Bounded Fixed-Parameter Tractability and log2 n Nondeterministic Bits

Fixed-parameter tractability (FPT) techniques have recently been successful in solving NP-complete problem instances of practical importance which were too large to be solved with previous methods. In this paper we show how to enhance this approach through the addition of parallelism, thereby allowing even larger problem instances to be solved in practice. More precisely, we demonstrate the potential of parallelism when applied to the bounded-tree search phase of FPT algorithms. We apply our methodology to the k-VERTEX COVER problem which has important applications, e.g., in multiple sequence alignments for computational biochemistry. We have implemented our parallel FPT method for the k-VERTEX COVER problem using C and the MPI communication library, and tested it on a PC cluster. This is the first experimental examination of parallel FPT techniques. We have tested our parallel k-VERTEX COVER method on protein sequences obtained from the National Center for Biotechnology Information. As part of our experiments, we solved larger instances of k-VERTEX COVER than in any previously reported implementations. For example, our code can solve problem instances with k /spl ges/ 400 in less than 1.5 hours. Since our parallel FPT algorithm requires only very little communication between processors, we expect our method to also perform well on Grids.

We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic Set Cover problem and the more recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers \(k_\ell \) and \(k_r\), and a family \(\mathcal F \) of \(\ell \) sets over U, the Gen-RBSC problem is to decide whether there is a subfamily \(\mathcal F '\subseteq \mathcal F \) of size at most \(k_\ell \) that covers all blue elements, but at most \(k_r\) of the red elements. This generalizes Set Cover and thus in full generality it is intractable in the parameterized setting, when parameterized by \(k_\ell +k_r\). In this paper, we study Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for the parameters \(k_\ell , k_r\), and \(k_\ell +k_r\). For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). Finally, for the parameter \(k_\ell +k_r\), for which Gen-RBSC-lines admits FPT algorithms, we show that the problem does not have a polynomial kernel unless \(\text { co-NP}\subseteq \text { NP}/\mathrm{poly}\). Further, we show that the FPT algorithm does not generalize to higher dimensions.

• NP-Completeness and APX-Hardness • We prove that the problems MinFixMask OPT and MinFixMask BOUND are NP-complete. • We establish that FixMasks + is NP-complete and APX-hard. • We show NP-completeness of DFixMasks + . • Exact, FPT, and Approximation Algorithms for DFixMasks + • We design a fixed-parameter tractable (FPT) algorithm running in O *(2 M ) time, parameterized by the number of shield configurations M . • We present a polynomial-time 1 M -approximation algorithm. • We propose an exponential k M -approximation method (for a treatment plan length k ). • We provide a tight ( 1 − 1 e ) -approximation algorithm by exploiting monotonicity and submodularity of the objective function. • Quasi-Polynomial Algorithms and Logarithmic Approximations • We develop quasi-polynomial-time algorithms for MinFixMask OPT , MinFixMask OPT + , MinFixMask BOUND , and MinFixMask BOUND + , assuming that the maximum prescribed dose d ^ max is polynomially bounded. • Under the same assumption, we achieve polynomial-time approximation algorithms with a logarithmic factor log ( d ^ max ) for MinFixMask OPT and MinFixMask OPT + . • Polynomial-Time Solvable Special Cases • We identify polynomial-time solvable cases: FixMask and FixMasks + when M = 1 . • We show that the problems MinFixMask BOUND and MinFixMask BOUND + can be solved in polynomial time when T max 1, N ;. In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.

Within the field of phylogenetics there is growing interest in measures for summarising the dissimilarity, or incongruence, of two or more phylogenetic trees. Many of these measures are NP-hard to compute and this has stimulated a considerable volume of research into fixed parameter tractable algorithms. In this article we use Monadic Second Order logic to give alternative, compact proofs of fixed parameter tractability for several well-known incongruence measures. In doing so we wish to demonstrate the considerable potential of MSOL - machinery still largely unknown outside the algorithmic graph theory community - within phylogenetics. A crucial component of this work is the observation that many measures, when bounded, imply the existence of an agreement forest of bounded size, which in turn implies that an auxiliary graph structure, the display graph, has bounded treewidth. It is this bound on treewidth that makes the machinery of MSOL available for proving fixed parameter tractability.

