29,893 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
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
A volume-ish theorem for the Jones polynomial of alternating knots
The Volume conjecture claims that the hyperbolic Volume of a knot is
determined by the colored Jones polynomial.
The purpose of this article is to show a Volume-ish theorem for alternating
knots in terms of the Jones polynomial, rather than the colored Jones
polynomial: The ratio of the Volume and certain sums of coefficients of the
Jones polynomial is bounded from above and from below by constants.
Furthermore, we give experimental data on the relation of the growths of the
hyperbolic volume and the coefficients of the Jones polynomial, both for
alternating and non-alternating knots.Comment: 14 page
Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations
We study complexities of 3-manifolds defined from triangulations, Heegaard
splittings, and surgery presentations. We show that these complexities are
related by linear inequalities, by presenting explicit geometric constructions.
We also show that our linear inequalities are asymptotically optimal. Our
results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov
-invariants in terms of geometric group theoretic and knot theoretic
data.Comment: 16 pages, 9 figures. An error in the triangulation argument found by
a referee has been fixed. Constants in Theorems A and B have been improved.
Minor remaining typos have been fixed. To appear in the Quarterly Journal of
Mathematic
Exceptional surgeries on alternating knots
We give a complete classification of exceptional surgeries on hyperbolic
alternating knots in the 3-sphere. As an appendix, we also show that the
Montesinos knots M (-1/2, 2/5, 1/(2q + 1)) with q at least 5 have no
non-trivial exceptional surgeries. This gives the final step in a complete
classification of exceptional surgery on arborescent knots.Comment: 30 pages, 19 figures. v2: recomputation performed via the newest
version of hikmot, v3: revised according to referees' comments, to appear in
Comm. Anal. Geo
Parallel Algorithms for Geometric Graph Problems
We give algorithms for geometric graph problems in the modern parallel models
inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem
over a set of points in the two-dimensional space, our algorithm computes a
-approximate MST. Our algorithms work in a constant number of
rounds of communication, while using total space and communication proportional
to the size of the data (linear space and near linear time algorithms). In
contrast, for general graphs, achieving the same result for MST (or even
connectivity) remains a challenging open problem, despite drawing significant
attention in recent years.
We develop a general algorithmic framework that, besides MST, also applies to
Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic
framework has implications beyond the MapReduce model. For example it yields a
new algorithm for computing EMD cost in the plane in near-linear time,
. We note that while recently Sharathkumar and Agarwal
developed a near-linear time algorithm for -approximating EMD,
our algorithm is fundamentally different, and, for example, also solves the
transportation (cost) problem, raised as an open question in their work.
Furthermore, our algorithm immediately gives a -approximation
algorithm with space in the streaming-with-sorting model with
passes. As such, it is tempting to conjecture that the
parallel models may also constitute a concrete playground in the quest for
efficient algorithms for EMD (and other similar problems) in the vanilla
streaming model, a well-known open problem
On the Enumeration of all Minimal Triangulations
We present an algorithm that enumerates all the minimal triangulations of a
graph in incremental polynomial time. Consequently, we get an algorithm for
enumerating all the proper tree decompositions, in incremental polynomial time,
where "proper" means that the tree decomposition cannot be improved by removing
or splitting a bag
- …