EFX exists for three agents

We study the problem of distributing a set of indivisible items among agents with additive valuations in a $\mathit{fair}$ manner. The fairness notion under consideration is Envy-freeness up to any item (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this paper, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture by Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some items are not allocated) with higher Nash welfare than that of any complete EFX allocation

Critical behavior of the Random-Field Ising Magnet with long range correlated disorder

We study the correlated-disorder driven zero-temperature phase transition of the Random-Field Ising Magnet using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportional to r^a, where r is the distance between two lattice sites and a<0. To obtain exact ground states, we use a well established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than two million spins. We use finite-size scaling analyses for values a={-1,-2,-3,-7} to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility and of the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with delta-correlated disorder, except for the finite-size exponent of the susceptibility and for the case a=-1, where the results are also compatible with a phase transition at infinitesimal disorder strength. A summary of this work can be found at the papercore database at www.papercore.org.Comment: 9 pages, 13 figure

Faster algorithms for the shortest path problem

We investigate efficient implementations of Dijkstra\u27s shortest path algorithm. We propose a new data structure, called the redistributive heap, for use in this algorithm. On a network with n vertices, m edges, and nonnegative integer arc costs bounded by C, a one-level form of redistributive heap gives a time bound for Dijkstra\u27s algorithm of O(m + nlogC). A two-level form of redistributive heap gives a bound of O(m + nlogC/loglogC). A combination of a redistributive heap and a previously known data structure called a Fibonacci heap gives a bound of O(m+ nsqrt{log C}). The best previously known bounds are O(m + nlogn) using Fibonacci heaps alone and O(mloglogC) using the priority queue structure of Van Emde Boas, Kaas, and Zijlstra

Nash Social Welfare for 2-value Instances

We study the problem of allocating a set of indivisible goods among agents with 2-value additive valuations. Our goal is to find an allocation with maximum Nash social welfare, i.e., the geometric mean of the valuations of the agents. We give a polynomial-time algorithm to find a Nash social welfare maximizing allocation when the valuation functions are integrally 2-valued, i.e., each agent has a value either $1$ or $p$ for each good, for some positive integer $p$. We then extend our algorithm to find a better approximation factor for general 2-value instances

Photon and neutron production as in-situ diagnostics of proton-boron fusion

Short-pulse, ultra high-intensity lasers have opened new regimes for studying fusion plasmas and creating novel ultra-short ion beams and neutron sources. Diagnosing the plasma in these experiments is important for optimizing the fusion yield but difficult due to the picosecond time scales, 10s of micron-cubed volumes and high densities. We propose to use the yields of photons and neutrons produced by parallel reactions involving the same reactants to diagnose the plasma conditions and predict the yields of specific reactions of interest. In this work, we focus on verifying the yield of the high-interest aneutronic proton-boron fusion reaction $^{11}{B}(p,2\alpha){}^4{He}$, which is difficult to measure directly due to the short stopping range of the produced $\alpha$s in most materials. We identify promising photon-producing reactions for this purpose and compute the ratios of the photon yield to the $\alpha$ yield as a function of plasma parameters. In beam fusion experiments, the ${}^{11}{C}$ yield is an easily-measurable observable to verify the $\alpha$ yield. In light of our results, improving and extending measurements of the cross sections for these parallel reactions are important steps to gaining greater control over these laser-driven fusion plasmas.Comment: 23 pages, 7 figures, revtex forma

EFX Allocations: Simplifications and Improvements

The existence of EFX allocations is a fundamental open problem in discretefair division. Given a set of agents and indivisible goods, the goal is todetermine the existence of an allocation where no agent envies anotherfollowing the removal of any single good from the other agent's bundle. Sincethe general problem has been illusive, progress is made on two fronts: $(i)$proving existence when the number of agents is small, $(ii)$ proving existenceof relaxations of EFX. In this paper, we improve results on both fronts (andsimplify in one of the cases). We prove the existence of EFX allocations with three agents, restricting onlyone agent to have an MMS-feasible valuation function (a strict generalizationof nice-cancelable valuation functions introduced by Berger et al. whichsubsumes additive, budget-additive and unit demand valuation functions). Theother agents may have any monotone valuation functions. Our proof technique issignificantly simpler and shorter than the proof by Chaudhury et al. onexistence of EFX allocations when there are three agents with additivevaluation functions and therefore more accessible. Secondly, we consider relaxations of EFX allocations, namely, approximate-EFXallocations and EFX allocations with few unallocated goods (charity). Chaudhuryet al. showed the existence of $(1-\epsilon)$-EFX allocation with$O((n/\epsilon)^{\frac{4}{5}})$ charity by establishing a connection to aproblem in extremal combinatorics. We improve their result and prove theexistence of $(1-\epsilon)$-EFX allocations with $\tilde{O}((n/\epsilon)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be usedto prove improved upper-bounds on a problem in zero-sum combinatoricsintroduced by Alon and Krivelevich.<br

On Fair Division of Indivisible Items

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of $e^{1/e} \approx 1.445$

Structure-guided mutational analysis reveals the functional requirements for product specificity of DOT1 enzymes

DOT1 enzymes are conserved methyltransferases that catalyse the methylation of lysine 79 on histone H3 (H3K79). Most eukaryotes contain one DOT1 enzyme, whereas African trypanosomes have two homologues, DOT1A and DOT1B, with different enzymatic activities. DOT1A mediates mono-and dimethylation of H3K76, the homologue of H3K79 in other organisms, whereas DOT1B additionally catalyses H3K76 trimethylation. However, it is unclear how these different enzymatic activities are achieved. Here we employ a trypanosomal nucleosome reconstitution system and structure-guided homology modelling to identify critical residues within and outside the catalytic centre that modulate product specificity. Exchange of these residues transfers the product specificity from one enzyme to the other, and reveals the existence of distinct regulatory domains adjacent to the catalytic centre. Our study provides the first evidence that a few crucial residues in DOT1 enzymes are sufficient to catalyse methyl-state-specific reactions. These results might also have far-reaching consequences for the functional understanding of homologous enzymes in higher eukaryotes

Structure-guided mutational analysis reveals the functional requirements for product specificity of DOT1 enzymes

DOT1 enzymes are conserved methyltransferases that catalyse the methylation of lysine 79 on histone H3 (H3K79). Most eukaryotes contain one DOT1 enzyme, whereas African trypanosomes have two homologues, DOT1A and DOT1B, with different enzymatic activities. DOT1A mediates mono-and dimethylation of H3K76, the homologue of H3K79 in other organisms, whereas DOT1B additionally catalyses H3K76 trimethylation. However, it is unclear how these different enzymatic activities are achieved. Here we employ a trypanosomal nucleosome reconstitution system and structure-guided homology modelling to identify critical residues within and outside the catalytic centre that modulate product specificity. Exchange of these residues transfers the product specificity from one enzyme to the other, and reveals the existence of distinct regulatory domains adjacent to the catalytic centre. Our study provides the first evidence that a few crucial residues in DOT1 enzymes are sufficient to catalyse methyl-state-specific reactions. These results might also have far-reaching consequences for the functional understanding of homologous enzymes in higher eukaryotes