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Abstract: We provide a new characterization of the NP-hard arc routing problem Rural Postman in terms of a constrained variant of minimum-weight perfect matching on bipartite graphs. To this end, we employ a parameterized equivalence between Rural Postman and Eulerian Extension, a natural arc addition problem in directed multigraphs. We indicate the NP-hardness of the introduced matching problem. In particular, we use the matching problem to make partial progress towards answering the open question about the parameterized complexity of Rural Postman with respect to the parameter “number of weakly connected components in the graph induced by the required arcs”. This is a more than thirty years open and long-neglected question with significant practical relevance. [Copyright &y& Elsevier]

Abstract: The NP-hard Interval Constrained Coloring (ICC) problem appears in the interpretation of experimental data in biochemistry dealing with protein fragments. Given a set of m integer intervals in the range 1 to n and a set of m associated multisets of colors (specifying for each interval the colors to be used for its elements), one asks whether there is a “consistent” coloring for all integer points from that complies with the constraints specified by the color multisets. We thoroughly analyze a known NP-hardness proof for ICC. In this way, we identify numerous parameters that naturally occur in ICC and strongly influence its practical solvability. Accordingly, we present several positive (fixed-parameter) tractability results exploiting various parameterizations. We substantiate the usefulness of this “multivariate algorithmics approach” by presenting experimental results with real-world data. [Copyright &y& Elsevier]

Abstract: Complementing recent progress on classical complexity and polynomial-time approximability of feedback set problems in (bipartite) tournaments, we extend and improve fixed-parameter tractability results for these problems. We show that Feedback Vertex Set in tournaments (FVST) is amenable to the novel iterative compression technique, and we provide a depth-bounded search tree for Feedback Arc Set in bipartite tournaments based on a new forbidden subgraph characterization. Moreover, we apply the iterative compression technique to d-Hitting Set, which generalizes Feedback Vertex Set in tournaments, and obtain improved upper bounds for the time needed to solve 4-Hitting Set and 5-Hitting Set. Using our parameterized algorithm for Feedback Vertex Set in tournaments, we also give an exact (not parameterized) algorithm for it running in time, where n is the number of input graph vertices, answering a question of Woeginger [G.J. Woeginger, Open problems around exact algorithms, Discrete Appl. Math. 156 (3) (2008) 397–405]. [Copyright &y& Elsevier]

Abstract: Set Cover problems are of core importance in many applications. In recent research, the “red-blue variants” where blue elements all need to be covered whereas red elements add further constraints on the optimality of a covering have received considerable interest. Application scenarios range from data mining to interference reduction in cellular networks. As a rule, these problem variants are computationally at least as hard as the original set cover problem. In this work we investigate whether and how the well-known consecutive ones property, restricting the structure of the input sets, makes the red-blue covering problems feasible. We explore a sharp border between polynomial-time solvability and NP-hardness for these problems. [Copyright &y& Elsevier]

Abstract: We introduce an NP-complete special case of the Weighted Set Cover problem and show its fixed-parameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given collection C of subsets of a base set S should be “tree-like”. That is, the subsets in C can be organized in a tree T such that every subset one-to-one corresponds to a tree node and, for each element s of S, the nodes corresponding to the subsets containing s induce a subtree of T. This is equivalent to the problem of finding a minimum edge cover in an edge-weighted acyclic hypergraph. Our main result is an algorithm running in time where k denotes the maximum subset size, , and . The algorithm also implies a fixed-parameter tractability result for the NP-complete Multicut in Trees problem, complementing previous approximation results. Our results find applications in computational biology in phylogenomics and for saving memory in tree decomposition based graph algorithms. [Copyright &y& Elsevier]

Given a collection C of subsets of size three of a finite set S and a positive integer k, the 3-Hitting Set problem is to determine a subset S′⊆S with &z.sfnc;S′&z.sfnc;⩽k, so that S′ contains at least one element from each subset in C. The problem is NP-complete, and is motivated, for example, by applications in computational biology. Improving previous work, we give an O(2.270k+n) time algorithm for 3-Hitting Set, which is efficient for small values of k, a typical occurrence in some applications. For d-Hitting Set we present an O(ck+n) time algorithm with c=d−1+O(d−1). [Copyright &y& Elsevier]

A classic result of Lenstra [Math. Oper. Res. 1983] says that an integer linear program can be solved in fixed-parameter tractable (FPT) time for the parameterization by the number of variables. We extend this result by incorporating piecewise linear convex or concave functions to our (mixed) integer programs. This general technique allows us to analyze the parameterized complexity of a number of classic NP -hard computational problems. In particular, we prove that Weighted Set Multicover is in FPT when parameterized by the number of elements to cover, and that there exists an FPT -time approximation scheme for Multiset Multicover for the same parameter—this is our most technical result. Further, we use our general technique to prove that a number of problems from computational social choice (e.g., problems related to bribery and control in elections) are in FPT when parameterized by the number of candidates. For bribery, this resolves a nearly 10-year old family of open problems, and for weighted electoral control of Approval voting, this improves some previously known XP -memberships to FPT -memberships. [ABSTRACT FROM AUTHOR]

