Greedy distributed optimization of multi-commodity flows

The multi-commodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connection-oriented network architectures. We consider solutions for distributed multi-commodity flow problems, which are solved by multiple agents operating in a cooperative but uncoordinated manner. We provide the first stateless greedy distributed algorithm for the concurrent multi-commodity flow problem with poly-logarithmic convergence. More precisely, our algorithm achieves ${1+\varepsilon}The multi-commodity flow problem is a classical combinatorial optimization problem that addresses a number of practically important issues of congestion and bandwidth management in connection-oriented network architectures. We consider solutions for distributed multi-commodity flow problems, which are solved by multiple agents operating in a cooperative but uncoordinated manner. We provide the first stateless greedy distributed algorithm for the concurrent multi-commodity flow problem with poly-logarithmic convergence. More precisely, our algorithm achieves 1+e{1+\varepsilon} approximation, with running time O(H· log^O(1)m· (1/e)^O(1)){O(H{\cdot} \log^{O(1)}m{\cdot} (1{/}\varepsilon)^{O(1)})} where H is the number of edges on any allowed flow-path. No prior results exist for our model. Our algorithm is a reasonable alternative to existing polynomial sequential approximation algorithms, such as Garg–K?nemann (Proceedings of the 39th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, USA, pp. 300–309, 1998). The algorithm is simple and can be easily implemented or taught in a classroom. Remarkably, our algorithm requires that the increase in the flow rate on a link is more aggressive than the decrease in the rate. Essentially all of the existing flow-control heuristics are variations of TCP, which uses a conservative cap on the increase (e.g., additive), and a rather liberal cap on the decrease (e.g., multiplicative). In contrast, our algorithm requires the increase to be multiplicative, and that this increase is dramatically more aggressive than the decrease.     

Models and methods for solving the problem of network vulnerability

The problem of analysis of the vulnerability of single-commodity and multi-commodity networks taking into account complete breakdown of one or several arcs is considered. The guaranteed estimate of the loss of users of this network is chosen as the criterion of the network performance. A number of formal statements of the problem are proposed for vulnerability analysis. These statements of the problem are studied by flow, graph theoretical, probabilistic, and other methods. The cases when these statements of the problem are polynomially solvable are specified, and the corresponding algorithms are given. In the other cases, it is proposed to use approximation, combinatorial, and other methods.     

Approximation Algorithms for Soft-Capacitated Facility Location in Capacitated Network Design

Answering an open question published in Operations Research (54, 73–91, 2006) in the area of network design and logistic optimization, we present the first constant-factor approximation algorithms for the problem combining facility location and cable installation in which capacity constraints are imposed on both facilities and cables. We study the problem of designing a minimum cost network to serve client demands by opening facilities for service provision and installing cables for service shipment. Both facilities and cables have capacity constraints and incur buy-at-bulk costs. This Max SNP-hard problem arises in diverse applications and is shown in this paper to admit a combinatorial 19.84-approximation algorithm of cubic running time. This is achieved by an integration of primal-dual schema, Lagrangian relaxation, demand clustering and bi-factor approximation. Our techniques extend to several variants of this problem, which include those with unsplitable demands or requiring network connectivity, and provide constant-factor approximate algorithms in strongly polynomial time. X. Chen is Visiting Fellow, University of Warwick.     

Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem

The capacitated multi-facility Weber problem is concerned with locating I capacitated facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. This is a nonconvex optimization problem and difficult to solve. In this work, we focus on a multi-commodity extension and consider the situation where K distinct commodities are shipped subject to capacity constraints between each customer and facility pair. Customer locations, demands and capacities for each commodity, and bundle restrictions are known a priori. The transportation costs, which are proportional to the distance between customers and facilities, depend on the commodity type. We address several location-allocation and discrete approximation heuristics using different strategies. Based on the obtained computational results we can say that the alternate solution of location and allocation problems is a very efficient strategy; but the discrete approximation has excellent accuracy.     

Optimal Routing Designs in Self-healing Communications Networks

Self-healing communication networks that allow re-routing of demands through switching processes at designated nodes are studied. It is shown how network utilization, demand throughput and reliability of such networks can be studied simultaneously to achieve an optimal design for all three. This is done through a max–min–max multi-commodity network flow formulation of the routing problem in which it is ensured that maximum network throughput is achieved with minimum loss of demands that are blocked due to single switching node failures. It is shown that a node-path linear programming approximation to the multi-commodity network flow formulation solves the problem for medium and large network sizes in moderate computational times.     

