A Variant of the Maximum Weight Independent Set Problem

We study a natural extension of the Maximum Weight Independent Set Problem (MWIS), one of the most studied optimization problems in Graph algorithms. We are given a graph $G=(V,E)$, a weight function $w: V \rightarrow \mathbb{R^+}$, a budget function $b: V \rightarrow \mathbb{Z^+}$, and a positive integer $B$. The weight (resp. budget) of a subset of vertices is the sum of weights (resp. budgets) of the vertices in the subset. A $k$-budgeted independent set in $G$ is a subset of vertices, such that no pair of vertices in that subset are adjacent, and the budget of the subset is at most $k$. The goal is to find a $B$-budgeted independent set in $G$ such that its weight is maximum among all the $B$-budgeted independent sets in $G$. We refer to this problem as MWBIS. Being a generalization of MWIS, MWBIS also has several applications in Scheduling, Wireless networks and so on. Due to the hardness results implied from MWIS, we study the MWBIS problem in several special classes of graphs. We design exact algorithms for trees, forests, cycle graphs, and interval graphs. In unweighted case we design an approximation algorithm for $d+1$-claw free graphs whose approximation ratio ($d$) is competitive with the approximation ratio ($\frac{d}{2}$) of MWIS (unweighted). Furthermore, we extend Baker's technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page

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