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

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 Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems

Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form ?? = ?, where ? is the Laplacian matrix of a weighted graph with weights w(i,j) > 0 on the edges. The solution ? of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i,j) is 1/w(i,j). Kelner, Orrechia, Sidford, and Zhu [Kelner et al., 2013] give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem (OMv), which has been conjectured to be difficult for dynamic algorithms [Henzinger et al., 2015]. The conjecture implies that the data structure does not have an O(n^{1-?}) time algorithm for any ? > 0, and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an O?(m^{1.5}) time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O(m^{1 + ?}) for any ? > 0. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart

Firmament: Fast, Centralized Cluster Scheduling at Scale

Centralized datacenter schedulers can make high-quality placement decisions when scheduling tasks in a cluster. Today, however, high-quality placements come at the cost of high latency at scale, which degrades response time for interactive tasks and reduces cluster utilization. This paper describes Firmament, a centralized scheduler that scales to over ten thousand machines at sub- second placement latency even though it continuously reschedules all tasks via a min-cost max-flow (MCMF) optimization. Firmament achieves low latency by using multiple MCMF algorithms, by solving the problem incrementally, and via problem-specific optimizations. Experiments with a Google workload trace from a 12,500-machine cluster show that Firmament improves placement latency by 20 x over Quincy [22], a prior centralized scheduler using the same MCMF optimiza- tion. Moreover, even though Firmament is centralized, it matches the placement latency of distributed schedulers for workloads of short tasks. Finally, Firmament exceeds the placement quality of four widely-used central- ized and distributed schedulers on a real-world cluster, and hence improves batch task response time by 6 x.This work was supported by a Google European Doc- toral Fellowship, by NSF award CNS-1413920, and by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory (AFRL), under contract FA8750-11-C-0249