%AChekuri, Chandra%AQuanrud, Kent%AMegow, Nicole Ed.%ASmith, Adam Ed.%D2023%ISchloss Dagstuhl – Leibniz-Zentrum für Informatik
%Kelimination graphs; independent set; submodular maximization; primal-dual; Theory of computation → Graph algorithms analysis; Theory of computation → Submodular optimization and polymatroids
%MOSTI ID: 10500359
%PMedium: X
%TIndependent Sets in Elimination Graphs with a Submodular Objective
%XMaximum weight independent set (MWIS) admits a 1/k-approximation in inductively k-independent graphs [Karhan Akcoglu et al., 2002; Ye and Borodin, 2012] and a 1/(2k)-approximation in k-perfectly orientable graphs [Kammer and Tholey, 2014]. These are a parameterized class of graphs that generalize k-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others [Ye and Borodin, 2012; Kammer and Tholey, 2014]. We consider a generalization of MWIS to a submodular objective. Given a graph G = (V,E) and a non-negative submodular function f: 2^V → ℝ_+, the goal is to approximately solve max_{S ∈ ℐ_G} f(S) where ℐ_G is the set of independent sets of G. We obtain an Ω(1/k)-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least 1/e(k+1). This approach also yields parallel (or low-adaptivity) approximations.
Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively k-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.
Country unknown/Code not availablehttps://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.24OSTI-MSA