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

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

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

The Moser-Tardos Framework with Partial Resampling

The resampling algorithm of Moser \& Tardos is a powerful approach to develop constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006) on graph transversals asymptotically, and obtain improved approximation ratios for a packet routing problem of Leighton, Maggs, & Rao (1994)

Parameter estimation for integer-valued Gibbs distributions

We consider Gibbs distributions, which are families of probability distributions over a discrete space $\Omega$ with probability mass function given by $\mu^\Omega_\beta(x) = \frac{e^{\beta H(x)}}{Z(\beta)}$. Here $H:\Omega\rightarrow\{0,..,n\}$ is a fixed function (called a Hamiltonian), $\beta$ is the parameter of the distribution, and the normalization factor $Z(\beta)=\sum_{x\in\Omega}e^{\beta H(x)}=\sum_{k=0}^nc_ke^{\beta k}$ is called the partition function. We study how function $Z$ can be estimated using an oracle that produces samples $x\sim\mu^\Omega_\beta(.)$ for a value $\beta$ in a given interval $[\beta_{min},\beta_{max}]$. We consider the problem of estimating the normalized coefficients $c_k$ for indices $k\in\cal K$ satisfying $\max_\beta\mu^\Omega_\beta(\{x|H(x)=k\})\ge\mu_*$, where $\mu_*\in(0,1)$ is a given parameter and $\cal K$ is a given subset of $\cal H$. We solve this using $\tilde O(\frac{\min\{q,n^2\}+\frac{\min\{\sqrt q,|\cal K|\}}{\mu_*}}{\epsilon^2})$ samples where $q=\log\frac{Z(\beta_{max})}{Z(\beta_{min})}$, and we show this is optimal up to logarithmic factors. We also improve the sample complexity to roughly $\tilde O(\frac{1/\mu_*+\min\{q+n,n^2\}}{\epsilon^2})$ for applications where the coefficients are log-concave (e.g. counting connected subgraphs of a given graph). As a key subroutine, we show how to estimate $q$ using $\tilde O(\frac{\min\{q,n^2\}}{\epsilon^2})$ samples. This improves over a prior algorithm of Kolmogorov (2018) that uses $\tilde O(\frac q{\epsilon^2})$ samples. We also show a "batched" version of this algorithm which simultaneously estimates $\frac{Z(\beta)}{Z(\beta_{min})}$ for many values of $\beta$, at essentially the same cost as for estimating just $\frac{Z(\beta_{max})}{Z(\beta_{min})}$ alone. We show matching lower bounds, demonstrating that this complexity is optimal as a function of $n,q$ up to logarithmic terms.Comment: Superseded by arXiv:2007.1082

Improved bounds and algorithms for graph cuts and network reliability

Karger (SIAM Journal on Computing, 1999) developed the first fully-polynomial approximation scheme to estimate the probability that a graph $G$ becomes disconnected, given that its edges are removed independently with probability $p$. This algorithm runs in $n^{5+o(1)} \epsilon^{-3}$ time to obtain an estimate within relative error $\epsilon$. We improve this run-time through algorithmic and graph-theoretic advances. First, there is a certain key sub-problem encountered by Karger, for which a generic estimation procedure is employed, we show that this has a special structure for which a much more efficient algorithm can be used. Second, we show better bounds on the number of edge cuts which are likely to fail. Here, Karger's analysis uses a variety of bounds for various graph parameters, we show that these bounds cannot be simultaneously tight. We describe a new graph parameter, which simultaneously influences all the bounds used by Karger, and obtain much tighter estimates of the cut structure of $G$. These techniques allow us to improve the runtime to $n^{3+o(1)} \epsilon^{-2}$, our results also rigorously prove certain experimental observations of Karger & Tai (Proc. ACM-SIAM Symposium on Discrete Algorithms, 1997). Our rigorous proofs are motivated by certain non-rigorous differential-equation approximations which, however, provably track the worst-case trajectories of the relevant parameters. A key driver of Karger's approach (and other cut-related results) is a bound on the number of small cuts: we improve these estimates when the min-cut size is "small" and odd, augmenting, in part, a result of Bixby (Bulletin of the AMS, 1974)

Tight bounds and conjectures for the isolation lemma

Given a hypergraph $H$ and a weight function $w: V \rightarrow \{1, \dots, M\}$ on its vertices, we say that $w$ is isolating if there is exactly one edge of minimum weight $w(e) = \sum_{i \in e} w(i)$. The Isolation Lemma is a combinatorial principle introduced in Mulmuley et. al (1987) which gives a lower bound on the number of isolating weight functions. Mulmuley used this as the basis of a parallel algorithm for finding perfect graph matchings. It has a number of other applications to parallel algorithms and to reductions of general search problems to unique search problems (in which there are one or zero solutions). The original bound given by Mulmuley et al. was recently improved by Ta-Shma (2015). In this paper, we show improved lower bounds on the number of isolating weight functions, and we conjecture that the extremal case is when $H$ consists of $n$ singleton edges. When $M \gg n$ our improved bound matches this extremal case asymptotically. We are able to show that this conjecture holds in a number of special cases: when $H$ is a linear hypergraph or is 1-degenerate, or when $M = 2$. We also show that it holds asymptotically when $M \gg n \gg 1$