85 research outputs found
On the Lovász theta function for independent sets in sparse graphs
We consider the maximum independent set problem on graphs with maximum degree~. We show that the integrality gap of the Lov\'asz -function based SDP is . This improves on the previous best result of , and almost matches the integrality gap of recently shown for stronger SDPs, namely those obtained using poly- levels of the semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of -free graphs for large values of . We also show how to obtain an algorithmic version of the above-mentioned -based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of matches the best unique-games-based hardness result up to lower-order poly- factors
On the Lovász theta function for independent sets in sparse graphs
We consider the maximum independent set problem on sparse graphs with maximum degree d. We show that the Lovász ϑ-function based semidefinite program (SDP) has an integrality gap of O(d/log3/2 d), improving on the previous best result of O(d/log d). This improvement is based on a new Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r. We also show that for stronger SDPs, namely, those obtained using polylog(d) levels of the SA+ semidefinite hierarchy, the integrality gap reduces to O(d/log2 d). This matches the best unique-games-based hardness result up to lower-order poly(log log d) factors. Finally, we give an algorithmic version of this SA+-based integrality gap result, albeit using d levels of SA+, via a coloring algorithm of Johansson
How to Hide a Clique?
In the well known planted clique problem, a clique (or alternatively, an independent set) of size k is planted at random in an Erdos-Renyi random G(n, p) graph, and the goal is to design an algorithm that finds the maximum clique (or independent set) in the resulting graph. We introduce a variation on this problem, where instead of planting the clique at random, the clique is planted by an adversary who attempts to make it difficult to find the maximum clique in the resulting graph. We show that for the standard setting of the parameters of the problem, namely, a clique of size k = ?n planted in a random G(n, 1/2) graph, the known polynomial time algorithms can be extended (in a non-trivial way) to work also in the adversarial setting. In contrast, we show that for other natural settings of the parameters, such as planting an independent set of size k = n/2 in a G(n, p) graph with p = n^{-1/2}, there is no polynomial time algorithm that finds an independent set of size k, unless NP has randomized polynomial time algorithms
Cooperative coloring of some graph families
Given a family of graphs on the vertex set , a
cooperative coloring of it is a choice of independent sets in
such that . For a graph class
, let be the minimum such that every
graph family with and for , has a cooperative coloring. For the class of
trees and the class of wheels, we get that
and . Also, we show that and , where
is the class of graphs whose components are balanced
complete bipartite graphs, and is the class of bipartite graphs
with one part size at most
On Minrank and Forbidden Subgraphs
The minrank over a field of a graph on the vertex set
is the minimum possible rank of a matrix such that for every , and
for every distinct non-adjacent vertices and in . For an
integer , a graph , and a field , let
denote the maximum possible minrank over of an -vertex graph
whose complement contains no copy of . In this paper we study this quantity
for various graphs and fields . For finite fields, we prove by
a probabilistic argument a general lower bound on , which
yields a nearly tight bound of for the triangle
. For the real field, we prove by an explicit construction that for
every non-bipartite graph , for some
. As a by-product of this construction, we disprove a
conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by
questions in information theory, circuit complexity, and geometry.Comment: 15 page
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