10 research outputs found
NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in -minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in -free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the -minor-free
graphs and -minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as . Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
Determining whether a given graph has a perfect matching and if so,
finding one.
Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
Finding a maximum -flow in the network, with arbitrary
capacities.
The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.Comment: 21 pages, 6 figure
Tight Bounds for Gomory-Hu-like Cut Counting
By a classical result of Gomory and Hu (1961), in every edge-weighted graph
, the minimum -cut values, when ranging over all ,
take at most distinct values. That is, these instances
exhibit redundancy factor . They further showed how to construct
from a tree that stores all minimum -cut values. Motivated
by this result, we obtain tight bounds for the redundancy factor of several
generalizations of the minimum -cut problem.
1. Group-Cut: Consider the minimum -cut, ranging over all subsets
of given sizes and . The redundancy
factor is .
2. Multiway-Cut: Consider the minimum cut separating every two vertices of
, ranging over all subsets of a given size . The
redundancy factor is .
3. Multicut: Consider the minimum cut separating every demand-pair in
, ranging over collections of demand pairs. The
redundancy factor is . This result is a bit surprising, as
the redundancy factor is much larger than in the first two problems.
A natural application of these bounds is to construct small data structures
that stores all relevant cut values, like the Gomory-Hu tree. We initiate this
direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which
have some overlap with our results), see Bibliographic Update 1.
Degree-3 Treewidth Sparsifiers
We study treewidth sparsifiers. Informally, given a graph of treewidth
, a treewidth sparsifier is a minor of , whose treewidth is close to
, is small, and the maximum vertex degree in is bounded.
Treewidth sparsifiers of degree are of particular interest, as routing on
node-disjoint paths, and computing minors seems easier in sub-cubic graphs than
in general graphs.
In this paper we describe an algorithm that, given a graph of treewidth
, computes a topological minor of such that (i) the treewidth of
is ; (ii) ; and (iii) the maximum
vertex degree in is . The running time of the algorithm is polynomial in
and . Our result is in contrast to the known fact that unless , treewidth does not admit polynomial-size kernels.
One of our key technical tools, which is of independent interest, is a
construction of a small minor that preserves node-disjoint routability between
two pairs of vertex subsets. This is closely related to the open question of
computing small good-quality vertex-cut sparsifiers that are also minors of the
original graph.Comment: Extended abstract to appear in Proceedings of ACM-SIAM SODA 201