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RNA-Seq technology offers new high-throughput ways for transcript identification and quantification based on short reads, and has recently attracted great interest. This is achieved by constructing a weighted DAG whose vertices stand for exons, and whose arcs stand for split alignments of the RNA-Seq reads to the exons. The task consists of finding a number of paths, together with their expression levels, which optimally explain the weights of the graph under various fitting functions, such as least sum of squared residuals. In (Tomescu et al. BMC Bioinformatics, 2013) we studied this genome-guided multi-assembly problem when the number of allowed solution paths was linear in the number of arcs. In this paper, we further refine this problem by asking for a bounded number $k$ of solution paths, which is the setting of most practical interest. We formulate this problem in very broad terms, and show that for many choices of the fitting function it becomes NP-hard. Nevertheless, we identify a natural graph parameter of a DAG $G$ , which we call arc-width and denote $\langle G\rangle$ , and give a dynamic programming algorithm running in time $O(W^k\langle G\rangle ^k(\langle G\rangle + k)n)$ , where $n$ is the number of vertices and $W$ is the maximum weight of $G$ . This implies that the problem is fixed-parameter tractable (FPT) in the parameters $W$ , $\langle G\rangle$ , and $k$ . We also show that the arc-width of DAGs constructed from simulated and real RNA-Seq reads is small in practice. Finally, we study the approximability of this problem, and, in particular, give a fully polynomial-time approximation scheme (FPTAS) for the case when the fitting function penalizes the maximum ratio between the weights of the arcs and their predicted coverage. [ABSTRACT FROM PUBLISHER]

In the PROPER INTERVAL COMPLETION problem we are given a graph G and an integer k, and the task is to turn G using at most k edge additions into a proper interval graph, i.e., a graph admitting an intersection model of equal-length intervals on a line. The study of PROPER INTERVAL COMPLETION from the viewpoint of parameterized complexity has been initiated by Kaplan, Shamir, and Tarjan [SIAM J. Comput., 28 (1999), pp. 1906-1922], who showed an algorithm for the problem working in O(16k (n+m)) time. In this paper we present an algorithm with running time kO(k2/3) + O(nm(kn + m)), which is the first subexponential parameterized algorithm for PROPER INTERVAL COMPLETION. [ABSTRACT FROM AUTHOR]

We study the computational complexity of the puzzle Quell. The goal is to collect pearls by sliding a droplet of water over them in a map that is a two-dimensional grid. Each cell of the grid is either a free space (possibly with a pearl in it) or an obstacle. In each move, the droplet slides in one of the four directions to the maximal extent, through the free spaces of the map and collecting all pearls along the way, until it is stopped by an obstacle. We show that any-Moves-all-Pearls (deciding whether it is possible to collect all the pearls using any number of moves) can be solved in polynomial time. In contrast, both any-Moves-max-Pearls (finding the maximum number of pearls that can be collected using any number of moves) and min-Moves-all-Pearls (finding the minimum number of moves required to collect all the pearls) are APX-hard, although the corresponding decision problems are in FPT. We also present a simple 2-approximation for any-Moves-max-Pearls , and leave open the question whether min-Moves-all-Pearls admits a polynomial-time constant approximation. [ABSTRACT FROM AUTHOR]

We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut parameterized above the Edwards-Erdős bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size in time 2⋅ n, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed a number of times over the past 15 years. Our algorithm has asymptotically optimal running time, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size. [ABSTRACT FROM AUTHOR]

The NP-hard 2- Club problem is, given an undirected graph G = ( V , E ) and ℓ ∈ N , to decide whether there is a vertex set S ⊆ V of size at least ℓ such that the induced subgraph G [ S ] has diameter at most two. We make progress towards a systematic classification of the complexity of 2- Club with respect to a hierarchy of prominent structural graph parameters. First, we present the following tight NP-hardness results: 2- Club is NP-hard on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter three. Then, we consider the parameter h - index of the input graph. The study of this parameter is motivated by real-world instances and the fact that 2- Club is fixed-parameter tractable when parameterized by the larger parameter maximum degree . We present an algorithm that solves 2- Club in | V | f ( k ) time with k being the h - index of G . By showing W[1]-hardness for this parameter, we provide evidence that the above algorithm cannot be improved to a fixed-parameter algorithm. Furthermore, the reduction used for this hardness result can be modified to show that 2- Club is NP-hard if the input graph has constant degeneracy . Finally, we show that 2- Club is fixed-parameter tractable when parameterized by distance to cographs . [ABSTRACT FROM AUTHOR]

