4,655 research outputs found
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
Restricted non-separable planar maps and some pattern avoiding permutations
Tutte founded the theory of enumeration of planar maps in a series of papers
in the 1960s. Rooted non-separable planar maps are in bijection with
West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori,
Jacquard and Schaeffer in 1997, as well as a family of permutations defined by
the avoidance of two four letter patterns. In this paper we give upper and
lower bounds on the number of multiple-edge-free rooted non-separable planar
maps. We also use the bijection between rooted non-separable planar maps and a
certain class of permutations, found by Claesson, Kitaev and Steingrimsson in
2009, to show that the number of 2-faces (excluding the root-face) in a map
equals the number of occurrences of a certain mesh pattern in the permutations.
We further show that this number is also the number of nodes in the
corresponding beta(1,0)-tree that are single children with maximum label.
Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
Obstructions to weak decomposability for simplicial polytopes
Provan and Billera introduced notions of (weak) decomposability of simplicial
complexes as a means of attempting to prove polynomial upper bounds on the
diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera
and Klee provided the first examples of simplicial polytopes that are not
weakly vertex-decomposable. These polytopes are polar to certain simple
transportation polytopes. In this paper, we refine their analysis to prove that
these -dimensional polytopes are not even weakly -decomposable.
As a consequence, (weak) decomposability cannot be used to prove a polynomial
version of the Hirsch conjecture
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
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