106 research outputs found
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
Parameterized Directed -Chinese Postman Problem and Arc-Disjoint Cycles Problem on Euler Digraphs
In the Directed -Chinese Postman Problem (-DCPP), we are given a
connected weighted digraph and asked to find non-empty closed directed
walks covering all arcs of such that the total weight of the walks is
minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128)
asked for the parameterized complexity of -DCPP when is the parameter.
We prove that the -DCPP is fixed-parameter tractable.
We also consider a related problem of finding arc-disjoint directed
cycles in an Euler digraph, parameterized by . Slivkins (ESA 2003) showed
that this problem is W[1]-hard for general digraphs. Generalizing another
result by Slivkins, we prove that the problem is fixed-parameter tractable for
Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler
digraphs remains W[1]-hard even for Euler digraphs
The Firefighter Problem: A Structural Analysis
We consider the complexity of the firefighter problem where b>=1 firefighters
are available at each time step. This problem is proved NP-complete even on
trees of degree at most three and budget one (Finbow et al.,2007) and on trees
of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this
paper, we provide further insight into the complexity landscape of the problem
by showing that the pathwidth and the maximum degree of the input graph govern
its complexity. More precisely, we first prove that the problem is NP-complete
even on trees of pathwidth at most three for any fixed budget b>=1. We then
show that the problem turns out to be fixed parameter-tractable with respect to
the combined parameter "pathwidth" and "maximum degree" of the input graph
Kernel Bounds for Structural Parameterizations of Pathwidth
Assuming the AND-distillation conjecture, the Pathwidth problem of
determining whether a given graph G has pathwidth at most k admits no
polynomial kernelization with respect to k. The present work studies the
existence of polynomial kernels for Pathwidth with respect to other,
structural, parameters. Our main result is that, unless NP is in coNP/poly,
Pathwidth admits no polynomial kernelization even when parameterized by the
vertex deletion distance to a clique, by giving a cross-composition from
Cutwidth. The cross-composition works also for Treewidth, improving over
previous lower bounds by the present authors. For Pathwidth, our result rules
out polynomial kernels with respect to the distance to various classes of
polynomial-time solvable inputs, like interval or cluster graphs. This leads to
the question whether there are nontrivial structural parameters for which
Pathwidth does admit a polynomial kernelization. To answer this, we give a
collection of graph reduction rules that are safe for Pathwidth. We analyze the
success of these results and obtain polynomial kernelizations with respect to
the following parameters: the size of a vertex cover of the graph, the vertex
deletion distance to a graph where each connected component is a star, and the
vertex deletion distance to a graph where each connected component has at most
c vertices.Comment: This paper contains the proofs omitted from the extended abstract
published in the proceedings of Algorithm Theory - SWAT 2012 - 13th
Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201
On Sparsification for Computing Treewidth
We investigate whether an n-vertex instance (G,k) of Treewidth, asking
whether the graph G has treewidth at most k, can efficiently be made sparse
without changing its answer. By giving a special form of OR-cross-composition,
we prove that this is unlikely: if there is an e > 0 and a polynomial-time
algorithm that reduces n-vertex Treewidth instances to equivalent instances, of
an arbitrary problem, with O(n^{2-e}) bits, then NP is in coNP/poly and the
polynomial hierarchy collapses to its third level.
Our sparsification lower bound has implications for structural
parameterizations of Treewidth: parameterizations by measures that do not
exceed the vertex count, cannot have kernels with O(k^{2-e}) bits for any e >
0, unless NP is in coNP/poly. Motivated by the question of determining the
optimal kernel size for Treewidth parameterized by vertex cover, we improve the
O(k^3)-vertex kernel from Bodlaender et al. (STACS 2011) to a kernel with
O(k^2) vertices. Our improved kernel is based on a novel form of
treewidth-invariant set. We use the q-expansion lemma of Fomin et al. (STACS
2011) to find such sets efficiently in graphs whose vertex count is
superquadratic in their vertex cover number.Comment: 21 pages. Full version of the extended abstract presented at IPEC
201
Parameterized Complexity of Graph Constraint Logic
Graph constraint logic is a framework introduced by Hearn and Demaine, which
provides several problems that are often a convenient starting point for
reductions. We study the parameterized complexity of Constraint Graph
Satisfiability and both bounded and unbounded versions of Nondeterministic
Constraint Logic (NCL) with respect to solution length, treewidth and maximum
degree of the underlying constraint graph as parameters. As a main result we
show that restricted NCL remains PSPACE-complete on graphs of bounded
bandwidth, strengthening Hearn and Demaine's framework. This allows us to
improve upon existing results obtained by reduction from NCL. We show that
reconfiguration versions of several classical graph problems (including
independent set, feedback vertex set and dominating set) are PSPACE-complete on
planar graphs of bounded bandwidth and that Rush Hour, generalized to boards, is PSPACE-complete even when is at most a constant
Tight Bounds for Connectivity Problems Parameterized by Cutwidth
In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. [Bas A. M. van Geffen et al., 2020] posed this question for Odd Cycle Transversal and Feedback Vertex Set. We answer it for these two and four further problems, namely Connected Vertex Cover, Connected Dominating Set, Steiner Tree, and Connected Odd Cycle Transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al. [Carla Groenland et al., 2022] to solve what we call coloring-like problems. Such problems are defined by a symmetric matrix M over ?? indexed by a set of colors. The goal is to count the number (modulo some prime p) of colorings of a graph such that M has a 1-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if M has small rank over ?_p. We apply this result to get our upper bounds for CVC and CDS. The upper bounds for OCT and FVS use a subdivision trick to get below the bounds that matrix rank would yield
Arrangement Problems Parameterized by Neighbourhood Diversity
We show that the arrangement problems BANDWIDTH, DISTORTION and IMBALANCE is FPT when parameterized by neighbourhood diversity, and establish some structural results for CUTWIDTH parameterized by neighbourhood diversity.Masteroppgave i informatikkINF39
The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems
Given a directed graph G and a list (s_1, t_1), ..., (s_k, t_k) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed s_i -> t_i path for every 1 <= i <= k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t_1, . . .t_k) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t_i to every other t_j ) is known to be W[1]-hard parameterized by the number of terminals. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if H is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list (s_1, t_1), ..., (s_k, t_k) of requests form a directed graph that is a member of H. Our main result is a complete characterization of the classes H resulting in fixed-parameter tractable special cases: we show that if every pattern in H has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable H not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], Strongly Connected Steiner Subgraph is W[1]-hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before
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