Some recent results in the analysis of greedy algorithms for assignment problems

We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rateɛ-balanced. Sharp bounds onɛ in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallestɛ in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,ɛ-balanced forɛ≈0.06

A Markovian Model for Joint Observations, Bell's Inequality and Hidden States

While the standard approach to quantum systems studies length preserving linear transformations of wave functions, the Markov picture focuses on trace preserving operators on the space of Hermitian (self-adjoint) matrices. The Markov approach extends the standard one and provides a refined analysis of measurements and quantum Markov chains. In particular, Bell's inequality becomes structurally clear. It turns out that hidden state models are natural in the Markov context. In particular, a violation of Bell's inequality is seen to be compatible with the existence of hidden states. The Markov model moreover clarifies the role of the "negative probabilities" in Feynman's analysis of the EPR paradox.Comment: 14 page

Submodular linear programs on forests

A general linear programming model for an order-theoretic analysis of both Edmonds' greedy algorithm for matroids and the NW-corner rule for transportation problems with Monge costs is introduced. This approach includes the model of Queyranne, Spieksma and Tardella (1993) as a special case. We solve the problem by optimal greedy algorithms for rooted forests as underlying structures. Other solvable cases are also discussed

On Hidden States in Quantum Random Walks

It was recently pointed out that identifiability of quantum random walks and hidden Markov processes underlie the same principles. This analogy immediately raises questions on the existence of hidden states also in quantum random walks and their relationship with earlier debates on hidden states in quantum mechanics. The overarching insight was that not only hidden Markov processes, but also quantum random walks are finitary processes. Since finitary processes enjoy nice asymptotic properties, this also encourages to further investigate the asymptotic properties of quantum random walks. Here, answers to all these questions are given. Quantum random walks, hidden Markov processes and finitary processes are put into a unifying model context. In this context, quantum random walks are seen to not only enjoy nice ergodic properties in general, but also intuitive quantum-style asymptotic properties. It is also pointed out how hidden states arising from our framework relate to hidden states in earlier, prominent treatments on topics such as the EPR paradoxon or Bell's inequalities.Comment: 26 page

Packing a bin online to maximize the total number of items

A bin of capacity 1 and a nite sequence of items of\ud sizes a1; a2; : : : are considered, where the items are given one by one\ud without information about the future. An online algorithm A must\ud irrevocably decide whether or not to put an item into the bin whenever\ud it is presented. The goal is to maximize the number of items collected.\ud A is f-competitive for some function f if n() f(nA()) holds for all\ud sequences , where n is the (theoretical) optimum and nA the number\ud of items collected by A.\ud A necessary condition on f for the existence of an f-competitive\ud (possibly randomized) online algorithm is given. On the other hand,\ud this condition is seen to guarantee the existence of a deterministic online\ud algorithm that is "almost" f-competitive in a well-dened sense

An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games

An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in ${\mathbb R}^n$. This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u

A Discrete Choquet Integral for Ordered Systems

A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lovász' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities.Choquet integral, belief function, measurability, set systems, Monge algorithm, supermodularity

An algorithmic characterization of antimatroids

In an article entitled “Optimal sequencing of a single machine subject to precedence constraints” E.L. Lawler presented a now classical minmax result for job scheduling. In essence, Lawler's proof demonstrated that the properties of partially ordered sets were sufficient to solve the posed scheduling problem. These properties are, in fact, common to a more general class of combinatorial structures known as antimatroids, which have recently received considerable attention in the literature. It is demonstrated that the properties of antimatroids are not only sufficient but necessary to solve the scheduling problem posed by Lawler, thus yielding an algorithmic characterization of antimatroids. Examples of problems solvable by the general result are provided