On the Lovász theta function for independent sets in sparse graphs

We consider the maximum independent set problem on graphs with maximum degree~$d$. We show that the integrality gap of the Lov\'asz $\vartheta$-function based SDP is $\widetilde{O}(d/\log^{3/2} d)$. This improves on the previous best result of $\widetilde{O}(d/\log d)$, and almost matches the integrality gap of $\widetilde{O}(d/\log^2 d)$ recently shown for stronger SDPs, namely those obtained using poly-$(\log(d))$ levels of the $SA^+$ semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of $K_r$-free graphs for large values of $r$. We also show how to obtain an algorithmic version of the above-mentioned $SA^+$-based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of $\widetilde{O}(d/\log^2 d)$ matches the best unique-games-based hardness result up to lower-order poly-$(\log\log d)$ 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 $\{G_1,\ldots, G_m\}$ on the vertex set $V$, a cooperative coloring of it is a choice of independent sets $I_i$ in $G_i$ $(1\leq i\leq m)$ such that $\bigcup^m_{i=1}I_i=V$. For a graph class $\mathcal{G}$, let $m_{\mathcal{G}}(d)$ be the minimum $m$ such that every graph family $\{G_1,\ldots,G_m\}$ with $G_j\in\mathcal{G}$ and $\Delta(G_j)\leq d$ for $j\in [m]$, has a cooperative coloring. For $\mathcal{T}$ the class of trees and $\mathcal{W}$ the class of wheels, we get that $m_\mathcal{T}(3)=4$ and $m_\mathcal{W}(4)=5$. Also, we show that $m_{\mathcal{B}_{bc}}(d)=O(\log_2 d)$ and $m_{\mathcal{B}_k}(d)=O\big(\frac{\log d}{\log\log d}\big)$, where $\mathcal{B}_{bc}$ is the class of graphs whose components are balanced complete bipartite graphs, and $\mathcal{B}_k$ is the class of bipartite graphs with one part size at most $k$

On Minrank and Forbidden Subgraphs

The minrank over a field $\mathbb{F}$ of a graph $G$ on the vertex set $\{1,2,\ldots,n\}$ is the minimum possible rank of a matrix $M \in \mathbb{F}^{n \times n}$ such that $M_{i,i} \neq 0$ for every $i$, and $M_{i,j}=0$ for every distinct non-adjacent vertices $i$ and $j$ in $G$. For an integer $n$, a graph $H$, and a field $\mathbb{F}$, let $g(n,H,\mathbb{F})$ denote the maximum possible minrank over $\mathbb{F}$ of an $n$-vertex graph whose complement contains no copy of $H$. In this paper we study this quantity for various graphs $H$ and fields $\mathbb{F}$. For finite fields, we prove by a probabilistic argument a general lower bound on $g(n,H,\mathbb{F})$, which yields a nearly tight bound of $\Omega(\sqrt{n}/\log n)$ for the triangle $H=K_3$. For the real field, we prove by an explicit construction that for every non-bipartite graph $H$, $g(n,H,\mathbb{R}) \geq n^\delta$ for some $\delta = \delta(H)>0$. 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