9,664 research outputs found
A Note on the Sparing Number of Graphs
An integer additive set-indexer is defined as an injective function
such that the induced function defined by is also
injective. An IASI is said to be a weak IASI if
for all . A graph which admits a
weak IASI may be called a weak IASI graph. The set-indexing number of an
element of a graph , a vertex or an edge, is the cardinality of its
set-labels. The sparing number of a graph is the minimum number of edges
with singleton set-labels, required for a graph to admit a weak IASI. In
this paper, we study the sparing number of certain graphs and the relation of
sparing number with some other parameters like matching number, chromatic
number, covering number, independence number etc.Comment: 10 pages, 10 figures, submitte
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
- β¦