity with a probability of at least 2/3. The algorithm runs in $O(d(\frac{c}{\epsilon d})^{k}\log\frac {1}{\epsilon d})$ time (c>1 is a constant) for (k?1)-vertex-connected graphs, and in $O(d(\frac{ck}{\epsilon d})^{k}\log\frac{k}{\epsilon d})$ time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4.     

Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems

In this paper we provide improved approximation algorithms for the Min-Max Tree Cover and Bounded Tree Cover problems. Given a graph G=(V,E) with weights w:E→?⁺, a set T ₁,T ₂,…,T _k of subtrees of G is called a tree cover of G if $V=\bigcup_{i=1}^{k} V(T_{i})$ . In the Min-Max k-tree Cover problem we are given graph G and a positive integer k and the goal is to find a tree cover with k trees, such that the weight of the largest tree in the cover is minimized. We present a 3-approximation algorithm for this improving the two different approximation algorithms presented in Arkin et al. (J. Algorithms 59:1–18, 2006) and Even et al. (Oper. Res. Lett. 32(4):309–315, 2004) with ratio 4. The problem is known to have an APX-hardness lower bound of $\frac{3}{2}$ (Xu and Wen in Oper. Res. Lett. 38:169–173, 2010). In the Bounded Tree Cover problem we are given graph G and a bound λ and the goal is to find a tree cover with minimum number of trees such that each tree has weight at most λ. We present a 2.5-approximation algorithm for this, improving the 3-approximation bound in Arkin et al. (J. Algorithms 59:1–18, 2006).     

Proper Interval Vertex Deletion

The NP-complete problem Proper Interval Vertex Deletion is to decide whether an input graph on n vertices and m edges can be turned into a proper interval graph by deleting at most k vertices. Van Bevern et al. (In: Proceedings WG 2010. Lecture notes in computer science, vol. 6410, pp. 232–243, 2010) showed that this problem can be solved in $\mathcal {O}((14k +14)^{k+1} kn^{6})$ time. We improve this result by presenting an $\mathcal {O}(6^{k} kn^{6})$ time algorithm for Proper Interval Vertex Deletion. Our fixed-parameter algorithm is based on a new structural result stating that every connected component of a {claw,net,tent,C ₄,C ₅,C ₆}-free graph is a proper circular arc graph, combined with a simple greedy algorithm that solves Proper Interval Vertex Deletion on {claw,net,tent,C ₄,C ₅,C ₆}-free graphs in $\mathcal {O}(n+m)$ time. Our approach also yields a polynomial-time 6-approximation algorithm for the optimization variant of Proper Interval Vertex Deletion.     

Orthogonal Graph Drawing with Flexibility Constraints

Traditionally, the quality of orthogonal planar drawings is quantified by either the total number of bends, or the maximum number of bends per edge. However, this neglects that in typical applications, edges have varying importance. In this work, we investigate an approach that allows to specify the maximum number of bends for each edge individually, depending on its importance. We consider a new problem called FlexDraw that is defined as follows. Given a planar graph G=(V,E) on n vertices with maximum degree 4 and a function $\operatorname{flex}: E \longrightarrow\mathbb{N}_{0}$ that assigns a flexibility to each edge, does G admit a planar embedding on the grid such that each edge e has at most $\operatorname{flex}(e)$ bends? Note that in our setting the combinatorial embedding of G is not fixed. FlexDraw directly extends the problem β-embeddability asking whether G can be embedded with at most β bends per edge. We give an algorithm with running-time O(n ²) solving FlexDraw when the flexibility of each edge is positive. This includes 1-embeddability as a special case and thus closes the complexity gap between 0-embeddability, which is $\mathcal{NP}$ -hard to decide, and 2-embeddability, which is efficiently solvable since every planar graph with maximum degree 4 admits a 2-embedding except for the octahedron. In addition to the polynomial-time algorithm we show that FlexDraw is $\mathcal{NP}$ -hard even if the edges with flexibility 0 induce a tree or a union of disjoint stars.     

