160 research outputs found
Computing Tree-depth Faster Than
A connected graph has tree-depth at most if it is a subgraph of the
closure of a rooted tree whose height is at most . We give an algorithm
which for a given -vertex graph , in time
computes the tree-depth of . Our algorithm is based on combinatorial results
revealing the structure of minimal rooted trees whose closures contain
Finding Induced Subgraphs via Minimal Triangulations
Potential maximal cliques and minimal separators are combinatorial objects
which were introduced and studied in the realm of minimal triangulations
problems including Minimum Fill-in and Treewidth. We discover unexpected
applications of these notions to the field of moderate exponential algorithms.
In particular, we show that given an n-vertex graph G together with its set of
potential maximal cliques Pi_G, and an integer t, it is possible in time |Pi_G|
* n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a
given graph F of treewidth t, to decide if G contains an induced subgraph
isomorphic to F. Combined with an improved algorithm enumerating all potential
maximal cliques in time O(1.734601^n), this yields that both problems are
solvable in time 1.734601^n * n^(O(t)).Comment: 14 page
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
Maximum cuts in edge-colored graphs
The input of the Maximum Colored Cut problem consists of a graph
with an edge-coloring and a positive integer ,
and the question is whether has a nontrivial edge cut using at least
colors. The Colorful Cut problem has the same input but asks for a nontrivial
edge cut using all colors. Unlike what happens for the classical Maximum
Cut problem, we prove that both problems are NP-complete even on complete,
planar, or bounded treewidth graphs. Furthermore, we prove that Colorful Cut is
NP-complete even when each color class induces a clique of size at most 3, but
is trivially solvable when each color induces a . On the positive side, we
prove that Maximum Colored Cut is fixed-parameter tractable when parameterized
by either or , by constructing a cubic kernel in both cases.Comment: 15 pages, 6 figure
Turan Problems and Shadows III: expansions of graphs
The expansion of a graph is the -uniform hypergraph obtained
from by enlarging each edge of with a new vertex disjoint from
such that distinct edges are enlarged by distinct vertices. Let
denote the maximum number of edges in a -uniform hypergraph with
vertices not containing any copy of a -uniform hypergraph . The study of
includes some well-researched problems, including the case that
consists of disjoint edges, is a triangle, is a path or cycle,
and is a tree. In this paper we initiate a broader study of the behavior of
. Specifically, we show whenever and . One of the main open problems
is to determine for which graphs the quantity is quadratic in
. We show that this occurs when is any bipartite graph with Tur\'{a}n
number where , and in
particular, this shows where is the
three-dimensional cube graph
Role colouring graphs in hereditary classes
We study the computational complexity of computing role colourings of graphs
in hereditary classes. We are interested in describing the family of hereditary
classes on which a role colouring with k colours can be computed in polynomial
time. In particular, we wish to describe the boundary between the "hard" and
"easy" classes. The notion of a boundary class has been introduced by Alekseev
in order to study such boundaries. Our main results are a boundary class for
the k-role colouring problem and the related k-coupon colouring problem which
has recently received a lot of attention in the literature. The latter result
makes use of a technique for generating regular graphs of arbitrary girth which
may be of independent interest
Graph Homomorphism Polynomials: Algorithms and Complexity
We study homomorphism polynomials, which are polynomials that enumerate all
homomorphisms from a pattern graph to -vertex graphs. These polynomials
have received a lot of attention recently for their crucial role in several new
algorithms for counting and detecting graph patterns, and also for obtaining
natural polynomial families which are complete for algebraic complexity classes
, , and . We discover that, in the
monotone setting, the formula complexity, the ABP complexity, and the circuit
complexity of such polynomial families are exactly characterized by the
treedepth, the pathwidth, and the treewidth of the pattern graph respectively.
Furthermore, we establish a single, unified framework, using our
characterization, to collect several known results that were obtained
independently via different methods. For instance, we attain superpolynomial
separations between circuits, ABPs, and formulas in the monotone setting, where
the polynomial families separating the classes all correspond to well-studied
combinatorial problems. Moreover, our proofs rediscover fine-grained
separations between these models for constant-degree polynomials. The
characterization additionally yields new space-time efficient algorithms for
several pattern detection and counting problems.Comment: This version fixes a mistake in the proof of Lemma 1. It also fixes
an incorrect citation. Thanks to Marc Roth for pointing out the mistake in
the citatio
A Note on Exponential-Time Algorithms for Linearwidth
In this note, we give an algorithm that computes the linearwidth of input
-vertex graphs in time , which improves a trivial -time
algorithm, where and the number of vertices and edges, respectively.Comment: 4 page
On the relation of separability, bandwidth and embedding
In this paper we construct a class of bounded degree bipartite graphs with a
small separator and large bandwidth. Furthermore, we also prove that graphs
from this class are spanning subgraphs of graphs with minimum degree just
slightly larger than .Comment: submitted for publicatio
Irrelevant vertices for the planar Disjoint Paths Problem
The Disjoint Paths Problem asks, given a graph G and a set of pairs of terminals (s1,t1),âŠ,(sk,tk)(s1,t1),âŠ,(sk,tk), whether there is a collection of k pairwise vertex-disjoint paths linking sisi and titi, for i=1,âŠ,ki=1,âŠ,k. In their f(k)â
n3f(k)â
n3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k)g(k) there is an âirrelevantâ vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem , whose â very technical â proof gives a function g(k)g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k)=O(k3/2â
2k)g(k)=O(k3/2â
2k). Our bound is radically better than the bounds known for general graphs
- âŠ