221 research outputs found
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
The Lovasz number of random graphs
We study the Lovasz number theta along with two further SDP relaxations
theta1, theta1/2 of the independence number and the corresponding relaxations
of the chromatic number on random graphs G(n,p). We prove that these
relaxations are concentrated about their means Moreover, extending a result of
Juhasz, we compute the asymptotic value of the relaxations for essentially the
entire range of edge probabilities p. As an application, we give an improved
algorithm for approximating the independence number in polynomial expected
time, thereby extending a result of Krivelevich and Vu. We also improve on the
analysis of an algorithm of Krivelevich for deciding whether G(n,p) is
k-colorable
The Lovász Number of Random Graphs
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.We study the Lovász number along with two related SDP relaxations , of the independence number and the corresponding relaxations , , of the chromatic number on random graphs . We prove that are concentrated about their means, and that in the case is a constant. As an application, we give improved algorithms for approximating the independence number of and for deciding -colourability in polynomial expected time.Peer Reviewe
Colouring Semirandom Graphs
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Peer Reviewe
Recommended from our members
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
- …