This work presents new results on flood-filling games, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. A flooding move consists of changing the color of the monochromatic component containing a vertex p (the pivot of the move). These games are originally played on grids; however, when played on trees, we have interesting applications and significant effects on problem complexity. In this paper, a complete mapping of the complexity of flood-filling games on trees is made, charting the consequences of single and aggregate parameterizations by: number of colors, number of moves, maximum distance from the pivot, number of occurrences of a color, number of leaves, and difference between number of moves and number of colors. We show that Flood-It on trees and Restricted Shortest Common Supersequence (RSCS) are analogous problems, in the sense that they can be translated from one to another, preserving complexity issues; this implies interesting FPT and W[1]-hard cases to Flood-It. Restricting attention to phylogenetic colored trees (where each color occurs at most once in any root-leaf path, in order to model phylogenetic sequences), we also show some impressive NP-hard and FPT results for both games. In addition, we prove that Flood-It and Free-Flood-It remain NP-hard when played on 3-colored trees, which closes an open question posed by Fleischer and Woeginger. Finally, we present a general framework for reducibility from Flood-It to Free-Flood-It; some NP-hard cases for Free-Flood-It can be derived using this approach. [ABSTRACT FROM AUTHOR]

The NP-hard RAINBOW SUBGRAPH problem, motivated from bioinformatics, is to find in an edge-colored graph a subgraph that contains each edge color exactly once and has at most k vertices. We examine the parameterized complexity of RAINBOW SUBGRAPH for paths, trees, and general graphs. We show that RAINBOW SUBGRAPH is W[1]-hard with respect to the parameter k and also with respect to the dual parameter ℓ := n - k where n is the number of vertices. Hence, we examine parameter combinations and show, for example, a polynomial-size problem kernel for the combined parameter ℓ and "maximum number of colors incident with any vertex". Additionally, we show APX-hardness even if the input graph is a properly edge-colored path in which every color occurs at most twice. [ABSTRACT FROM AUTHOR]

Homogeneous team formation is the task of grouping individuals into teams, each of which consists of members who fulfill the same set of prespecified properties. In this theoretical work, we propose, motivate, and analyze a combinatorial model where, given a matrix over a finite alphabet whose rows correspond to individuals and columns correspond to attributes of individuals, the user specifies lower and upper bounds on team sizes as well as combinations of attributes that have to be homogeneous (that is, identical) for all members of the corresponding teams. Furthermore, the user can define a cost for assigning any individual to a certain team. We show that some special cases of our new model lead to NP-hard problems while others allow for (fixed-parameter) tractability results. For example, the problem is already NP-hard even if (i) there are no lower and upper bounds on the team sizes, (ii) all costs are zero, and (iii) the matrix has only two columns. In contrast, the problem becomes fixed-parameter tractable for the combined parameter 'number of possible teams' and 'number of different individuals', the latter being upper-bounded by the number of rows. [ABSTRACT FROM AUTHOR]

We study parameterized algorithms and approximation algorithms for the maximum agreement forest problem, which, for two given leaf-labeled trees, is to find a maximum forest that is a subgraph of both trees. The problem was motivated by research in phylogenetics. For parameterized algorithms, while the problem is known to be fixed-parameter tractable for binary trees, it was an open problem whether the problem is still fixed-parameter tractable for general trees. We resolve this open problem by developing an O ( 3 k n ) -time parameterized algorithm for general trees. Our techniques on tree structures also lead to a polynomial-time approximation algorithm of ratio 3 for the problem, giving the first constant-ratio approximation algorithm for general trees. [ABSTRACT FROM AUTHOR]

Phylogenetic networks are used to display the relationship among different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as maximum parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of maximum parsimony on networks, "hardwired" and "softwired," and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multiterminal Cut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and the softwired parsimony scores can be computed efficiently using integer linear programming. The software has been made freely available. [ABSTRACT FROM AUTHOR]

