324 research outputs found
Fractional total colourings of graphs of high girth
Reed conjectured that for every epsilon>0 and Delta there exists g such that
the fractional total chromatic number of a graph with maximum degree Delta and
girth at least g is at most Delta+1+epsilon. We prove the conjecture for
Delta=3 and for even Delta>=4 in the following stronger form: For each of these
values of Delta, there exists g such that the fractional total chromatic number
of any graph with maximum degree Delta and girth at least g is equal to
Delta+1
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails
First-order limits, an analytical perspective
In this paper we present a novel approach to graph (and structural) limits
based on model theory and analysis. The role of Stone and Gelfand dualities is
displayed prominently and leads to a general theory, which we believe is
naturally emerging. This approach covers all the particular examples of
structural convergence and it put the whole in new context. As an application,
it leads to new intermediate examples of structural convergence and to a "grand
conjecture" dealing with sparse graphs. We survey the recent developments
Research in structural graph theory
Issued as final reportNational Science Foundation (U.S.
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
A graph is -free if it does not contain pairwise vertex-disjoint and
non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in
-free graphs can be solved in quasi-polynomial time. As a main technical
result, we establish that "sparse" (here, not containing large complete
bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback
vertex set number) at most logarithmic in the number of vertices. This is
proven sharp as there is an infinite family of -free graphs without
-subgraph and whose treewidth is (at least) logarithmic.
Other consequences include that most of the central NP-complete problems
(such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set,
Minimum Coloring) can be solved in polynomial time in sparse -free graphs,
and that deciding the -freeness of sparse graphs is polynomial time
solvable.Comment: 28 pages, 6 figures. v3: improved complexity result
Approximating the Largest Root and Applications to Interlacing Families
We study the problem of approximating the largest root of a real-rooted
polynomial of degree using its top coefficients and give nearly
matching upper and lower bounds. We present algorithms with running time
polynomial in that use the top coefficients to approximate the maximum
root within a factor of and when and respectively. We also prove corresponding
information-theoretic lower bounds of and
, and show strong lower
bounds for noisy version of the problem in which one is given access to
approximate coefficients.
This problem has applications in the context of the method of interlacing
families of polynomials, which was used for proving the existence of Ramanujan
graphs of all degrees, the solution of the Kadison-Singer problem, and bounding
the integrality gap of the asymmetric traveling salesman problem. All of these
involve computing the maximum root of certain real-rooted polynomials for which
the top few coefficients are accessible in subexponential time. Our results
yield an algorithm with the running time of for all
of them
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
From the Ising and Potts models to the general graph homomorphism polynomial
In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
We also identify the smallest graphs which are not determined by their
homomorphism polynomials for and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
Finding combinatorial structures
In this thesis we answer questions in two related areas of combinatorics:
Ramsey theory and asymptotic enumeration.
In Ramsey theory we introduce a new method for finding desired structures.
We find a new upper bound on the Ramsey number of a path against a kth
power of a path.
Using our new method and this result we obtain a new upper bound on the
Ramsey number of the kth power of a long cycle.
As a corollary we show that, while graphs on n vertices with maximum
degree k may in general have Ramsey numbers as large as ckn, if the stronger
restriction that the bandwidth should be at most k is given, then the Ramsey
numbers are bounded by the much smaller value.
We go on to attack an old conjecture of Lehel: by using our new method
we can improve on a result of Luczak, Rodl and Szemeredi [60]. Our new
method replaces their use of the Regularity Lemma, and allows us to prove
that for any n > 218000, whenever the edges of the complete graph on n
vertices are two-coloured there exist disjoint monochromatic cycles covering
all n vertices.
In asymptotic enumeration we examine first the class of bipartite graphs
with some forbidden induced subgraph H. We obtain some results for every
H, with special focus on the cases where the growth speed of the class is
factorial, and make some comments on a connection to clique-width. We
then move on to a detailed discussion of 2-SAT functions. We find the correct
asymptotic formula for the number of 2-SAT functions
on n variables (an improvement on a result of Bollob´as, Brightwell and
Leader [13], who found the dominant term in the exponent), the first error
term for this formula, and some bounds on smaller error terms. Finally
we obtain various expected values in the uniform model of random 2-SAT
functions
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