7,176 research outputs found
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
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
Dynamic Geometric Independent Set
We present fully dynamic approximation algorithms for the Maximum Independent
Set problem on several types of geometric objects: intervals on the real line,
arbitrary axis-aligned squares in the plane and axis-aligned -dimensional
hypercubes.
It is known that a maximum independent set of a collection of intervals
can be found in time, while it is already \textsf{NP}-hard for a
set of unit squares. Moreover, the problem is inapproximable on many important
graph families, but admits a \textsf{PTAS} for a set of arbitrary pseudo-disks.
Therefore, a fundamental question in computational geometry is whether it is
possible to maintain an approximate maximum independent set in a set of dynamic
geometric objects, in truly sublinear time per insertion or deletion. In this
work, we answer this question in the affirmative for intervals, squares and
hypercubes.
First, we show that for intervals a -approximate maximum
independent set can be maintained with logarithmic worst-case update time. This
is achieved by maintaining a locally optimal solution using a constant number
of constant-size exchanges per update.
We then show how our interval structure can be used to design a data
structure for maintaining an expected constant factor approximate maximum
independent set of axis-aligned squares in the plane, with polylogarithmic
amortized update time. Our approach generalizes to -dimensional hypercubes,
providing a -approximation with polylogarithmic update time.
Those are the first approximation algorithms for any set of dynamic arbitrary
size geometric objects; previous results required bounded size ratios to obtain
polylogarithmic update time. Furthermore, it is known that our results for
squares (and hypercubes) cannot be improved to a
-approximation with the same update time
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
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