95,399 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
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and 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
approximation algorithm using randomized time and 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 -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or 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 as a function
of the number of vertices , the number of edges , 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 with edges and vertices, this
takes time and 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 Color-Magnitude Diagram for a Globular Cluster In the Giant Elliptical Galaxy NGC 5128
The Hubble Space Telescope has been used to obtain WFPC2 (V,I) photometry for
a large sample of stars in the outer halo of the giant elliptical NGC 5128 (d =
4 Mpc). The globular cluster N5128-C44, at the center of the Planetary Camera
field, is well enough resolved to permit the construction of a color-magnitude
diagram (CMD) for it which covers the brightest two magnitudes of the giant
branch. The CMD is consistent with that of a normal old, moderately
low-metallicity ([Fe/H] = -1.30 globular cluster, distinctly more metal-poor
than most of the field halo stars at the same projected location (which average
[Fe/H] ~ -0.5). This is the most distant globular cluster in which direct
color-magnitude photometry has been achieved to date, and the first one
belonging to a giant E galaxy.Comment: 12 pages, LaTeX, including 5 postscript figures; submitted to
Astronomical Journa
V-Mail Written by Robert G. Harris to the Bryant College Service Club Dated June 5, 1943
[Transcription begins] Cpl. Robt. Harris 31119022 3406 Ord. MM Co. (Q) APO # 758, c/o Postmaster, N. Y. June 5, 1943
BRYANT SERVICE CLUB c/o BRYANT COLLEGE YOUNG & ORCHARD STS. PROVIDENCE, RHODE ISLAND U. S. A.
Dear Bryant Service Club:
It was a very pleasant surprise for me to receive a letter from my old Alma Mater. I think the idea of you kids back home keeping in touch with the old grads. is a swell jesture [sic]. Things like that make a fellow over here realize that he as [sic] a few friends in the world after all. Being away from home and in with a group of fellows you never new [sic], gives a soldier the impression he is almost alone in the world. Keep up your good work and let me in the latest news more often if possible. However, I realize that you must have a very difficult time trying to contact all of the boys.
I am situated in North Africa, the exact spot can’t be mentioned because of the censor. This is a land of sun, sand and palm trees but mostly sun and sand. It is indeed [sic] a very educational part of the world, but as for myself they can give it back to the Arabs. Speaking of Arabs, there is a subject that [I] could write a book about but I’m afried [sic] it would not be published. I can say though that the movies builds you up for a big let-down, when it shows you some scenes of the country. There is one thing that interests me about these natives, and that is the work that they turn out by hand. They are still using methods that have been in practise [sic] for generation after generation, but the results obtained are wonderful. The difficulty is that a person needs a fortune to buy any of the things that they make. The chief mode of transportation is the burro and the less fortunates walk. It strikes me funny to see them walking down the street with their shoes in their hands. Personally I don’t get the point. Have you any ideas?
Perhaps you would like to know my duties as a soldier. Well there isn’t much to write about on that subject. Every Company has an Orderly room in which the administrative work is done, and that is where I work. Being an accountant, they thought I would make a good clerk. Any one who has worked in civilian life as a clerk or accoutant [sic] I’M sure would find the Army administration interesting and amuseing [sic].
That seems to give you a very brief idea of some of the points as seen from a man on overseas duty. If anyone in school would like a little advise [sic] on planning for entrance into the service, I would suggest that they try and get into the Air Corps, either as a flyer or ground man. The Air Corps needs plenty of men for administration and is about the best branch of service in my estimation.
Good luck to you in your Bryant Service Club and as I have written before write me a word whenever you can. I was in the class of ’38, and any new [sic] of the boys that were in that class and are now in the army would be of great interest to me. Write and let me know about them.
May we all be celebrating a complete victory very soon,
Sincerely, Robert G. Harris [Transcription ends
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