The NP-hard Subset Interconnection Design problem, also known as Minimum Topic-Connected Overlay, is motivated by numerous applications including the design of scalable overlay networks and vacuum systems. It has as input a finite set V and a collection of subsets V1, V2, . . ., Vm ⊆ V, and asks for a minimum-cardinality edge set E such that for the graph G = (V,E) all induced subgraphs G[V1],G[V2], . . .,G[Vm] are connected. We study Subset Interconnection Design in the context of polynomial-time data reduction rules that preserve the possibility of constructing optimal solutions. Our contribution is threefold: First, we show the incorrectness of earlier polynomial-time data reduction rules. Second, we show linear-time solvability in case of a constant number m of subsets, implying fixed-parameter tractability for the parameter m. Third, we provide a fixed-parameter tractability result for small subset sizes and tree-like output graphs. To achieve our results, we elaborate on polynomial-time data reduction rules which also may be of practical use in solving Subset Interconnection Design. [ABSTRACT FROM AUTHOR]

The Multicut problem, given a graph G, a set of terminal pairs T = {(si, ti) Ι 1 ≤ i ≤ r}, and an integer p, asks whether one can find a cutset consisting of at most p nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, ti is not reachable from si for each 1 ≤ i ≤ r. The fixed-parameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355--388] and, independently, by Bousquet, Daligault, and Thomassé [Proceedings of STOC, ACM, 2011, pp. 459-468], after resisting attacks as a long-standing open problem. In this paper we prove that Multicut is fixed-parameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains W[1]-hard. [ABSTRACT FROM AUTHOR]

We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph H as a minor to graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either "almost embeddable" to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter |H|. We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a "typical" application of the structure theorem, we show that on graphs excluding H as a topological subgraph, Partial Dominating Set (find k vertices whose closed neighborhood has maximum size) can be solved in time f(H,k)· nO(1). More significantly, we show that on graphs excluding H as a topological subgraph, Graph Isomorphism can be solved in time nf(H). This result unifies and generalizes two previously known important polynomial time solvable cases of Graph Isomorphism: boundeddegree graphs [E. M. Luks, J. Comput. System Sci., 25 (1982), pp. 42-65] and H-minor free graphs [I. N. Ponomarenko, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 174 (1988), pp. 147-177, 182]. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition. [ABSTRACT FROM AUTHOR]

For any integer k ⩾ 3 , we consider the following decision problem. Given a simple graph, does there exist a partition of its vertices into two disjoint sets such that every simple k -cycle of G contains vertices in both of these sets? This problem is NP-hard because it admits a polynomial reduction from NAE 3-SAT. We construct a reduction that is polynomial both in the length of the instance and in k , which answers a recent question of Karpiński. [ABSTRACT FROM AUTHOR]

Co-clustering, that is partitioning a numerical matrix into “homogeneous” submatrices, has many applications ranging from bioinformatics to election analysis. Many interesting variants of co-clustering are NP-hard. We focus on the basic variant of co-clustering where the homogeneity of a submatrix is defined in terms of minimizing the maximum distance between two entries. In this context, we spot several NP-hard, as well as a number of relevant polynomial-time solvable special cases, thus charting the border of tractability for this challenging data clustering problem. For instance, we provide polynomial-time solvability when having to partition the rows and columns into two subsets each (meaning that one obtains four submatrices). When partitioning rows and columns into three subsets each, however, we encounter NP-hardness, even for input matrices containing only values from {0, 1, 2}.

The Induced Graph Matching problem asks to find $$k$$ disjoint induced subgraphs isomorphic to a given graph $$H$$ in a given graph $$G$$ such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in $$k$$ on claw-free graphs when $$H$$ is a fixed connected graph, and even admits a polynomial kernel when $$H$$ is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove $$\mathsf {W}[1]$$ -hardness of Induced Graph Matching on graphs excluding $$K_{1,4}$$ as an induced subgraph, for any fixed complete graph $$H$$ . In particular, we show that Independent Set is $$\mathsf {W}[1]$$ -hard on $$K_{1,4}$$ -free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or $$\mathsf {NP}$$ -complete, depending on the connectivity of $$H$$ and the structure of $$G$$ . [ABSTRACT FROM AUTHOR]