A Cubic-Vertex Kernel for Flip Consensus Tree

Given a bipartite graph G=(V _c ,V _t ,E) and a nonnegative integer k, the NP-complete Minimum-Flip Consensus Tree problem asks whether G can be transformed, using up to k edge insertions and deletions, into a graph that does not contain an induced P ₅ with its first vertex in V _t (a so-called M-graph or Σ-graph). This problem plays an important role in computational phylogenetics, V _c standing for the characters and V _t standing for taxa. Chen et al. (IEEE/ACM Trans. Comput. Biol. Bioinform. 3:165–173, 2006). showed that Minimum-Flip Consensus Tree is NP-complete and presented a parameterized algorithm with running time O(6 ^k ?|V _t |?|V _c |). Subsequently, Böcker et al. (ACM Trans. Algorithms 8:7:1–7:17, 2012) presented a refined search tree algorithm with running time O(4.42 ^k (|V _t |+|V _c |)+|V _t |?|V _c |). We continue the study of Minimum-Flip Consensus Tree parameterized by k. Our main contribution are polynomial-time executable data reduction rules yielding a problem kernel with O(k ³) vertices. In addition, we present an improved search tree algorithm with running time O(3.68 ^k ?|V _c |²|V _t |).     

Maximum Series-Parallel Subgraph

Consider the NP-hard problem of, given a simple graph?G, to find a series-parallel subgraph of?G with the maximum number of edges. The algorithm that, given a connected graph?G, outputs a spanning tree of?G, is a $\frac{1}{2}$ -approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has?n?1 edges and any series-parallel graph on?n vertices has at most?2n?3 edges. We present a $\frac{7}{12}$ -approximation for this problem and results showing the limits of our approach.     

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.     

An Exact Exponential Time Algorithm for Power Dominating Set

The Power Dominating Set problem is an extension of the well-known domination problem on graphs in a way that we enrich it by a second propagation rule: given a graph G(V,E), a set P?V is a power dominating set if every vertex is observed after the exhaustive application of the following two rules. First, a vertex is observed if v∈P or it has a neighbor in P. Secondly, if an observed vertex has exactly one unobserved neighbor u, then also u will be observed, as well. We show that Power Dominating Set remains $\mathcal{NP}$ -hard on cubic graphs. We design an algorithm solving this problem in time $\mathcal{O}^{*}(1.7548^{n})$ on general graphs, using polynomial space only. To achieve this, we introduce so-called reference search trees that can be seen as a compact representation of usual search trees, providing non-local pointers in order to indicate pruned subtrees.     

2-Layer Right Angle Crossing Drawings

A 2-layer drawing represents a bipartite graph where each vertex is a point on one of two parallel lines, no two vertices on the same line are adjacent, and the edges are straight-line segments. In this paper we study 2-layer drawings where any two crossing edges meet at right angle. We characterize the graphs that admit this type of drawing, provide linear-time testing and embedding algorithms, and present a polynomial-time crossing minimization technique. Also, for a given graph G and a constant k, we prove that it is $\mathcal{NP}$ -complete to decide whether G contains a subgraph of at least k edges having a 2-layer drawing with right angle crossings.     

Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs

Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E _r ^k ) whereE _r ^k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE _r ¹⁷ . We also prove that ¦E _r ^k ¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n ²) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)^0.5 m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n ^1.5 log^0.5 n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n ² +n ^1.5 log^0.5 n).     

Algorithms for Placing Monitors in a Flow Network

In the Flow Edge-Monitor Problem, we are given an undirected graph G=(V,E), an integer k>0 and some unknown circulation ψ on G. We want to find a set of k edges in G, so that if we place k monitors on those edges to measure the flow along them, the total number of edges for which the flow can be uniquely determined is maximized. In this paper, we first show that the Flow Edge-Monitor Problem is NP-hard. Then we study an algorithm called σ-Greedy that, in each step, places monitors on σ edges for which the number of edges where the flow is determined is maximized. We show that the approximation ratio of 1-Greedy is 3 and that the approximation ratio of 2-Greedy is 2.     

Finding Maximum Induced Matchings in Subclasses of Claw-Free and <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs,and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P ₅,D _m )-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.     

Optimal 1-edge fault-tolerant designs for ladders

A graph G∗ is 1-edge fault-tolerant with respect to a graph G, denoted by 1-EFT(G), if every graph obtained by removing any edge from G∗ contains G. A 1-EFT(G) graph is optimal if it contains the minimum number of edges among all 1-EFT(G) graphs. The kth ladder graph, L_k, is defined to be the cartesian product of the P_k and P₂ where P_n is the n-vertex path graph. In this paper, we present several 1-edge fault-tolerant graphs with respect to ladders. Some of these graphs are proven to be optimal.