We investigate for temporal graphs the computational complexity of separating two distinct vertices s and z by vertex deletion. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s , z) -Separation problem is NP -complete, it is natural to investigate whether relevant special cases exist that are computationally tractable. To this end, we study restrictions of the underlying (static) graph—there we observe polynomial-time solvability in the case of bounded treewidth—as well as restrictions concerning the "temporal evolution" along the time steps. Systematically studying partially novel concepts in this direction, we identify sharp borders between tractable and intractable cases. [ABSTRACT FROM AUTHOR]

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path , that is, given an undirected graph, find a maximum-length path in G . Longest Path is NP-hard in general but known to be solvable in O ( n 4 ) time on n -vertex interval graphs. We show how to solve Longest Path on Interval Graphs , parameterized by vertex deletion number k to proper interval graphs, in O ( k 9 n ) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial-time solvable problems similarly to what classical parameterized complexity analysis does for NP-hard problems. [ABSTRACT FROM AUTHOR]

Abstract: Given an undirected graph  and an integer , the NP-hard Bounded-Degree Vertex Deletion problem asks to delete as few vertices as possible from such that the resulting graph has maximum vertex degree . Our main result is to prove that Bounded-Degree Vertex Deletion is W[1]-hard with respect to the parameter treewidth. As a side result, we obtain that the NP-hard Vector Dominating Set problem is W[1]-hard with respect to the parameter treewidth. On the positive side, we show that Bounded-Degree Vertex Deletion becomes fixed-parameter tractable when parameterized by the combined parameter treewidth and number of vertices to delete, and when parametrized by the feedback edge set number. [Copyright &y& Elsevier]

Abstract: Many NP-complete problems on planar graphs are “fixed-parameter tractable:” Recent theoretical work provided tree decomposition-based fixed-parameter algorithms exactly solving various parameterized problems on planar graphs, among others VERTEX COVER, in time . Here, is some constant depending on the graph problem to be solved, is the number of graph vertices, and is the problem parameter (for VERTEX COVER this is the size of the vertex cover). In this paper, we present an experimental study for such tree decomposition-based algorithms focusing on VERTEX COVER. We demonstrate that the tree decomposition-based approach provides a valuable way of exactly solving VERTEX COVER on planar graphs. Doing so, we also demonstrate the impressive power of the so-called Nemhauser/Trotter theorem which provides a VERTEX COVER-specific, extremely useful data reduction through polynomial time preprocessing. Altogether, this underpins the practical importance of the underlying theory. [Copyright &y& Elsevier]

The maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L)·2K/5, where K is the number of clauses and L is their total length. In fact, the running time is only poly(L)·2K2/5, where K2 is the number of clauses containing two literals. This bound implies the bound poly(L)·2L/10. Our results significantly improve previous bounds: poly(L)·2K/2.88 (J. Algorithms 36 (2000) 62–88) and poly(L)·2K/3.44 (implicit in Bansal and Raman (Proceedings of the 10th Annual Conference on Algorithms and Computation, ISAAC’99, Lecture Notes in Computer Science, Vol. 1741, Springer, Berlin, 1999, pp. 247–258.))As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly(M)·2M/3, where M is the number of edges in the graph. This is of special interest for graphs with low vertex degree. [Copyright &y& Elsevier]

Time-evolving or temporal graphs gain more and more popularity when exploring complex networks. In this context, the multistage view on computational problems is among the most natural frameworks. Roughly speaking, herein one studies the different (time) layers of a temporal graph (effectively meaning that the edge set may change over time, but the vertex set remains unchanged), and one searches for a solution of a given graph problem for each layer. The twist in the multistage setting is that the solutions found must not differ too much between subsequent layers. We relax on this already established notion by introducing a global instead of the local budget view studied so far. More specifically, we allow for few disruptive changes between subsequent layers but request that overall, that is, summing over all layers, the degree of change is moderate. Studying several classical graph problems (both NP-hard and polynomial-time solvable ones) from a parameterized complexity angle, we encounter both fixed-parameter tractability and parameterized hardness results. Surprisingly, we find that sometimes the global multistage versions of NP-hard problems such as Vertex Cover turn out to be computationally more tractable than the ones of polynomial-time solvable problems such as Matching. In addition to time complexity, we also analyze the space efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

Abstract: The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most  edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we give the first algorithmic result for Closest 4-Leaf Power, showing that Closest 4-Leaf Power is fixed-parameter tractable with respect to the parameter . [Copyright &y& Elsevier]