The problem of allocating divisible goods has enjoyed a lot of attention in both mathematics (e.g. the cake-cutting problem) and economics (e.g. market equilibria). On the other hand, the natural requirement of indivisible goods has been somewhat neglected, perhaps because of its more complicated nature. In this work we study a fairness criterion, called the Max-Min Fairness problem, for k players who want to allocate among themselves m indivisible goods. Each player has a specified valuation function on the subsets of the goods and the goal is to split the goods between the players so as to maximize the minimum valuation. Viewing the problem from a game theoretic perspective, we show that for two players and additive valuations the expected minimum of the (randomized) cut-and-choose mechanism is a 1/2-approximation of the optimum. To complement this result we show that no truthful mechanism can compute the exact optimum.We also consider the algorithmic perspective when the (true) additive valuation functions are part of the input. We present a simple 1/( m - k + 1) approximation algorithm which allocates to every player at least 1/ k fraction of the value of all but the k - 1 heaviest items. We also give an algorithm with additive error against the fractional optimum bounded by the value of the largest item. The two approximation algorithms are incomparable in the sense that there exist instances when one outperforms the other.

Energy is often the most constrained resource in networks of battery-powered devices, and as devices become smaller, they spend a larger fraction of their energy on communication (transceiver usage) not computation. As an imperfect proxy for true energy usage, we define energy complexity to be the number of time slots a device transmits/listens; idle time and computation are free.In this paper we investigate the energy complexity of fundamental communication primitives such as Broadcast in multi-hop radio networks. We consider models with collision detection (CD) and without (No-CD), as well as both randomized and deterministic algorithms. Some take-away messages from this work include:• The energy complexity of Broadcast in a multi-hop network is intimately connected to the time complexity of LeaderElection in a single-hop (clique) network. Many existing lower bounds on time complexity immediately transfer to energy complexity. For example, in the CD and No-CD models, we need Ω(log n) and Ω(log 2 n) energy, respectively.• The energy lower bounds above can almost be achieved, given sufficient (Ω(n)) time. In the CD and No-CD models we can solve Broadcast using O( log n log log n log log log n ) energy and O(log 3 n) energy, respectively. • The complexity measures of Energy and Time are in conflict, and it is an open problem whether both can be minimized simultaneously. We give a tradeoff showing it is possible to be nearly optimal in both measures simultaneously. For any constant > 0, Broadcast can be solved in O(D 1+ log O(1/ ) n) time with O(log O(1/ ) n) energy, where D is the diameter of the network.

We introduce a reduction-based model for analyzing supervised learning tasks. We use this model to devise a new reduction from cost-sensitive classification to binary classification with the following guarantee: If the learned binary classifier has error rate at most then the cost-sensitive classifier has cost at most 2 times the expected sum of costs of all choices. Since cost-sensitve classification can embed any bounded loss finite choice supervised learning task, this result shows that any such task can be solved using a binary classification oracle. Finally, we present experimental results showing that our new reduction outperforms existing algorithms for multi-class cost-sensitive learning. Preliminary work. Under review by the International Conference on Machine Learning (ICML). Do not distribute.

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We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes K variables that must be assigned one out of q colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs (K = 2) and bicoloring of random hypergraphs (q = 2), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size q of the domain of the variables and the arity K of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem (q = 2) where we find that for K = 3 and K = 4 the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of q = 2, K ≥ 4. Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where q and/or K diverge.