On the existence of subexponential parameterized algorithms

The existence of subexponential-time parameterized algorithms is examined for various parameterized problems solvable in time O(2^O(k)p(n)). It is shown that for each t?1, there are parameterized problems in FPT for which the existence of O(2^o(k)p(n))-time parameterized algorithms implies the collapse of W[t] to FPT. Evidence is demonstrated that Max-SNP-hard optimization problems do not admit subexponential-time parameterized algorithms. In particular, it is shown that each Max-SNP-complete problem is solvable in time O(2^o(k)p(n)) if and only if 3-SAT∈DTIME(2^o(n)). These results are also applied to show evidence for the non-existence of -time parameterized algorithms for a number of other important problems such as Dominating Set, Vertex Cover, and Independent Set on planar graph instances.     

Implicit branching and parameterized partial cover problems

Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well-known problems like Set Cover, Vertex Cover, Dominating Set and Facility Location to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (non-partial) version of all these problems has been intensively studied in planar graphs and in graphs excluding a fixed graph H as a minor. However, the techniques developed for parameterized version of non-partial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems.     

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.     

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.     

Labeled Search Trees and Amortized Analysis: Improved Upper Bounds for NP-Hard Problems

A sequence of exact algorithms to solve the Vertex Cover and Maximum Independent Set problems have been proposed in the literature. All these algorithms appeal to a very conservative analysis that considers the size of the search tree, under a worst-case scenario, to derive an upper bound on the running time of the algorithm. In this paper we propose a different approach to analyze the size of the search tree. We use amortized analysis to show how simple algorithms, if analyzed properly, may perform much better than the upper bounds on their running time derived by considering only a worst-case scenario. This approach allows us to present a simple algorithm of running time O(1.194^kk² + n) for the parameterized Vertex Cover problem on degree-3 graphs, and a simple algorithm of running time O(1.1255ⁿ) for the Maximum Independent Set problem on degree-3 graphs. Both algorithms improve the previous best algorithms for the problems.     

Dominating set is fixed parameter tractable in claw-free graphs

We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K_1,3, the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of Dominating Set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general.The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of K_i,j-free graphs, which exclude, for some fixed i,j∈N, the complete bipartite graph K_i,j as a subgraph. For i,j≥2, the class of claw-free graphs and any class of K_i,j-free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable.We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any t≥4, the Dominating Set problem parameterized by the solution size is W[2]-hard in graphs that exclude the t-claw K_1,t as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W[2]-hard in these graph classes.Finally, we show that for any t∈N, the Clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.     

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.     

Parameterized Measure & Conquer for Problems with No Small Kernels

Measure & Conquer (M&C) is a prominent technique for analyzing exact algorithms for computationally hard problems, in particular, graph problems. It tries to balance worse and better situations within the algorithm analysis. This has led, e.g., to algorithms for Minimum Vertex Cover with a running time of $\mathcal{O}(c^{n})$ for some constant c??1.2, where n is the number of vertices in the graph. Several obstacles prevent the application of this technique in parameterized algorithmics, making it rarely applied in this area. However, these difficulties can be handled in some situations. We will exemplify this with two problems related to Vertex Cover, namely Connected Vertex Cover and Edge Dominating Set. For both problems, several parameterized algorithms have been published, all based on the idea of first enumerating minimal vertex covers. Using M&C in this context will allow us to improve on the hitherto published running times. In contrast to some of the earlier suggested algorithms, ours will use polynomial space.     

Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG

We give improved parameterized algorithms for two “edge” problems MAXCUT and MAXDAG, where the solution sought is a subset of edges. MAXCUT of a graph is a maximum set of edges forming a bipartite subgraph of the given graph. On the other hand, MAXDAG of a directed graph is a set of arcs of maximum size such that the graph induced on these arcs is acyclic. Our algorithms are obtained through new kernelization and efficient exact algorithms for the optimization versions of the problems. More precisely our results include:

(i)

a kernel with at most αk vertices and βk edges for MAXCUT. Here 0<α?1 and 1<β?2. Values of α and β depends on the number of vertices and the edges in the graph;

(ii)

a kernel with at most 4k/3 vertices and 2k edges for MAXDAG;

(iii)

an O^∗(k^1.2418) parameterized algorithm for MAXCUT in undirected graphs. This improves the O^∗(k^1.4143)¹ algorithm presented in [E. Prieto, The method of extremal structure on the k-maximum cut problem, in: The Proceedings of Computing: The Australasian Theory Symposium (CATS), 2005, pp. 119-126];

(iv)

an O^∗(n²) algorithm for optimization version of MAXDAG in directed graphs. This is the first such algorithm to the best of our knowledge;

(v)

an O^∗(k²) parameterized algorithm for MAXDAG in directed graphs. This improves the previous best of O^∗(k⁴) presented in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446-458];

(vi)

an O^∗(k¹⁶) parameterized algorithm to determine whether an oriented graph having m arcs has an acyclic subgraph with at least m/2+k arcs. This improves the O^∗(k²) algorithm given in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446-458].

In addition, we show that if a directed graph has minimum out degree at least f(n) (some function of n) then Directed Feedback Arc Set problem is fixed parameter tractable. The parameterized complexity of Directed Feedback Arc Set is a well-known open problem.     

Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs

The densest k-subgraph problem asks for a k-vertex subgraph with the maximum number of edges. This problem is NP-hard on bipartite graphs, chordal graphs, and planar graphs. A 3-approximation algorithm is known for chordal graphs. We present -approximation algorithms for proper interval graphs and bipartite permutation graphs. The latter result relies on a new characterisation of bipartite permutation graphs which may be of independent interest.     

Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  $O(2^{6.903\sqrt{n}})We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  O(2^6.903?n)O(2^{6.903\sqrt{n}}) time algorithm solving weighted Hamiltonian Cycle on an n-vertex planar graph. Similar technique solves Planar Graph Travelling Salesman Problem with n cities in time O(2^9.8594?n)O(2^{9.8594\sqrt{n}}) . Our approach can be used to design parameterized algorithms as well. For example, we give an algorithm that for a given k decides if a planar graph on n vertices has a cycle of length at least k in time O(2^13.6?kn+n³)O(2^{13.6\sqrt{k}}n+n^{3}) .     

Computational study on planar dominating set problem

Recently, there has been significant theoretical progress towards fixed-parameter algorithms for the DOMINATING SET problem of planar graphs. It is known that the problem on a planar graph with n vertices and dominating number k can be solved in time using tree/branch-decomposition based algorithms. In this paper, we report computational results of Fomin and Thilikos algorithm which uses the branch-decomposition based approach. The computational results show that the algorithm can solve the DOMINATING SET problem of large planar graphs in a practical time and memory space for the class of graphs with small branchwidth. For the class of graphs with large branchwidth, the size of instances that can be solved by the algorithm in practice is limited to about one thousand edges due to a memory space bottleneck. The practical performances of the algorithm coincide with the theoretical analysis of the algorithm. The results of this paper suggest that the branch-decomposition based algorithms can be practical for some applications on planar graphs.     

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.     

Inclusion/Exclusion Meets Measure and Conquer

Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times. In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number. This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.     

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51–63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction:

• Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution.
• Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs.
• If T is a given tree, deciding whether a circle graph G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t=|V(T)|. We prove that the FPT algorithm runs in subexponential time, namely $2^{\mathcal{O}(t \cdot\frac{\log\log t}{\log t})} \cdot n^{\mathcal{O}(1)}$ , where n=|V(G)|.

     

Visibility representation of plane graphs via canonical ordering tree

In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [Rectilinear planar layouts and bipolar orientations of planar graphs, Discrete Comput. Geom. 1 (1986) 343], Tamassia and Tollis [An unified approach to visibility representations of planar graphs, Discrete Comput. Geom. 1 (1986) 321] independently gave linear time VR algorithms for 2-connected plane graph. Afterwards, one of the main concerns for VR is the size of the representation. In this paper, we prove that any plane graph G has a VR with height bounded by . This improves the previously known bound . We also construct a plane graph G with n vertices where any VR of G requires a size of . Our result provides an answer to Kant's open question about whether there exists a plane graph G such that all of its VR require width greater that cn, where c>1 [G. Kant, A more compact visibility representation, Internat. J. Comput. Geom. Appl. 7 (1997) 197].     

Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques

We give substantially improved exact exponential-time algorithms for a number of NP-hard problems. These algorithms are obtained using a variety of techniques. These techniques include: obtaining exact algorithms by enumerating maximal independent sets in a graph, obtaining exact algorithms from parameterized algorithms and a variant of the usual branch-and-bound technique which we call the "colored" branch-and-bound technique. These techniques are simple in that they avoid detailed case analyses and yield algorithms that can be easily implemented. We show the power of these techniques by applying them to several NP-hard problems and obtaining new improved upper bounds on the running time. The specific problems that we tackle are: (1) the Odd Cycle Transversal problem in general undirected graphs, (2) the Feedback Vertex Set problem in directed graphs of maximum degree 4, (3) Feedback Arc Set problem in tournaments, (4) the 4-Hitting Set problem and (5) the Minimum Maximal Matching and the Edge Dominating Set problems. The algorithms that we present for these problems are the best known and are a substantial improvement over previous best results. For example, for the Minimum Maximal Matching we give an O^*(1.4425ⁿ) algorithm improving the previous best result of O^*(1.4422^m) [35]. For the Odd Cycle Transversal problem, we give an O^*(1.62ⁿ) algorithm which improves the previous time bound of O^*(1.7724ⁿ) [3].     

Parameterized complexity of connected even/odd subgraph problems

In 2011, Cai an Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs: for a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about:

•

a connected k-edge subgraph with all vertices of odd degrees, the problem known as k-Edge Connected Odd Subgraph; and     

A generalization of Nemhauser and Trotter?s local optimization theorem

The Nemhauser–Trotter local optimization theorem applies to the NP-hard Vertex Cover problem and has applications in approximation as well as parameterized algorithmics. We generalize Nemhauser and Trotter?s result to vertex deletion problems, introducing a novel algorithmic strategy based on purely combinatorial arguments (not referring to linear programming as the Nemhauser–Trotter result originally did). The essence of our strategy can be understood as a doubly iterative process of cutting away “easy parts” of the input instance, finally leaving a “hard core” whose size is (almost) linearly related to the cardinality of the solution set. We exhibit our approach using a generalization of Vertex Cover, called Bounded-Degree Vertex Deletion. For some fixed d?0, Bounded-Degree Vertex Deletion asks to delete at most k vertices from a graph in order to transform it into a graph with maximum vertex degree at most d. Vertex Cover is the special case of d=0. Our generalization of the Nemhauser–Trotter-Theorem implies that Bounded-Degree Vertex Deletion, parameterized by k, admits an O(k)-vertex problem kernel for d?1 and, for any ?>0, an O(k1+?)-vertex problem kernel for d?2. Finally, we provide a W[2]-completeness result for Bounded-Degree Vertex Deletion in case of unbounded d-values.