760,872 research outputs found
Algorithms and Bounds for Very Strong Rainbow Coloring
A well-studied coloring problem is to assign colors to the edges of a graph
so that, for every pair of vertices, all edges of at least one shortest
path between them receive different colors. The minimum number of colors
necessary in such a coloring is the strong rainbow connection number
(\src(G)) of the graph. When proving upper bounds on \src(G), it is natural
to prove that a coloring exists where, for \emph{every} shortest path between
every pair of vertices in the graph, all edges of the path receive different
colors. Therefore, we introduce and formally define this more restricted edge
coloring number, which we call \emph{very strong rainbow connection number}
(\vsrc(G)).
In this paper, we give upper bounds on \vsrc(G) for several graph classes,
some of which are tight. These immediately imply new upper bounds on \src(G)
for these classes, showing that the study of \vsrc(G) enables meaningful
progress on bounding \src(G). Then we study the complexity of the problem to
compute \vsrc(G), particularly for graphs of bounded treewidth, and show this
is an interesting problem in its own right. We prove that \vsrc(G) can be
computed in polynomial time on cactus graphs; in contrast, this question is
still open for \src(G). We also observe that deciding whether \vsrc(G) = k
is fixed-parameter tractable in and the treewidth of . Finally, on
general graphs, we prove that there is no polynomial-time algorithm to decide
whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor
, unless PNP
Solving Partition Problems Almost Always Requires Pushing Many Vertices Around
A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G[A] and G[B] satisfy properties Pi_A and Pi_B, respectively. This so-called (Pi_A,Pi_B)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (Pi_A,Pi_B)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A\u27,B\u27), and pushes appropriate vertices from A\u27 to B\u27 and vice versa to eventually arrive at a correct bipartition.
In this paper, we study whether (Pi_A,Pi_B)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where Pi_A is the set of P_3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and Pi_B is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP not subseteq coNP/poly, (Pi_A,Pi_B)-Recognition admits a polynomial kernel if and only if H contains a graph of order at most 2. In both the kernelization and the lower bound results, we make crucial use of the pushing process
Complexity Framework for Forbidden Subgraphs
For any finite set H={H1,…,Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,…,Hp as a subgraph. Similar to known meta-classifications for the minor and topological minor relations, we give a meta-classification for the subgraph relation. Our framework classifies if problems are "efficiently solvable" or "computationally hard" for H-subgraph-free graphs. The conditions are that the problem should be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness is preserved under edge subdivision. We show that all problems satisfying these conditions are efficiently solvable if H contains a disjoint union of one or more paths and subdivided claws, and are computationally hard otherwise. To illustrate the broad applicability of our framework, we study partitioning, covering and packing problems, network design problems and width parameter problems. We apply the framework to obtain a dichotomy between polynomial-time solvability and NP-completeness. For other problems we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). Along the way we unify and strengthen known results from the literature
Particle Computation: Complexity, Algorithms, and Logic
We investigate algorithmic control of a large swarm of mobile particles (such
as robots, sensors, or building material) that move in a 2D workspace using a
global input signal (such as gravity or a magnetic field). We show that a maze
of obstacles to the environment can be used to create complex systems. We
provide a wide range of results for a wide range of questions. These can be
subdivided into external algorithmic problems, in which particle configurations
serve as input for computations that are performed elsewhere, and internal
logic problems, in which the particle configurations themselves are used for
carrying out computations. For external algorithms, we give both negative and
positive results. If we are given a set of stationary obstacles, we prove that
it is NP-hard to decide whether a given initial configuration of unit-sized
particles can be transformed into a desired target configuration. Moreover, we
show that finding a control sequence of minimum length is PSPACE-complete. We
also work on the inverse problem, providing constructive algorithms to design
workspaces that efficiently implement arbitrary permutations between different
configurations. For internal logic, we investigate how arbitrary computations
can be implemented. We demonstrate how to encode dual-rail logic to build a
universal logic gate that concurrently evaluates and, nand, nor, and or
operations. Using many of these gates and appropriate interconnects, we can
evaluate any logical expression. However, we establish that simulating the full
range of complex interactions present in arbitrary digital circuits encounters
a fundamental difficulty: a fan-out gate cannot be generated. We resolve this
missing component with the help of 2x1 particles, which can create fan-out
gates that produce multiple copies of the inputs. Using these gates we provide
rules for replicating arbitrary digital circuits.Comment: 27 pages, 19 figures, full version that combines three previous
conference article
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