263 research outputs found
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
Counting Carambolas
We give upper and lower bounds on the maximum and minimum number of geometric
configurations of various kinds present (as subgraphs) in a triangulation of
points in the plane. Configurations of interest include \emph{convex
polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also
consider related problems for \emph{directed} planar straight-line graphs.Comment: update reflects journal version, to appear in Graphs and
Combinatorics; 18 pages, 13 figure
Eight-Fifth Approximation for TSP Paths
We prove the approximation ratio 8/5 for the metric -path-TSP
problem, and more generally for shortest connected -joins.
The algorithm that achieves this ratio is the simple "Best of Many" version
of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys
(2012), which consists in determining the best Christofides -tour out
of those constructed from a family \Fscr_{>0} of trees having a convex
combination dominated by an optimal solution of the fractional
relaxation. They give the approximation guarantee for
such an -tour, which is the first improvement after the 5/3 guarantee
of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao
(2012) extended this result to a 13/8-approximation of shortest connected
-joins, for .
The ratio 8/5 is proved by simplifying and improving the approach of An,
Kleinberg and Shmoys that consists in completing in order to dominate
the cost of "parity correction" for spanning trees. We partition the edge-set
of each spanning tree in \Fscr_{>0} into an -path (or more
generally, into a -join) and its complement, which induces a decomposition
of . This decomposition can be refined and then efficiently used to
complete without using linear programming or particular properties of
, but by adding to each cut deficient for an individually tailored
explicitly given vector, inherent in .
A simple example shows that the Best of Many Christofides algorithm may not
find a shorter -tour than 3/2 times the incidentally common optima of
the problem and of its fractional relaxation.Comment: 15 pages, corrected typos in citations, minor change
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