2,167 research outputs found
Exact Algorithms for Maximum Independent Set
We show that the maximum independent set problem (MIS) on an -vertex graph
can be solved in time and polynomial space, which even is
faster than Robson's -time exponential-space algorithm
published in 1986. We also obtain improved algorithms for MIS in graphs with
maximum degree 6 and 7, which run in time of and
, respectively. Our algorithms are obtained by using fast
algorithms for MIS in low-degree graphs in a hierarchical way and making a
careful analyses on the structure of bounded-degree graphs
Minimum-weight Cycle Covers and Their Approximability
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L.
We investigate how well L-cycle covers of minimum weight can be approximated.
For undirected graphs, we devise a polynomial-time approximation algorithm that
achieves a constant approximation ratio for all sets L. On the other hand, we
prove that the problem cannot be approximated within a factor of 2-eps for
certain sets L.
For directed graphs, we present a polynomial-time approximation algorithm
that achieves an approximation ratio of O(n), where is the number of
vertices. This is asymptotically optimal: We show that the problem cannot be
approximated within a factor of o(n).
To contrast the results for cycle covers of minimum weight, we show that the
problem of computing L-cycle covers of maximum weight can, at least in
principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2007). Minor change
Exponential Time Complexity of Weighted Counting of Independent Sets
We consider weighted counting of independent sets using a rational weight x:
Given a graph with n vertices, count its independent sets such that each set of
size k contributes x^k. This is equivalent to computation of the partition
function of the lattice gas with hard-core self-repulsion and hard-core pair
interaction. We show the following conditional lower bounds: If counting the
satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time
2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in
time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph
needs time 2^{\Omega(n)} and weighted counting of independent sets needs time
2^{\Omega(n/log^3 n)} for all rational weights x\neq 0.
We have two technical ingredients: The first is a reduction from 3SAT to
independent sets that preserves the number of solutions and increases the
instance size only by a constant factor. Second, we devise a combination of
vertex cloning and path addition. This graph transformation allows us to adapt
a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by
a family of reductions, each of which increases the instance size only
polylogarithmically.Comment: Introduction revised, differences between versions of counting
independent sets stated more precisely, minor improvements. 14 page
A QPTAS for Maximum Weight Independent Set of Polygons with Polylogarithmically Many Vertices
The Maximum Weight Independent Set of Polygons problem is a fundamental
problem in computational geometry. Given a set of weighted polygons in the
2-dimensional plane, the goal is to find a set of pairwise non-overlapping
polygons with maximum total weight. Due to its wide range of applications, the
MWISP problem and its special cases have been extensively studied both in the
approximation algorithms and the computational geometry community. Despite a
lot of research, its general case is not well-understood. Currently the best
known polynomial time algorithm achieves an approximation ratio of n^(epsilon)
[Fox and Pach, SODA 2011], and it is not even clear whether the problem is
APX-hard. We present a (1+epsilon)-approximation algorithm, assuming that each
polygon in the input has at most a polylogarithmic number of vertices. Our
algorithm has quasi-polynomial running time.
We use a recently introduced framework for approximating maximum weight
independent set in geometric intersection graphs. The framework has been used
to construct a QPTAS in the much simpler case of axis-parallel rectangles. We
extend it in two ways, to adapt it to our much more general setting. First, we
show that its technical core can be reduced to the case when all input polygons
are triangles. Secondly, we replace its key technical ingredient which is a
method to partition the plane using only few edges such that the objects
stemming from the optimal solution are evenly distributed among the resulting
faces and each object is intersected only a few times. Our new procedure for
this task is not more complex than the original one, and it can handle the
arising difficulties due to the arbitrary angles of the polygons. Note that
already this obstacle makes the known analysis for the above framework fail.
Also, in general it is not well understood how to handle this difficulty by
efficient approximation algorithms
The Cost of Perfection for Matchings in Graphs
Perfect matchings and maximum weight matchings are two fundamental
combinatorial structures. We consider the ratio between the maximum weight of a
perfect matching and the maximum weight of a general matching. Motivated by the
computer graphics application in triangle meshes, where we seek to convert a
triangulation into a quadrangulation by merging pairs of adjacent triangles, we
focus mainly on bridgeless cubic graphs. First, we characterize graphs that
attain the extreme ratios. Second, we present a lower bound for all bridgeless
cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic
graphs, most of which are shown to be tight. Additionally, we present tight
bounds for the class of regular bipartite graphs
Computing a maximum clique in geometric superclasses of disk graphs
In the 90's Clark, Colbourn and Johnson wrote a seminal paper where they
proved that maximum clique can be solved in polynomial time in unit disk
graphs. Since then, the complexity of maximum clique in intersection graphs of
d-dimensional (unit) balls has been investigated. For ball graphs, the problem
is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient
polynomial time approximation scheme (EPTAS) for disk graphs. However, the
complexity of maximum clique in this setting remains unknown. In this paper, we
show the existence of a polynomial time algorithm for a geometric superclass of
unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS
for intersection graphs of convex pseudo-disks
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