42 research outputs found
An Algorithmic Weakening of the Erd?s-Hajnal Conjecture
We study the approximability of the Maximum Independent Set (MIS) problem in H-free graphs (that is, graphs which do not admit H as an induced subgraph). As one motivation we investigate the following conjecture: for every fixed graph H, there exists a constant ? > 0 such that MIS can be n^{1-?}-approximated in H-free graphs, where n denotes the number of vertices of the input graph. We first prove that a constructive version of the celebrated Erd?s-Hajnal conjecture implies ours. We then prove that the set of graphs H satisfying our conjecture is closed under the so-called graph substitution. This, together with the known polynomial-time algorithms for MIS in H-free graphs (e.g. P?-free and fork-free graphs), implies that our conjecture holds for many graphs H for which the Erd?s-Hajnal conjecture is still open. We then focus on improving the constant ? for some graph classes: we prove that the classical Local Search algorithm provides an OPT^{1-1/t}-approximation in K_{t, t}-free graphs (hence a ?{OPT}-approximation in C?-free graphs), and, while there is a simple ?n-approximation in triangle-free graphs, it cannot be improved to n^{1/4-?} for any ? > 0 unless NP ? BPP. More generally, we show that there is a constant c such that MIS in graphs of girth ? cannot be n^{c/(?)}-approximated. Up to a constant factor in the exponent, this matches the ratio of a known approximation algorithm by Monien and Speckenmeyer, and by Murphy. To the best of our knowledge, this is the first strong (i.e., ?(n^?) for some ? > 0) inapproximability result for Maximum Independent Set in a proper hereditary class
Degrees of nonlinearity in forbidden 0â1 matrix problems
AbstractA 0â1 matrix A is said to avoid a forbidden 0â1 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of DavenportâSchinzel sequences and their generalizations, Stanley and Wilfâs permutation avoidance problem, and TurĂĄn-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0â1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. FĂŒredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an nĂn matrix avoiding P is O(nlogn). In the first part of the article we refute of this conjecture. We exhibit nĂn matrices with weight Î(nlognloglogn) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 0â1 matrices. In the second part of the article we simplify one aspect of Keszegh and Genesonâs proof that there are infinitely many minimal nonlinear forbidden 0â1 matrices. In the last part of the article we investigate the relationship between 0â1 matrices and generalized DavenportâSchinzel sequences. We prove that all forbidden subsequences formed by concatenating two permutations have a linear extremal function
Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons
The Brownian separable permutons are a one-parameter family -- indexed by
-- of universal limits of random constrained permutations. We show
that for each , there are explicit constants such that the length of the longest increasing subsequence
in a random permutation of size sampled from the Brownian separable
permuton is between and with
probability tending to 1 as . In the symmetric case , we
have and . We present
numerical simulations which suggest that the lower bound is close
to optimal in the whole range .
Our results work equally well for the closely related Brownian cographons. In
this setting, we show that for each , the size of the largest
clique (resp. independent set) in a random graph on vertices sampled from
the Brownian cographon is between and (resp. and ) with
probability tending to 1 as .
Our proofs are based on the analysis of a fragmentation process embedded in a
Brownian excursion introduced by Bertoin (2002). We expect that our techniques
can be extended to prove similar bounds for uniform separable permutations and
uniform cographs.Comment: New version before journal submissio
Clique number of tournaments
We introduce the notion of clique number of a tournament and investigate its
relation with the dichromatic number. In particular, it permits defining
\dic-bounded classes of tournaments, which is the paper's main topic
HalmazelmĂ©let; PartĂciĂł kalkulus, VĂ©gtelen grĂĄfok elmĂ©lete = Set Theory; Partition Calculus , Theory of Infinite Graphs
ElĆzetes tervĂŒnknek megfelelĆen a halmazelmĂ©let alĂĄbbi terĂŒletein vĂ©geztĂŒnk kutatĂĄst Ă©s Ă©rtĂŒnk el szĂĄmos eredmĂ©nyt: I. Kombinatorika II. A valĂłsak szĂĄmsossĂĄginvariĂĄnsai Ă©s ideĂĄlelmĂ©let III. HalmazelmĂ©leti topolĂłgia Ezek mellett SĂĄgi GĂĄbor kiterjedt kutatĂĄst vĂ©gzett a modellelmĂ©let terĂŒletĂ©n , amely eredmĂ©nyek kapcsolĂłdnak a kombinatorikĂĄhoz is. EredmĂ©nyeinket 38 közlemĂ©nyben publikĂĄltuk, amelyek majdnem mind az adott terĂŒlet vezetĆ nemzetközi lapjaiban jelentel meg (5 cikket csak benyĂșjtottunk). SzĂĄmos nemzetközi konferenciĂĄn is rĂ©sztvettĂŒnk, Ă©s hĂĄrman közƱlĂŒnk (JuhĂĄsz, SĂĄdi, Soukup) plenĂĄris/meghĂvott elĆadĂłk voltak szĂĄmos alkalommal. | Following our research plan, we have mainly done research -- and established a number of significant results -- in several areas of set theory: I. Combinatorics II. Cardinal invariants of the continuum and ideal theory III. Set-theoretic topology In addition to these, G. SĂĄgi has done extended research in model theory that had ramifications to combinatorics. We presented our results in 38 publications, almost all of which appeared or will appear in the leading international journals of these fields (5 of these papers have been submitted but not accepted as yet). We also participated at a number of international conferences, three of us (JuhĂĄsz, SĂĄgi, Soukup) as plenary and/or invited speakers at many of these
Towards the Erd\H{o}s-Hajnal conjecture for -free graphs
The Erd\H{o}s-Hajnal conjecture is one of the most classical and well-known
problems in extremal and structural combinatorics dating back to 1977. It
asserts that in stark contrast to the case of a general -vertex graph if one
imposes even a little bit of structure on the graph, namely by forbidding a
fixed graph as an induced subgraph, instead of only being able to find a
polylogarithmic size clique or an independent set one can find one of
polynomial size. Despite being the focus of considerable attention over the
years the conjecture remains open. In this paper we improve the best known
lower bound of on this question, due to Erd\H{o}s
and Hajnal from 1989, in the smallest open case, namely when one forbids a
, the path on vertices. Namely, we show that any -free vertex
graph contains a clique or an independent set of size at least . Our methods also lead to the same improvement for an infinite
family of graphs