Answer Set Solving with Bounded Treewidth Revisited

Parameterized algorithms are a way to solve hard problems more efficiently, given that a specific parameter of the input is small. In this paper, we apply this idea to the field of answer set programming (ASP). To this end, we propose two kinds of graph representations of programs to exploit their treewidth as a parameter. Treewidth roughly measures to which extent the internal structure of a program resembles a tree. Our main contribution is the design of parameterized dynamic programming algorithms, which run in linear time if the treewidth and weights of the given program are bounded. Compared to previous work, our algorithms handle the full syntax of ASP. Finally, we report on an empirical evaluation that shows good runtime behaviour for benchmark instances of low treewidth, especially for counting answer sets.Comment: This paper extends and updates a paper that has been presented on the workshop TAASP'16 (arXiv:1612.07601). We provide a higher detail level, full proofs and more example

Transformée de radon lissée et utilisation dans une procédure de détection en contrôle non destructif

- Cet article introduit une variante de la transformée de Radon qui autorise la mise en oeuvre d'une procédure de détection rapide non pénalisée par la rotation du segment recherché dans une image à horizon glissant. Cette transformée de Radon dite lissée utilise une fonction de voisinage sur lequel s'effectue l'intégration au lieu de la projection standard sur droite pivotante. Elle généralise la transformée de Radon et permet l'introduction de connaissances a priori sur la structure de l'image à traiter et la nature des bruits qui la dégrade. Son application à une procédure de contrôle non destructif de rail par ultrasons en mode B-Scan est présentée en fin d'article

Ionic modeling of lithium manganese spinel materials for use in rechargeable batteries

In order to understand and evaluate materials for use in Li ion rechargeable battery electrodes, we have modeled the crystal structures of various Mn oxide and Li Mn oxide compounds. We have modeled the MnO{sub 2} polymorphs and several spinels with intermediate compositions based on the amount of Li inserted into the tetrahedral site. 3-D representations of the structures provide a basis for identifying site occupancies, coordinations, Mn valence, order-disorder, and potentially new dopants for enhanced cathode behavior. XRD simulations of the crystal structures provide good agreement with observed patterns for synthesized samples. Ionic modeling of these materials consists of an energy minimization approach using Coulombic, repulsive, and van der Waals interactions. Modeling using electronic polarizabilities (shell model) allows a systematic analysis of changes in lattice energy, cell volume, and the relative stability of doped structures using ions such as Al, Ti, Ni, and Co

Structurally Parameterized d-Scattered Set

In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any $d\ge2$, an $O^*(d^{\textrm{tw}})$-time algorithm, where $\textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - $d$-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if $k$ is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A $2^{O(\textrm{td}^2)}$-time algorithm parameterized by tree-depth ($\textrm{td}$), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter $\epsilon > 0$, runs in time $O^*((\textrm{tw}/\epsilon)^{O(\textrm{tw})})$ and returns a $d/(1+\epsilon)$-scattered set of size $k$, if a $d$-scattered set of the same size exists

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of G\"oz\"upek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure

Fixed-Parameter Tractability of Multicut in Directed Acyclic Graphs

The Multicut problem, given a graph G, a set of terminal pairs $\mathcal{T}=\{(s_i,t_i)\ |\ 1\leq i\leq r\}$, 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, $t_i$ is not reachable from $s_i$ for each $1\leq i\leq r$. 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

Parametrised Complexity of Model Checking and Satisfiability in Propositional Dependence Logic

In this paper, we initiate a systematic study of the parametrised complexity in the field of Dependence Logics which finds its origin in the Dependence Logic of V\"a\"an\"anen from 2007. We study a propositional variant of this logic (PDL) and investigate a variety of parametrisations with respect to the central decision problems. The model checking problem (MC) of PDL is NP-complete. The subject of this research is to identify a list of parametrisations (formula-size, treewidth, treedepth, team-size, number of variables) under which MC becomes fixed-parameter tractable. Furthermore, we show that the number of disjunctions or the arity of dependence atoms (dep-arity) as a parameter both yield a paraNP-completeness result. Then, we consider the satisfiability problem (SAT) showing a different picture: under team-size, or dep-arity SAT is paraNP-complete whereas under all other mentioned parameters the problem is in FPT. Finally, we introduce a variant of the satisfiability problem, asking for teams of a given size, and show for this problem an almost complete picture.Comment: Update includes refined result

Graph editing to a given degree sequence

We investigate the parameterized complexity of the graph editing problem called Editing to a Graph with a Given Degree Sequence where the aim is to obtain a graph with a given degree sequence σ by at most k vertex deletions, edge deletions and edge additions. We show that the problem is W[1]-hard when parameterized by k for any combination of the allowed editing operations. From the positive side, we show that the problem can be solved in time 2O(k(Δ⁎+k)2)n2log⁡n for n -vertex graphs, where Δ⁎=max⁡σ, i.e., the problem is FPT when parameterized by k+Δ⁎. We also show that Editing to a Graph with a Given Degree Sequence has a polynomial kernel when parameterized by k+Δ⁎ if only edge additions are allowed, and there is no polynomial kernel unless NP⊆co-NP/poly for all other combinations of the allowed editing operations

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

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 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 Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size 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 vc(G) since fvs(G) <= 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 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')| <= 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 is in coNP/poly and the polynomial hierarchy collapses to the third level.Comment: Published in "Theory of Computing Systems" as an Open Access publicatio