77 research outputs found
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
Integer Vector Addition Systems with States
This paper studies reachability, coverability and inclusion problems for
Integer Vector Addition Systems with States (ZVASS) and extensions and
restrictions thereof. A ZVASS comprises a finite-state controller with a finite
number of counters ranging over the integers. Although it is folklore that
reachability in ZVASS is NP-complete, it turns out that despite their
naturalness, from a complexity point of view this class has received little
attention in the literature. We fill this gap by providing an in-depth analysis
of the computational complexity of the aforementioned decision problems. Most
interestingly, it turns out that while the addition of reset operations to
ordinary VASS leads to undecidability and Ackermann-hardness of reachability
and coverability, respectively, they can be added to ZVASS while retaining
NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure
Rewriting Systems for Reachability in Vector Addition Systems with Pairs
15 pagesInternational audienceWe adapt hypergraph rewriting system to a generalization of Vector Addition Systems with States (VASS) that we call vector addition systems with pairs (VASP). We give rewriting systems and strategies, that allow us to obtain reachability equivalence results between some classes of VASP and VASS. Reachability for the later is well known be equivalent to reachability in Petri nets. VASP generalize also Branching Extension of VASS (BVASS) for which it is unknown if they are more expressive than VASS. We consider here a more restricted notion of reachability for VASP than that for BVASS. However the reachability decision problem corresponding is already equivalent to decidability of the provability in Multiplicative and Exponential Linear Logic (MELL), a question left open for more than 20 years
The 4C spectrum of fundamental behavioral relations for concurrent systems
The design of concurrent software systems, in particular process-aware information systems, involves behavioral modeling at various stages. Recently, approaches to behavioral analysis of such systems have been based on declarative abstractions defined as sets of behavioral relations. However, these relations are typically defined in an ad-hoc manner. In this paper, we address the lack of a systematic exploration of the fundamental relations that can be used to capture the behavior of concurrent systems, i.e., co-occurrence, conflict, causality, and concurrency. Besides the definition of the spectrum of behavioral relations, which we refer to as the 4C spectrum, we also show that our relations give rise to implication lattices. We further provide operationalizations of the proposed relations, starting by proposing techniques for computing relations in unlabeled systems, which are then lifted to become applicable in the context of labeled systems, i.e., systems in which state transitions have semantic annotations. Finally, we report on experimental results on efficiency of the proposed computations
Output Stability and Semilinear Sets in Chemical Reaction Networks and Deciders
Abstract. We study the set of output stable configurations of chemical reaction deciders (CRDs). It turns out that CRDs with only bimolecular reactions (which are almost equivalent to population protocols) have a special structure that allows for an algorithm to efficiently calculate the (finite) set of minimal output stable configurations. As a consequence, a relatively large sequence of configurations may be efficiently checked for output stability. We also provide a number of observations regarding the semilinearity result of Angluin et al. [Distrib. Comput., 2007] from the context of population protocols (which is a central result for output stable CRDs). In particular, we observe that the computation-friendly class of totally stable CRDs has equal expressive power as the larger class of output stable CRDs.
On structural parameterizations of hitting set: hitting paths in graphs using 2-SAT
Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T. We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results
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