21 research outputs found
An Algorithmic Metatheorem for Directed Treewidth
The notion of directed treewidth was introduced by Johnson, Robertson,
Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as
a first step towards an algorithmic metatheory for digraphs. They showed that
some NP-complete properties such as Hamiltonicity can be decided in polynomial
time on digraphs of constant directed treewidth. Nevertheless, despite more
than one decade of intensive research, the list of hard combinatorial problems
that are known to be solvable in polynomial time when restricted to digraphs of
constant directed treewidth has remained scarce. In this work we enrich this
list by providing for the first time an algorithmic metatheorem connecting the
monadic second order logic of graphs to directed treewidth. We show that most
of the known positive algorithmic results for digraphs of constant directed
treewidth can be reformulated in terms of our metatheorem. Additionally, we
show how to use our metatheorem to provide polynomial time algorithms for two
classes of combinatorial problems that have not yet been studied in the context
of directed width measures. More precisely, for each fixed , we show how to count in polynomial time on digraphs of directed
treewidth , the number of minimum spanning strong subgraphs that are the
union of directed paths, and the number of maximal subgraphs that are the
union of directed paths and satisfy a given minor closed property. To prove
our metatheorem we devise two technical tools which we believe to be of
independent interest. First, we introduce the notion of tree-zig-zag number of
a digraph, a new directed width measure that is at most a constant times
directed treewidth. Second, we introduce the notion of -saturated tree slice
language, a new formalism for the specification and manipulation of infinite
sets of digraphs.Comment: 41 pages, 6 figures, Accepted to Discrete Applied Mathematic
A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting
We present the first polynomial-time approximation scheme (PTAS), i.e., (1 + ϵ)-approximation algorithm for any constant ϵ > 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(logn(loglogn)O(1)). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing lightweight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems. Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which reduces the graph to have total weight within a factor of the optimal solution while approximately preserving the optimal solution. This is particularly challenging because group Steiner tree requires deciding which terminal in each group to connect by the tree, making it much harder than recent previous approaches to construct spanners for planar TSP by Klein [SIAM J. Computing 2008], subset TSP by Klein [STOC 2006], Steiner tree by Borradaile, Klein, and Mathieu [ACM Trans. Algorithms 2009], and Steiner forest by Bateni, Hajiaghayi, and Marx [J. ACM 2011] (and its improvement to an efficient PTAS by Eisenstat, Klein, and Mathieu [SODA 2012]. The main conceptual contribution here is realizing that selecting which terminals may be relevant is essentially a complicated prize-collecting process: we have to carefully weigh the cost and benefits of reaching or avoiding certain terminals in the spanner. Via a sequence of involved prize-collecting procedures, we can construct a spanner that reaches a set of terminals that is sufficient for an almost-optimal solution. Our PTAS for planar group Steiner tree implies the first PTAS for geometric Euclidean group Steiner tree with obstacles, as well as a (2 + ϵ)-approximation algorithm for group TSP with obstacles, improving over the best previous constant-factor approximation algorithms. By contrast, we show that planar group Steiner forest, a slight generalization of planar group Steiner tree, is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge
On the Treewidth of Dynamic Graphs
Dynamic graph theory is a novel, growing area that deals with graphs that
change over time and is of great utility in modelling modern wireless, mobile
and dynamic environments. As a graph evolves, possibly arbitrarily, it is
challenging to identify the graph properties that can be preserved over time
and understand their respective computability.
In this paper we are concerned with the treewidth of dynamic graphs. We focus
on metatheorems, which allow the generation of a series of results based on
general properties of classes of structures. In graph theory two major
metatheorems on treewidth provide complexity classifications by employing
structural graph measures and finite model theory. Courcelle's Theorem gives a
general tractability result for problems expressible in monadic second order
logic on graphs of bounded treewidth, and Frick & Grohe demonstrate a similar
result for first order logic and graphs of bounded local treewidth.