The Workflow Satisfiability Problem (WSP) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan - an assignment of tasks to authorized users - such that all constraints are satisfied. It is natural to see the WSP as a subclass of the Constraint Satisfaction Problem (CSP) in which the variables are tasks and the domain is the set of users. What makes the WSP distinctive is that the number of tasks is usually very small compared to the number of users, so it is appropriate to ask for which constraint languages the WSP is fixed-parameter tractable (FPT), parameterized by the number of tasks. This novel approach to the WSP, using techniques from CSP, has enabled us to design a generic algorithm which is FPT for several families of workflow constraints considered in the literature. Furthermore, we prove that the union of FPT languages remains FPT if they satisfy a simple compatibility condition. Lastly, we identify a new FPT constraint language, user-independent constraints, that includes many of the constraints of interest in business processing systems. We demonstrate that our generic algorithm has provably optimal running time O*(2^(klog k)), for this language, where k is the number of tasks.

In the Cycle Packing problem, we are given an undirected graph G, a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. In this paper, we study Cycle Packing with respect to a structural parameter, namely, distance to proper interval graphs (indifference graphs). In particular, we show that Cycle Packing is fixed-parameter tractable (FPT) when parameterized by t, the size of a proper interval deletion set. For this purpose, we design an algorithm with \(\mathcal {O}(2^{\mathcal {O}(t \log t)} n^{\mathcal {O}(1)})\) running time. Several structural parameterizations for Cycle Packing have been studied in the literature and our FPT algorithm fills a gap in the ecology of such parameterizations. We combine color coding, greedy strategy and dynamic programming based on structural properties of proper interval graphs in a non-trivial fashion to obtain the FPT algorithm.

Balanced stable marriage: How close is close enough?

Comparing genomes in terms of gene order is a classical combinatorial optimization problem in computational biology. Some of the popular distances include translocation, reversal, and double-cut-and-join (abbreviated as DCJ), which have been extensively used while comparing two genomes. Let \(d_x\), \(x\in \{\)translocation, reversal, DCJ\(\}\), be the distance between two genomes such that one can be sorted/converted into the other using the minimum number of x-operations. All these problems are NP-hard when the genomes are unsigned. Computing \(d_x\), \(x\in \{\)translocation, reversal, DCJ\(\}\), between two unsigned genomes involves computing a proper alternating cycle decomposition of its breakpoint graph, which becomes the bottleneck for computing the genomic distance under almost all types of genome rearrangement operations and prohibits to obtain approximation factors better than 1.375 in polynomial time. In this paper, we devise an FPT (fixed-parameter tractable) approximation algorithm for computing the DCJ and translocation distances with an approximation factor 4/3+\(\varepsilon \), and the running time is \(O^*(2^{d^*})\), where \(d^*\) represents the optimal DCJ or translocation distance. The algorithm is randomized and it succeeds with a high probability. This technique is based on a new randomized method to generate approximate maximum alternating cycle decomposition.

We study the classical Bandwidth problem from the viewpoint of parameterized algorithms. In the Bandwidth problem we are given a graph G = (V,E) together with a positive integer k, and asked whether there is an bijective function β: {1, ..., n} →V such that for every edge uv ∈ E, |β − 1(u) − β − 1(v)| ≤ k. The problem is notoriously hard, and it is known to be NP-complete even on very restricted subclasses of trees. The best known algorithm for Bandwidth for small values of k is the celebrated algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980 ], which runs in time \(2^{{\mathcal{O}}(k)}n^{k+1}\). In a seminal paper, Bodlaender, Fellows and Hallet [STOC 1994 ] ruled out the existence of an algorithm with running time of the form \(f(k)n^{{\mathcal{O}}(1)}\) for any function f even for trees, unless the entire W-hierarchy collapses.We initiate the search for classes of graphs where Bandwidth is fixed parameter tractable (FPT), that is, solvable in time \(f(k)n^{{\mathcal{O}}(1)}\) for some function f. In this paper we present an algorithm with running time \(2^{{\mathcal O}(k \log k)} n^2\) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete.KeywordsInterval GraphGraph ClassHair LengthPermutation GraphVertex PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given a graph G , a matching is a subset of edges of G that do not share an endpoint. A matching M is uniquely restricted if the subgraph induced by the endpoints of the edges of M has exactly one perfect matching. Given a graph G and a positive integer ℓ , Uniquely Restricted Matching asks whether G has a uniquely restricted matching of size at least ℓ . In this paper, we study the parameterized complexity of Uniquely Restricted Matching under various parameters. Specifically, we show that Uniquely Restricted Matching admits a fixed-parameter tractable (FPT) algorithm on line graphs when parameterized by the solution size. We also establish that the problem is FPT when parameterized by the treewidth of the input graph. Furthermore, we show that Uniquely Restricted Matching does not admit a polynomial kernel with respect to the vertex cover number plus the size of the matching unless NP ⊆ coNP/poly.

