Approximating the single source unsplittable min-cost flow problem

In the single source unsplittable min-cost flow problem, commodities must be routed simultaneously from a common source vertex to certain destination vertices in a given graph with edge capacities and costs; the demand of each commodity must be routed along a single path and the total cost must not exceed a given budget. This problem has been introduced by Kleinberg and generalizes several NP-complete problems from various areas in combinatorial optimization such as packing, partitioning, scheduling, load balancing and virtual-circuit routing

Scheduling precedence-constrained jobs with stochastic processing times on parallel machines

We consider parallel machine scheduling problems where the jobs are subject to precedence constraints, and the processing times of jobs are governed by independent probability distributions. The objective is to minimize the weighted sum of job completion times &;j wjC_j in expectation, where wj&; 0. Building upon an LP-relaxation by Möhring, Schulz, and Uetz (J.ACM 46 (1999), pp.924-942) and an idle time charging scheme by Chekuri, Motwani, Natarajan, and Stein (SIAM J. Comp., to appear) we derive the first approximation algorithms for this model

An Improved Upper Bound for the Ring Loading Problem

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances

Congestion-Free Rerouting of Flows on DAGs

Changing a given configuration in a graph into another one is known as a reconfiguration problem. Such problems have recently received much interest in the context of algorithmic graph theory. We initiate the theoretical study of the following reconfiguration problem: How to reroute k unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner? This problem finds immediate applications, e.g., in traffic engineering in computer networks. We show that the problem is generally NP-hard already for k=2 flows, which motivates us to study rerouting on a most basic class of flow graphs, namely DAGs. Interestingly, we find that for general k, deciding whether an unsplittable multi-commodity flow rerouting schedule exists, is NP-hard even on DAGs. Our main contribution is a polynomial-time (fixed parameter tractable) algorithm to solve the route update problem for a bounded number of flows on DAGs. At the heart of our algorithm lies a novel decomposition of the flow network that allows us to express and resolve reconfiguration dependencies among flows

Extending partial suborders and implication classes

We consider the following problem called transitive ordering with precedence constraints (TOP): Given a partial order P=(V,&;) and an (undirected) graph G=(V,E) such that all relations in P are represented by edges in G. Is there a transitive orientation D=(V,A) of G, such that P is contained in D

An origin-based model for unique shortest path routing

Link weights are the main parameters of shortest path routing protocols, the most commonly used protocols for IP networks. The problem of optimally setting link weights for unique shortest path routing is addressed. Due to the complexity of the constraints involved, there exist challenges to formulate the problem in such a way based on which a more efficient solution algorithm than the existing ones may be developed. In this paper, an exact formulation is first introduced and then mathematically proved correct. It is further illustrated that the formulation has advantages over a prior one in terms of both constraint structure and model size for a proposed decomposition method to solve the problem

FASTER ALGORITHMS FOR STABLE ALLOCATION PROBLEMS

We consider a high-multiplicity generalization of the classical stable matching problem known as the stable allocation problem, introduced by Baiou and Balinski in 2002. By leveraging new structural properties and sophisticated data structures, we show how to solve this problem in O(mlog n) time on an bipartite instance with n nodes and m edges, improving the best known running time of O(mn). Our approach simplifies the algorithmic landscape for this problem by providing a common generalization of two different approaches from the literature -- the classical Gale-Shapley algorithm, and a recent algorithm of Baiou and Balinski. Building on this algorithm, we provide an O(m log n) algorithm for the non-bipartite stable allocation problem that introduces a new and useful transformation from non-bipartite to bipartite instances. We also give a polynomial-time algorithm for solving the \u27optimal\u27 variant of the bipartite stable allocation problem, as well as a 2-approximation algorithm for the NP-hard \u27optimal\u27 variant of the non-bipartite stable allocation problem. Finally, we highlight some important connections between the stable allocation problem and the maximum flow problem

Scheduling Scarce Resources in Chemical Engineering

The efficient utilization of scarce resources, such as machines or manpower, is major challenge within production planning in the chemical industry. We describe solution methods for a resource-constrained scheduling problem which arises at a production facility at BASF AG in Ludwigshafen. We have developed and implemented two different algorithms to solve this problem, a novel approach which is based upon Lagrangian relaxation, as well as a branch-and-bound procedure. Since the Lagrangian approach is applicable for a whole variety of resource-constrained scheduling problems, it is of interest not only for the specific problem we describe, but is of interest also for many other industrial applications. In this paper, we describe both approaches, and also report on computational results, based upon practical problem instances as well as benchmark test sets