16,874 research outputs found
Bounded Max-Colorings of Graphs
In a bounded max-coloring of a vertex/edge weighted graph, each color class
is of cardinality at most and of weight equal to the weight of the heaviest
vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask
for such a coloring minimizing the sum of all color classes' weights.
In this paper we present complexity results and approximation algorithms for
those problems on general graphs, bipartite graphs and trees. We first show
that both problems are polynomial for trees, when the number of colors is
fixed, and approximable for general graphs, when the bound is fixed.
For the bounded max-vertex-coloring problem, we show a 17/11-approximation
algorithm for bipartite graphs, a PTAS for trees as well as for bipartite
graphs when is fixed. For unit weights, we show that the known 4/3 lower
bound for bipartite graphs is tight by providing a simple 4/3 approximation
algorithm. For the bounded max-edge-coloring problem, we prove approximation
factors of , for general graphs, , for
bipartite graphs, and 2, for trees. Furthermore, we show that this problem is
NP-complete even for trees. This is the first complexity result for
max-coloring problems on trees.Comment: 13 pages, 5 figure
Improved Inapproximability Results for Maximum k-Colorable Subgraph
We study the maximization version of the fundamental graph coloring problem.
Here the goal is to color the vertices of a k-colorable graph with k colors so
that a maximum fraction of edges are properly colored (i.e. their endpoints
receive different colors). A random k-coloring properly colors an expected
fraction 1-1/k of edges. We prove that given a graph promised to be
k-colorable, it is NP-hard to find a k-coloring that properly colors more than
a fraction ~1-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2)
was known. Our result pins down the correct asymptotic dependence of the
approximation factor on k. Along the way, we prove that approximating the
Maximum 3-colorable subgraph problem within a factor greater than 32/33 is
NP-hard. Using semidefinite programming, it is known that one can do better
than a random coloring and properly color a fraction 1-1/k +2 ln k/k^2 of edges
in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to
properly color (using k colors) more than a fraction 1-1/k + O(ln k/ k^2) of
edges of a k-colorable graph.Comment: 16 pages, 2 figure
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly
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