BackgroundCombinatorial works on genome rearrangements have so far ignored the influence of intergene sizes, i.e. the number of nucleotides between consecutive genes, although it was recently shown decisive for the accuracy of inference methods (Biller et al. in Genome Biol Evol 8:1427–39, 2016; Biller et al. in Beckmann A, Bienvenu L, Jonoska N, editors. Proceedings of Pursuit of the Universal-12th conference on computability in Europe, CiE 2016, Lecture notes in computer science, vol 9709, Paris, France, June 27–July 1, 2016. Berlin: Springer, p. 35–44, 2016). In this line, we define a new genome rearrangement model called wDCJ, a generalization of the well-known double cut and join (or DCJ) operation that modifies both the gene order and the intergene size distribution of a genome.ResultsWe first provide a generic formula for the wDCJ distance between two genomes, and show that computing this distance is strongly NP-complete. We then propose an approximation algorithm of ratio 4/3, and two exact ones: a fixed-parameter tractable (FPT) algorithm and an integer linear programming (ILP) formulation.ConclusionsWe provide theoretical and empirical bounds on the expected growth of the parameter at the center of our FPT and ILP algorithms, assuming a probabilistic model of evolution under wDCJ, which shows that both these algorithms should run reasonably fast in practice.

Out-branchings with Maximal Number of Leaves or Internal Vertices: Algorithmic Results and Open Problems

Poljak and Turzik (Discrete Math. 1986) introduced the notion of lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0 < lambda < 1 and lambda-extendible property Pi, any connected graph G on n vertices and m edges contains a spanning subgraph H in Pi with at least lambda m+ (1-lambda)/2 (n-1) edges. The property of being bipartite is lambda-extendible for lambda=1/2, and thus the Poljak-Turzik bound generalizes the well-known Edwards-Erdos bound for MAXCUT. We define a variant, namely strong lambda-extendibility, to which the Poljak-Turzik bound applies. For a strong lambda-extendible graph property \Pi, we define the parameterized Above Poljak-Turzik problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H in Pi and H has at least lambda m+ (1-lambda)/2 (n-1)+k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzik bound. We consider properties Pi for which the Above Poljak-Turzik problem is fixed-parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzik is FPT for all 0< lambda <1. Our results hold for properties of oriented graphs and graphs with edge labels. Our results generalize the recent result of Crowston et al. (ICALP 2012) on MAXCUT parameterized above the Edwards-Erdos, and yield FPT algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee Max q-Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

The Generalized Traveling Salesman (Path) Problem involves finding a minimum-cost tour (or path) through exactly one location from each of a set of generalized location categories (e.g., gas stations, coffee shops). This problem type has many practical applications in personal navigation and logistics. While NP-hard in general, this problem also admits fixed-parameter tractable (FPT) algorithms with run times of the form f(k)nO(1) for some function f (independent of the problem size, n) with respect to the number of location categories, k (typically very small in practice). We present both exact and approximate FPT algorithms for this problem. Experimental results on the road network of North America (with over 50 million edges) show that we can optimally solve nationwide queries with up to 7 categories and millions of optional category locations in sub-second time. Our approximate solutions improve this even further down to millisecond query times, resulting in only negligible relative error with respect to optimality, on average.

The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems which are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions and provide FPT algorithms for the showcase problems Capacitated Vertex Cover, Capacitated Dominating Set and Imbalance parameterized by the tree-cut width of an input graph G. On the other hand, we show that List Coloring, Precoloring Extension and Boolean CSP (the latter parameterized by the tree-cut width of the incidence graph) are W[1]-hard and hence unlikely to be fixed-parameter tractable when parameterized by tree-cut width.