 NSFPAR ID:
 10348209
 Date Published:
 Journal Name:
 Proceedings of the 2022 ACMSIAM Symposium on Discrete Algorithms
 Page Range / eLocation ID:
 1531  1555
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Megow, Nicole ; Smith, Adam (Ed.)Maximum weight independent set (MWIS) admits a 1/kapproximation in inductively kindependent graphs [Karhan Akcoglu et al., 2002; Ye and Borodin, 2012] and a 1/(2k)approximation in kperfectly orientable graphs [Kammer and Tholey, 2014]. These are a parameterized class of graphs that generalize kdegenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudodisks, 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 nonnegative 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 lowadaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively kindependent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primaldual 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 pseudodisks.more » « less

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Abstract For a subgraph
of the blowup of a graph$G$ , we let$F$ be the smallest minimum degree over all of the bipartite subgraphs of$\delta ^*(G)$ induced by pairs of parts that correspond to edges of$G$ . Johansson proved that if$F$ is a spanning subgraph of the blowup of$G$ with parts of size$C_3$ and$n$ , then$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$ contains$G$ vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$n$ is a spanning subgraph of the blowup of$G$ with parts of size$C_k$ and$n$ , then$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$ contains$G$ vertex disjoint copies of$n$ such that each$C_k$ intersects each of the$C_k$ parts exactly once. A similar conjecture was also made by Fischer and the case$k$ was proved for large$k=3$ by Magyar and Martin.$n$ In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of
to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.$G$ 
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