ε-Optimality for bicriteria programs and its application to minimum cost flows

A subsetS?X of feasible solutions of a multicriteria optimization problem is called ε-optimal w.r.t. a vector-valued functionf:X→Y \( \subseteq \) ? ^K if for allx∈X there is a solutionz _x∈S so thatf _k(z _x)≤(1+ε)f _k (x) for allk=1,...,K. For a given accuracy ε>0, a pseudopolynomial approximation algorithm for bicriteria linear programming using the lower and upper approximation of the optimal value function is given. Numerical results for the bicriteria minimum cost flow problem on NETGEN-generated examples are presented.     

Combinatorial Algorithms for the Unsplittable Flow Problem

We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.     

On the complexity of bandwidth allocation in radio networks

We define and study an optimization problem that is motivated by bandwidth allocation in radio networks. Because radio transmissions are subject to interference constraints in radio networks, physical space is a common resource that the nodes have to share in such a way, that concurrent transmissions do not interfere. The bandwidth allocation problem we study under these constraints is the following. Given bandwidth (traffic) demands between the nodes of the network, the objective is to schedule the radio transmissions in such a way that the traffic demands are satisfied. The problem is similar to a multicommodity flow problem, where the capacity constraints are replaced by the more complex notion of non-interfering transmissions. We provide a formal specification of the problem that we call round weighting  . By modeling non-interfering radio transmissions as independent sets, we relate the complexity of round weighting to the complexity of various independent set problems (e.g. maximum weight independent set, vertex coloring, fractional coloring). From this relation, we deduce that in general, round weighting is hard to approximate within n^1−ε$n^{1-\epsilon }$ (n$n$ being the size of the radio network). We also provide polynomial (exact or approximation) algorithms e.g. in the following two cases: (a) when the interference constraints are specific (for instance for a network whose vertices belong to the Euclidean space), or (b) when the traffic demands are directed towards a unique node in the network (also called gathering, analogous to single commodity flow).     

Combinatorial algorithms for feedback problems in directed graphs

Given a weighted directed graph G=(V,A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A′⊆A such that the directed graph (V,A?A′) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple combinatorial algorithms for these problems that achieve an approximation ratio bounded by the length, in terms of number of arcs, of a longest simple cycle of the digraph.     

Grade of Service Steiner Minimum Trees in the Euclidean Plane

Abstract. Let P = {p ₁ , p ₂ , \ldots, p _n } be a set of n {\lilsf terminal points} in the Euclidean plane, where point p _i has a {\lilsf service request of grade} g(p _i ) ∈ {1, 2, \ldots, n} . Let 0 < c(1) < c(2) < ?s < c(n) be n real numbers. The {\lilsf Grade of Service Steiner Minimum Tree (GOSST)} problem asks for a minimum cost network interconnecting point set P and some {\lilsf Steiner points} with a service request of grade 0 such that (1) between each pair of terminal points p _i and p _j there is a path whose minimum grade of service is at least as large as \min(g(p _i ), g(p _j )) ; and (2) the cost of the network is minimum among all interconnecting networks satisfying (1), where the cost of an edge with service of grade g is the product of the Euclidean length of the edge with c(g) . The GOSST problem is a generalization of the Euclidean Steiner minimum tree problem where all terminal points have the same grade of service request. When there are only two (three, respectively) different grades of service request by the terminal points, we present a polynomial time approximation algorithm with performance ratio \frac 4 3 ρ (((5+4\sqrt 2 )/7)ρ , respectively), where ρ is the performance ratio achieved by an approximation algorithm for the Euclidean Steiner minimum tree problem. For the general case, we prove that there exists a GOSST that is the minimum cost network under a full Steiner topology or its degeneracies. A powerful interior-point algorithm is used to find a (1+ε) -approximation to the minimum cost network under a given topology or its degeneracies in O(n ^1.5 (log n + log (1/ε))) time. We also prove a lower bound theorem which enables effective pruning in a branch-and-bound method that partially enumerates the full Steiner topologies in search for a GOSST. We then present a k -optimal heuristic algorithm to compute good solutions when the problem size is too large for the branch-and-bound algorithm. Preliminary computational results are presented.     