We extend these theorems by showing that dynamic graphs of bounded (local)
treewidth where the length of time over which the graph evolves and is observed
is finite and bounded can be modelled in such a way that the (local) treewidth
of the underlying graph is maintained. We show the application of these results
to problems in dynamic graph theory and dynamic extensions to static problems.
In addition we demonstrate that certain widely used dynamic graph classes
naturally have bounded local treewidth
Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large
We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC\u2708]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT\u2721] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP
Between Treewidth and Clique-width
Many hard graph problems can be solved efficiently when restricted to graphs
of bounded treewidth, and more generally to graphs of bounded clique-width. But
there is a price to be paid for this generality, exemplified by the four
problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that
are all FPT parameterized by treewidth but none of which can be FPT
parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7,
8]. We therefore seek a structural graph parameter that shares some of the
generality of clique-width without paying this price. Based on splits, branch
decompositions and the work of Vatshelle [18] on Maximum Matching-width, we
consider the graph parameter sm-width which lies between treewidth and
clique-width. Some graph classes of unbounded treewidth, like
distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph
Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized
by sm-width
An Improvement of Reed's Treewidth Approximation
We present a new approximation algorithm for the treewidth problem which
constructs a corresponding tree decomposition as well. Our algorithm is a
faster variation of Reed's classical algorithm. For the benefit of the reader,
and to be able to compare these two algorithms, we start with a detailed time
analysis for Reed's algorithm. We fill in many details that have been omitted
in Reed's paper. Computing tree decompositions parameterized by the treewidth
is fixed parameter tractable (FPT), meaning that there are algorithms
running in time where is a computable function, is a
polynomial function, and is the number of vertices. An analysis of Reed's
algorithm shows and for a
5-approximation. Reed simply claims time for bounded for his
constant factor approximation algorithm, but the bound of is well known. From a practical point of view, we notice that the
time of Reed's algorithm also contains a term of ,
which for small is much worse than the asymptotically leading term of
. We analyze more precisely, because the
purpose of this paper is to improve the running times for all reasonably small
values of .
Our algorithm runs in too, but with a much
smaller dependence on . In our case, . This
algorithm is simple and fast, especially for small values of . We should
mention that Bodlaender et al.\ [2016] have an asymptotically faster algorithm
running in time . It relies on a very sophisticated data
structure and does not claim to be useful for small values of
Width Parameterizations for Knot-Free Vertex Deletion on Digraphs
A knot in a directed graph G is a strongly connected subgraph Q of G with at least two vertices, such that no vertex in V(Q) is an in-neighbor of a vertex in V(G)V(Q). Knots are important graph structures, because they characterize the existence of deadlocks in a classical distributed computation model, the so-called OR-model. Deadlock detection is correlated with the recognition of knot-free graphs as well as deadlock resolution is closely related to the Knot-Free Vertex Deletion (KFVD) problem, which consists of determining whether an input graph G has a subset S subseteq V(G) of size at most k such that G[VS] contains no knot. Because of natural applications in deadlock resolution, KFVD is closely related to Directed Feedback Vertex Set. In this paper we focus on graph width measure parameterizations for KFVD. First, we show that: (i) KFVD parameterized by the size of the solution k is W[1]-hard even when p, the length of a longest directed path of the input graph, as well as kappa, its Kenny-width, are bounded by constants, and we remark that KFVD is para-NP-hard even considering many directed width measures as parameters, but in FPT when parameterized by clique-width; (ii) KFVD can be solved in time 2^{O(tw)} x n, but assuming ETH it cannot be solved in 2^{o(tw)} x n^{O(1)}, where tw is the treewidth of the underlying undirected graph. Finally, since the size of a minimum directed feedback vertex set (dfv) is an upper bound for the size of a minimum knot-free vertex deletion set, we investigate parameterization by dfv and we show that (iii) KFVD can be solved in FPT-time parameterized by either dfv+kappa or dfv+p. Results of (iii) cannot be improved when replacing dfv by k due to (i)