108 research outputs found

    Independent Sets in Elimination Graphs with a Submodular Objective

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    Maximum weight independent set (MWIS) admits a 1k\frac1k-approximation in inductively kk-independent graphs and a 12k\frac{1}{2k}-approximation in kk-perfectly orientable graphs. These are a a parameterized class of graphs that generalize kk-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others. We consider a generalization of MWIS to a submodular objective. Given a graph G=(V,E)G=(V,E) and a non-negative submodular function f:2V→R+f: 2^V \rightarrow \mathbb{R}_+, the goal is to approximately solve max⁥S∈IGf(S)\max_{S \in \mathcal{I}_G} f(S) where IG\mathcal{I}_G is the set of independent sets of GG. We obtain an Ω(1k)\Omega(\frac1k)-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 1e(k+1)\frac{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 kk-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.Comment: Extended abstract to appear in Proceedings of APPROX 2023. v2 corrects technical typos in few place

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Independent Sets in Elimination Graphs with a Submodular Objective

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    Maximum 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 ? ?_+, the goal is to approximately solve max_{S ? ?_G} f(S) where ?_G is the set of independent sets of G. We obtain an ?(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

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Exploring structural properties of kk-trees and block graphs

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    We present a new characterization of kk-trees based on their reduced clique graphs and (k+1)(k+1)-line graphs, which are block graphs. We explore structural properties of these two classes, showing that the number of clique-trees of a kk-tree GG equals the number of spanning trees of the (k+1)(k+1)-line graph of GG. This relationship allows to present a new approach for determining the number of spanning trees of any connected block graph. We show that these results can be accomplished in linear time complexity.Comment: 6 pages, 1 figur

    Treewidth versus clique number. II. Tree-independence number

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    In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call (tw,ω)(\mathrm{tw},\omega)-bounded. While (tw,ω)(\mathrm{tw},\omega)-bounded graph classes are known to enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem whether (tw,ω)(\mathrm{tw},\omega)-boundedness also has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by means of a new min-max graph invariant related to tree decompositions. We define the independence number of a tree decomposition T\mathcal{T} of a graph as the maximum independence number over all subgraphs of GG induced by some bag of T\mathcal{T}. The tree-independence number of a graph GG is then defined as the minimum independence number over all tree decompositions of GG. Generalizing a result on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is given together with a tree decomposition with bounded independence number, then the Maximum Weight Independent Packing problem can be solved in polynomial time. Applications of our general algorithmic result to specific graph classes will be given in the third paper of the series [Dallard, Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure].Comment: 33 pages; abstract has been shortened due to arXiv requirements. A previous version of this arXiv post has been reorganized into two parts; this is the first of the two parts (the second one is arXiv:2206.15092

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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