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

Improved algorithms and analysis for the laminar matroid secretary problem

In a matroid secretary problem, one is presented with a sequence of objects of various weights in a random order, and must choose irrevocably to accept or reject each item. There is a further constraint that the set of items selected must form an independent set of an associated matroid. Constant-competitive algorithms (algorithms whose expected solution weight is within a constant factor of the optimal) are known for many types of matroid secretary problems. We examine the laminar matroid and show an algorithm achieving provably 0.053 competitive ratio

The Liberian Truth and Reconciliation Commission: reconciling or re-dividing Liberia?

After 14 years of civil war and violence followed by the momentous and rather unusual elections of 2005, in which a woman defeated a footballer for the presidency, Liberia has seen over six years of state reconstruction and relative peace. Two recent announcements have, however, served as a warning to the extent of progress. The most recent is President Ellen Johnson-Sirleaf’s declaration that she will, despite previous statements to the contrary, stand for re-election in 2011 due to shortcomings in progress. The announcement preceding Johnson-Sirleaf’s was made in the form of the report of the Liberian Truth and Reconciliation Commission (TRC). It recommended that Johnson-Sirleaf, and indeed many others accused of involvement in the war, should be barred from public office for the next 30 years, and still more should stand trial on charges of war crimes. Four important questions arise. First, what was the mandate and findings of the TRC? Second, how has Liberia and the wider international community reacted to the final report? Third, has the TRC fulfilled its mandate and contributed to a process of reconciliation? Finally, and in a much broader sense, where does the TRC stand relative to the much wider liberal peace model

Poliovirus mutant that contains a cold-sensitive defect in viral RNA synthesis

By manipulating an infectious cDNA clone of poliovirus, we have introduced a single-codon insertion into the 3A region of the viral genome which has been proposed to encode a functional precursor of the virion-linked protein VPg. The resulting mutant was cold sensitive in monkey kidney cells. Viral RNA synthesis was poor at 32.5 degrees C, although no other function of the virus was obviously affected. The synthesis of both positive and negative strands was severely depressed. Temperature shift experiments suggest that a normal level of production of the affected function was required only during the early (exponential) phase of RNA synthesis. Analysis of viral polyprotein processing at the nonpermissive temperature revealed that some of the normal cleavages were not made, most likely as a consequence of the defect in RNA synthesis or as a result of the concomitant reduction in the level of virally encoded proteases

Agricultural Trade Reform and Industry Adjustment in Indonesia

This paper presents a component of a project on industry adjustment to agricultural trade reform in selected developing countries. The aim of the project is to examine the issues affecting the development of industry adjustment policies to manage the impact of trade reform. It will evaluate specific developing country examples of industries that are likely to face significant adjustment pressures from trade policy reform. The study is focused on industry specific policy responses for two reasons. First, many LDC's are concerned about the consequences of future WTO reforms for adjustment in 'sensitive' industries. Governments in developing countries have received advice and assistance on how to comply with the requirements of their WTO commitments from the Uruguay Round of trade negotiations. However, very little attention has been devoted to the domestic effects of trade reform. Second, the implementation of international trade commitments is likely to lead to industry requests for assistance. Adjustment policies used by developed countries may not be directly applicable to LDC situations. Differences in structural characteristics, institutional arrangements and the level of industry development require an investigation of the issues affecting adjustment in developing countries.International Relations/Trade,

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 $M \gg n \gg 1$

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)