Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E} C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a strongly polynomial algorithm for the case when all $C_{ij}$'s are convex quadratic functions, settling an open problem raised e.g. by Hochbaum [1994]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities, that can be formulated in this framework (see Shmyrev [2009], Devanur et al. [2011]). For the latter class this resolves an open question raised by Vazirani [2010]. The running time is $O(m^4\log m)$ for quadratic costs, $O(n^4+n^2(m+n\log n)\log n)$ for Fisher's markets with linear utilities and $O(mn^3 +m^2(m+n\log n)\log m)$ for spending constraint utilities. All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [1994]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings

A strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.Comment: minor correction

Exact and Efficient Simulation of Concordant Computation

Concordant computation is a circuit-based model of quantum computation for mixed states, that assumes that all correlations within the register are discord-free (i.e. the correlations are essentially classical) at every step of the computation. The question of whether concordant computation always admits efficient simulation by a classical computer was first considered by B. Eastin in quant-ph/1006.4402v1, where an answer in the affirmative was given for circuits consisting only of one- and two-qubit gates. Building on this work, we develop the theory of classical simulation of concordant computation. We present a new framework for understanding such computations, argue that a larger class of concordant computations admit efficient simulation, and provide alternative proofs for the main results of quant-ph/1006.4402v1 with an emphasis on the exactness of simulation which is crucial for this model. We include detailed analysis of the arithmetic complexity for solving equations in the simulation, as well as extensions to larger gates and qudits. We explore the limitations of our approach, and discuss the challenges faced in developing efficient classical simulation algorithms for all concordant computations.Comment: 16 page

The computational complexity of the Chow form

We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system defining the variety. In particular, it provides an alternative algorithm for the equidimensional decomposition of a variety. As an application we obtain an algorithm for the computation of a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As a further application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined equation system.Comment: 60 pages, Latex2

A Linux PC cluster for lattice QCD with exact chiral symmetry

A computational system for lattice QCD with exact chiral symmetry is described. The platform is a home-made Linux PC cluster, built with off-the-shelf components. At present this system constitutes of 64 nodes, with each node consisting of one Pentium 4 processor (1.6/2.0/2.5 GHz), one Gbyte of PC800/PC1066 RDRAM, one 40/80/120 Gbyte hard disk, and a network card. The computationally intensive parts of our program are written in SSE2 codes. The speed of this system is estimated to be 70 Gflops, and its price/performance is better than $1.0/Mflops for 64-bit (double precision) computations in quenched QCD. We discuss how to optimize its hardware and software for computing quark propagators via the overlap Dirac operator.Comment: 24 pages, LaTeX, 2 eps figures, v2:a note and references added, the version published in Int. J. Mod. Phys.