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

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