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems. [ABSTRACT FROM AUTHOR]

The Maximum Agreement Forest problem ( MAF ) asks for a largest common subforest of a collection of phylogenetic trees. The MAF problem on two binary phylogenetic trees has been studied extensively in the literature. In this paper, we present a group of fixed-parameter tractable algorithms for the MAF problem on multiple (i.e., two or more) binary phylogenetic trees. Our techniques work fine for the problem for both rooted trees and unrooted trees. The computational complexity of our algorithms is comparable with that of the known algorithms for two trees, and is independent of the number of phylogenetic trees for which a maximum agreement forest is constructed. [ABSTRACT FROM AUTHOR]

We study an extensive class of movement minimization problems that arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents (representing robots, people, map labels, network messages, etc.) to achieve a global property in the network while minimizing the maximum or average movement (expended energy). The only previous theoretical results about this class of problems are about approximation and are mainly negative: many movement problems of interest have polynomial inapproximability. Given that the number of mobile agents is typically much smaller than the complexity of the environment, we turn to fixed-parameter tractability. We characterize the boundary between tractable and intractable movement problems in a very general setup: it turns out the complexity of the problem fundamentally depends on the treewidth of the minimal configurations. Thus, the complexity of a particular problem can be determined by answering a purely combinatorial question. Using our general tools, we determine the complexity of several concrete problems and fortunately show that many movement problems of interest can be solved efficiently. [ABSTRACT FROM AUTHOR]

We describe an algorithm for the maximum clique problem that is parameterized by the graph's degeneracy $$d$$ . The algorithm runs in $$O\left( nm+n T_d \right) $$ time, where $$T_d$$ is the time to solve the maximum clique problem in an arbitrary graph on $$d$$ vertices. The best bound as of now is $$T_d=O(2^{d/4})$$ by Robson. This shows that the maximum clique problem is solvable in $$O(nm)$$ time in graphs for which $$d \le 4 \log _2 m + O(1)$$ . The analysis of the algorithm's runtime is simple; the algorithm is easy to implement when given a subroutine for solving maximum clique in small graphs; it is easy to parallelize. In the case of Bianconi-Marsili power-law random graphs, it runs in $$2^{O(\sqrt{n})}$$ time with high probability. We extend the approach for a graph invariant based on common neighbors, generating a second algorithm that has a smaller exponent at the cost of a larger polynomial factor. [ABSTRACT FROM AUTHOR]

A popular clustering algorithm for biological networks which was proposed by Hartuv and Shamir refid="ref5"/ identifies nonoverlapping highly connected components. We extend the approach taken by this algorithm by introducing the combinatorial optimization problem Highly Connected Deletion, which asks for removing as few edges as possible from a graph such that the resulting graph consists of highly connected components. We show that Highly Connected Deletion is NP-hard and provide a fixed-parameter algorithm and a kernelization. We propose exact and heuristic solution strategies, based on polynomial-time data reduction rules and integer linear programming with column generation. The data reduction typically identifies 75  percent of the edges that are deleted for an optimal solution; the column generation method can then optimally solve protein interaction networks with up to 6,000 vertices and 13,500 edges within five hours. Additionally, we present a new heuristic that finds more clusters than the method by Hartuv and Shamir. [ABSTRACT FROM PUBLISHER]

Abstract: A -cocoloring of a graph is a partition of the vertex set of into at most independent sets and at most cliques. It is known that determining the cochromatic number and the split chromatic number, which are respectively the minimum and the minimum such that is -cocolorable, is NP-hard problem. A -graph is a graph such that every subset of at most vertices induces at most distinct ’s. In 2011, Bravo et al. obtained a polynomial time algorithm to decide if a -graph is -cocolorable (Bravo et al., 2011). In this paper, we extend this result by obtaining polynomial time algorithms to decide the -cocolorability and to determine the cochromatic number and the split chromatic number for -graphs for every fixed and for graphs with bounded treewidth. We also obtain a polynomial time algorithm to obtain the maximum -cocolorable subgraph of a -graph for every fixed . All these algorithms are fixed parameter tractable. [Copyright &y& Elsevier]

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, ) 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 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. [ABSTRACT FROM AUTHOR]