Online Mixed Packing and Covering

In many problems, the inputs arrive over time, and must be dealt with irrevocably when they arrive. Such problems are online problems. A common method of solving online problems is to first solve the corresponding linear program, and then round the fractional solution online to obtain an integral solution. We give algorithms for solving linear programs with mixed packing and covering constraints online. We first consider mixed packing and covering linear programs, where packing constraints are given offline and covering constraints are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. No prior sublinear competitive algorithms are known for this problem. We give the first such --- a polylogarithmic-competitive algorithm for solving mixed packing and covering linear programs online. We also show a nearly tight lower bound. Our techniques for the upper bound use an exponential penalty function in conjunction with multiplicative updates. While exponential penalty functions are used previously to solve linear programs offline approximately, offline algorithms know the constraints beforehand and can optimize greedily. In contrast, when constraints arrive online, updates need to be more complex. We apply our techniques to solve two online fixed-charge problems with congestion. These problems are motivated by applications in machine scheduling and facility location. The linear program for these problems is more complicated than mixed packing and covering, and presents unique challenges. We show that our techniques combined with a randomized rounding procedure give polylogarithmic-competitive integral solutions. These problems generalize online set-cover, for which there is a polylogarithmic lower bound. Hence, our results are close to tight

When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture, which states that when the supply graph $G$ is series-parallel, the pair $(G,H)$ is cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of $G$ and delete edges of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with $p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of degree 2, and an edge connecting the two vertices of degree $p$. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012

A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions

This paper presents the first combinatorial polynomial-time algorithm for minimizing submodular set functions, answering an open question posed in 1981 by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the largest length of the function value. The paper also presents a strongly polynomial-time version that runs in time bounded by a polynomial in the size of the underlying set independent of the function value.Comment: 17 page

A Stackelberg Strategy for Routing Flow over Time

Routing games are used to to understand the impact of individual users' decisions on network efficiency. Most prior work on routing games uses a simplified model of network flow where all flow exists simultaneously, and users care about either their maximum delay or their total delay. Both of these measures are surrogates for measuring how long it takes to get all of a user's traffic through the network. We attempt a more direct study of how competition affects network efficiency by examining routing games in a flow over time model. We give an efficiently computable Stackelberg strategy for this model and show that the competitive equilibrium under this strategy is no worse than a small constant times the optimal, for two natural measures of optimality

Linear tolls suffice: New bounds and algorithms for tolls in single source networks

AbstractWe show that tolls that are linear in the latency of the maximum latency path are necessary and sufficient to induce heterogeneous network users to independently choose routes that lead to traffic with minimum average latency. This improves upon the earlier bound of O(n3lmax) given by Cole, Dodis, and Roughgarden in STOC 03. (Here, n is the number of nodes in the network; and lmax is the maximum latency of any edge.) Our proof is also simpler, relating the Nash flow to the optimal flow as flows rather than cuts.We model the set of users as the set [0,1] ordered by their increasing willingness to pay tolls to reduce latency—their valuation of time. Cole et al. give an algorithm that computes optimal tolls for a bounded number of agent valuations, under the very strong assumption that they know which path each user type takes in the Nash flow imposed by these (unknown) tolls. We show that in series parallel graphs, the set of paths traveled by users in any Nash flow with optimal tolls is independent of the distribution of valuations of time of the users. In particular, for any continuum of users (not restricted to a finite number of valuation classes) in series parallel graphs, we show how to compute these paths without knowing α.We give a simple example to demonstrate that if the graph is not series parallel, then the set of paths traveled by users in the Nash flow depends critically on the distribution of users’ valuations of time

Quickest Flows Over Time

Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in time‐expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time‐expanded network. We present several approaches for coping with this difficulty. First, inspired by the work of Ford and Fulkerson on maximal s‐t‐flows over time (or “maximal dynamic s‐t‐flows”), we show that static length‐bounded flows lead to provably good multicommodity flows over time. Second, we investigate “condensed” time‐expanded networks which rely on a rougher discretization of time. We prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed time‐expanded network of polynomial size. In particular, our approach yields fully polynomial‐time approximation schemes for the NP‐hard quickest min‐cost and multicommodity flow problems. For single commodity problems, we show that storage of flow at intermediate nodes is unnecessary, and our approximation schemes do not use any