35 research outputs found
Cartesian product of hypergraphs: properties and algorithms
Cartesian products of graphs have been studied extensively since the 1960s.
They make it possible to decrease the algorithmic complexity of problems by
using the factorization of the product. Hypergraphs were introduced as a
generalization of graphs and the definition of Cartesian products extends
naturally to them. In this paper, we give new properties and algorithms
concerning coloring aspects of Cartesian products of hypergraphs. We also
extend a classical prime factorization algorithm initially designed for graphs
to connected conformal hypergraphs using 2-sections of hypergraphs
Energy Efficient Scheduling of MapReduce Jobs
MapReduce is emerged as a prominent programming model for data-intensive
computation. In this work, we study power-aware MapReduce scheduling in the
speed scaling setting first introduced by Yao et al. [FOCS 1995]. We focus on
the minimization of the total weighted completion time of a set of MapReduce
jobs under a given budget of energy. Using a linear programming relaxation of
our problem, we derive a polynomial time constant-factor approximation
algorithm. We also propose a convex programming formulation that we combine
with standard list scheduling policies, and we evaluate their performance using
simulations.Comment: 22 page
Regular Matroids with Graphic Cocircuits
We introduce the notion of graphic cocircuits and show that a large class of
regular matroids with graphic cocircuits belongs to the class of signed-graphic
matroids. Moreover, we provide an algorithm which determines whether a
cographic matroid with graphic cocircuits is signed-graphic or not
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
Complexity of Strong Implementability
We consider the question of implementability of a social choice function in a
classical setting where the preferences of finitely many selfish individuals
with private information have to be aggregated towards a social choice. This is
one of the central questions in mechanism design. If the concept of weak
implementation is considered, the Revelation Principle states that one can
restrict attention to truthful implementations and direct revelation
mechanisms, which implies that implementability of a social choice function is
easy to check. For the concept of strong implementation, however, the
Revelation Principle becomes invalid, and the complexity of deciding whether a
given social choice function is strongly implementable has been open so far. In
this paper, we show by using methods from polyhedral theory that strong
implementability of a social choice function can be decided in polynomial space
and that each of the payments needed for strong implementation can always be
chosen to be of polynomial encoding length. Moreover, we show that strong
implementability of a social choice function involving only a single selfish
individual can be decided in polynomial time via linear programming
Improved Approximation Algorithms for the Max-Edge Coloring Problem
The max edge-coloring problem asks for a proper edge-coloring of an edge-weighted graph minimizing the sum of the weights of the heaviest edge in each color class. In this paper we present a PTAS for trees and an 1.74-approximation algorithm for bipartite graphs; we also adapt the last algorithm to one for general graphs of the same, asymptotically, approximation ratio. Up to now, no approximation algorithm of ratio 2 − δ, for any constant δ> 0, was known for general or bipartite graphs, while the complexity of the problem on trees remains an open question.ou