108 research outputs found
Independent Sets in Elimination Graphs with a Submodular Objective
Maximum weight independent set (MWIS) admits a -approximation in
inductively -independent graphs and a -approximation in
-perfectly orientable graphs. These are a a parameterized class of graphs
that generalize -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 and a non-negative submodular function , the goal is to approximately solve where is the set of independent sets of . We obtain an
-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 . 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 -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
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Independent Sets in Elimination Graphs with a Submodular Objective
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
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Exploring structural properties of -trees and block graphs
We present a new characterization of -trees based on their reduced clique
graphs and -line graphs, which are block graphs. We explore structural
properties of these two classes, showing that the number of clique-trees of a
-tree equals the number of spanning trees of the -line graph of
. 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
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 -bounded. While -bounded graph
classes are known to enjoy some good algorithmic properties related to clique
and coloring problems, it is an interesting open problem whether
-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 of a graph as the maximum independence
number over all subgraphs of induced by some bag of . The
tree-independence number of a graph is then defined as the minimum
independence number over all tree decompositions of . 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
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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