1,057 research outputs found
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
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
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
Parameterized Algorithms for Modular-Width
It is known that a number of natural graph problems which are FPT
parameterized by treewidth become W-hard when parameterized by clique-width. It
is therefore desirable to find a different structural graph parameter which is
as general as possible, covers dense graphs but does not incur such a heavy
algorithmic penalty.
The main contribution of this paper is to consider a parameter called
modular-width, defined using the well-known notion of modular decompositions.
Using a combination of ILPs and dynamic programming we manage to design FPT
algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path
and Hamiltonian cycle), which are W-hard for both clique-width and its recently
introduced restriction, shrub-depth. We thus argue that modular-width occupies
a sweet spot as a graph parameter, generalizing several simpler notions on
dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with
arXiv:1304.5479 by other author
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
enjoy similar properties as their Euclidean counterparts, but in one dimension
lower: many standard graph problems, such as Independent Set, Dominating Set,
Steiner Tree, and Hamiltonian Cycle can be solved in
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require time under ETH in constant maximum degree
Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in
noisy uniform disk graphs with a matching lower bound
under ETH. This shows that the hyperbolic plane is a potential source of
NP-intermediate problems.Comment: Short version appears in SODA 202
Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class
We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set.
To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)
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