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  1. We consider the Max-3-Section problem, where we are given an undirected graph G = (V, E) equipped with non-negative edge weights w : E → R+ and the goal is to find a partition of V into three equisized parts while maximizing the total weight of edges crossing between different parts. Max-3-Section is closely related to other well-studied graph partitioning problems, e.g., Max-Cut, Max-3-Cut, and Max-Bisection. We present a polynomial time algorithm achieving an approximation of 0.795, that improves upon the previous best known approximation of 0.673. The requirement of multiple parts that have equal sizes renders Max-3-Section much harder to cope with compared to, e.g., Max-Bisection. We show a new algorithm that combines the existing approach of Lassere hierarchy along with a random cut strategy that suffices to give our result. 
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    Free, publicly-accessible full text available August 30, 2024
  2. Synopsis

    Genome size varies ∼100,000-fold across eukaryotes and has long been hypothesized to be influenced by metamorphosis in animals. Transposable element accumulation has been identified as a major driver of increase, but the nature of constraints limiting the size of genomes has remained unclear, even as traits such as cell size and rate of development co-vary strongly with genome size. Salamanders, which possess diverse metamorphic and non-metamorphic life histories, join the lungfish in having the largest vertebrate genomes—3 to 40 times that of humans—as well as the largest range of variation in genome size. We tested 13 biologically-inspired hypotheses exploring how the form of metamorphosis imposes varying constraints on genome expansion in a broadly representative phylogeny containing 118 species of salamanders. We show that metamorphosis during which animals undergo the most extensive and synchronous remodeling imposes the most severe constraint against genome expansion, with the severity of constraint decreasing with reduced extent and synchronicity of remodeling. More generally, our work demonstrates the potential for broader interpretation of phylogenetic comparative analysis in exploring the balance of multiple evolutionary pressures shaping phenotypic evolution.

     
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  3. null (Ed.)
  4. Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay delta is incurred during which no data can be transmitted. For the offline version of the problem we present a (1-(1/e)-epsilon) approximation ratio (for any constant epsilon >0). Since the natural linear programming relaxation for the problem has an unbounded integrality gap, we adopt a hybrid approach that combines the combinatorial greedy with randomized rounding of a different suitable linear program. For the online version of the problem we present a (bi-criteria) ((e-1)/(2e-1)-epsilon)-competitive ratio (for any constant epsilon >0 ) that exceeds time by an additive factor of O(delta/epsilon). We note that no uni-criteria online algorithm is possible. Surprisingly, we obtain the result by reducing the online version to the offline one. 
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  5. Semi-definite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [23] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations. 
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