94,042 research outputs found

### Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
$r$. Our main result is a deterministic algorithm to generate a matching which
is an $O(r)$-approximation to the maximum weight matching, running in $\tilde
O(r \log \Delta + \log^2 \Delta + \log^* n)$ rounds. (Here, the $\tilde O()$
notations hides $\text{polyloglog } \Delta$ and $\text{polylog } r$ factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
$(1+\epsilon)$ approximation algorithm using $\tilde O(\log \Delta / \epsilon^3
+ \text{polylog}(1/\epsilon, \log \log n))$ randomized time and $\tilde
O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon)$ deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for $(2 \Delta - 1)$-edge-list
coloring in $\tilde O(\log^2 \Delta \log n)$ rounds deterministically or
$\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most $\lceil (1+\epsilon) \lambda \rceil$ for a graph of
arboricity $\lambda$; for fixed $\epsilon$ this runs in $\tilde O(\log^6 n)$
rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly

### Universality of the Future Chronological Boundary

The purpose of this note is to establish, in a categorical manner, the
universality of the Geroch-Kronheimer-Penrose causal boundary when considering
the types of causal structures that may profitably be put on any sort of
boundary for a spacetime. Actually, this can only be done for the future causal
boundary (or the past causal boundary) separately; furthermore, only the
chronology relation, not the causality relation, is considered, and the GKP
topology is eschewed. The final result is that there is a unique map, with the
proper causal properties, from the future causal boundary of a spacetime onto
any ``reasonable" boundary which supports some sort of chronological structure
and which purports to consist of a future completion of the spacetime.
Furthermore, the future causal boundary construction is categorically unique in
this regard.Comment: 25 pages, AMS-TeX; 2 figures, PostScript (separate); captions
(separate); submitted to Class. Quantum Grav, slight revision: bottom lines
legible, figures added, expanded discussion and example

### Some results on chromatic number as a function of triangle count

A variety of powerful extremal results have been shown for the chromatic
number of triangle-free graphs. Three noteworthy bounds are in terms of the
number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994),
and Johansson. There have been comparatively fewer works extending these types
of bounds to graphs with a small number of triangles. One noteworthy exception
is a result of Alon et. al (1999) bounding the chromatic number for graphs with
low degree and few triangles per vertex; this bound is nearly the same as for
triangle-free graphs. This type of parametrization is much less rigid, and has
appeared in dozens of combinatorial constructions.
In this paper, we show a similar type of result for $\chi(G)$ as a function
of the number of vertices $n$, the number of edges $m$, as well as the triangle
count (both local and global measures). Our results smoothly interpolate
between the generic bounds true for all graphs and bounds for triangle-free
graphs. Our results are tight for most of these cases; we show how an open
problem regarding fractional chromatic number and degeneracy in triangle-free
graphs can resolve the small remaining gap in our bounds

### Deterministic parallel algorithms for bilinear objective functions

Many randomized algorithms can be derandomized efficiently using either the
method of conditional expectations or probability spaces with low independence.
A series of papers, beginning with work by Luby (1988), showed that in many
cases these techniques can be combined to give deterministic parallel (NC)
algorithms for a variety of combinatorial optimization problems, with low time-
and processor-complexity.
We extend and generalize a technique of Luby for efficiently handling
bilinear objective functions. One noteworthy application is an NC algorithm for
maximal independent set. On a graph $G$ with $m$ edges and $n$ vertices, this
takes $\tilde O(\log^2 n)$ time and $(m + n) n^{o(1)}$ processors, nearly
matching the best randomized parallel algorithms. Other applications include
reduced processor counts for algorithms of Berger (1997) for maximum acyclic
subgraph and Gale-Berlekamp switching games.
This bilinear factorization also gives better algorithms for problems
involving discrepancy. An important application of this is to automata-fooling
probability spaces, which are the basis of a notable derandomization technique
of Sivakumar (2002). Our method leads to large reduction in processor
complexity for a number of derandomization algorithms based on
automata-fooling, including set discrepancy and the Johnson-Lindenstrauss
Lemma

### A Two Term Truncation of the Multiple Ising Model Coupled to 2d Gravity

We consider a model of p independent Ising spins on a dynamical planar
phi-cubed graph. Truncating the free energy to two terms yields an exactly
solvable model that has a third order phase transition from a pure gravity
region (gamma=-1/2) to a tree-like region (gamma=1/2), with gamma=1/3 on the
critical line. We are able to make an order of magnitude estimate of the value
of p above which there exists a branched polymer (ie tree-like) phase in the
full model, that is, p is approximately 13-23, which corresponds to a central
charge c of about 6-12.Comment: 18 pages, LaTeX, 12 figure

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