Maximizing the minimum load for selfish agents

We consider the problem of maximizing the minimum load (completion time) for machines that are controlled by selfish agents, who are only interested in maximizing their own profit. Unlike the classical load balancing problem, this problem has not been considered for selfish agents until now. The goal is to design a truthful mechanism, i.e., one in which all users have an incentive to tell the truth about the speeds of their machines. This then allows us to find good job assignments. It is known that this requires monotone approximation algorithms, in which the amount of work assigned to an agent does not increase if its bid (claimed cost per unit work) increases.For a constant number of machines, m, we show a monotone polynomial-time approximation scheme (PTAS) with running time that is linear in the number of jobs. It uses a new technique for reducing the number of jobs while remaining close to the optimal solution. We use an FPTAS for the classical problem, i.e., where no selfish agents are involved, to give a monotone FPTAS.Additionally, we give a monotone approximation algorithm with approximation ratio min(m,(2+ε)s₁/s_m) where ε>0 can be chosen arbitrarily small and s_i is the (real) speed of machine i. Finally we give improved results for two machines.     

Complexity of approximation of 3-edge-coloring of graphs

The problem to find a 4-edge-coloring of a 3-regular graph is solvable in polynomial time but an analogous problem for 3-edge-coloring is NP-hard. To make the gap more precise, we study complexity of approximation algorithms for invariants measuring how far is a 3-regular graph from having a 3-edge-coloring. We show that it is an NP-hard problem to approximate such invariants with an error O(n1−ε), where n denotes the order of the graph and 0<ε<1 is a constant.     

An efficient network flow code for finding all minimum cost s–t cutsets

Cutset algorithms have been well documented in the operations research literature. A directed graph is used to model the network, where each node and arc has an associated cost to cut or remove it from the graph. The problem considered in this paper is to determine all minimum cost sets of nodes and/or arcs to cut so that no directed paths exist from a specified source node s to a specified sink node t. By solving the dual maximum flow problem, it is possible to construct a binary relation associated with an optimal maximum flow such that all minimum cost s–t cutsets are identified through the set of closures for this relation. The key to our implementation is the use of graph theoretic techniques to rapidly enumerate this set of closures. Computational results are presented to suggest the efficiency of our approach.Scope and purposeThis paper describes the technical details of a network flow algorithm used to find all minimum cost s–t cutsets in any network topology. The motivation for this work was to provide additional automated analysis capability to a military network targeting system. Specifically, the problem is to identify a minimum cost set of nodes and/or arcs that when removed from the network, will disconnect a selected pair of origin–destination nodes. Algorithms for solving this problem are well understood, with an active research thrust in both the operations research and computer science academic communities in developing more efficient algorithms for larger networks. The main contribution of this paper is in extending these algorithms to quickly find all minimum cost cutset solutions. The implementation described in this paper outperformed conventional methods by several orders of magnitude on networks having thousands of nodes and arcs, with empirical solution times that grew linearly with the network size. The results translate to a real-time cutset analysis capability to support military targeting applications.     

Approximation Algorithms for a Capacitated Network Design Problem

We study a capacitated network design problem with applications in local access network design. Given a network, the problem is to route flow from several sources to a sink and to install capacity on the edges to support the flow at minimum cost. Capacity can be purchased only in multiples of a fixed quantity. All the flow from a source must be routed in a single path to the sink. This NP-hard problem generalizes the Steiner tree problem and also more effectively models the applications traditionally formulated as capacitated tree problems. We present an approximation algorithm with performance ratio (ρ_ST + 2) where ρ_ST is the performance ratio of any approximation algorithm for the minimum Steiner tree problem. When all sources have unit demand, the ratio improves to ρ_ST + 1) and, in particular, to 2 when all nodes in the graph are sources.     

The k-Splittable Flow Problem

In traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, no restriction is imposed on the number of paths used for delivering each commodity; it is thus feasible to spread the flow over a large number of different paths. Motivated by routing problems arising in real-life applications, e.g., telecommunication, unsplittable flows have moved into the focus of research. Here, the demand of each commodity may not be split but has to be sent along a single path. In this paper a generalization of this problem is studied. In the considered flow model, a commodity can be split into a bounded number of chunks which can then be routed on different paths. In contrast to classical (splittable) flows and unsplittable flows, the single-commodity case of this problem is already NP-hard and even hard to approximate. We present approximation algorithms for the single- and multi-commodity case and point out strong connections to unsplittable flows. Moreover, results on the hardness of approximation are presented. In particular, we show that some of our approximation results are in fact best possible, unless P = NP.     

Approximation algorithms for time-dependent orienteering

The time-dependent orienteering problem is dual to the time-dependent traveling salesman problem. It consists of visiting a maximum number of sites within a given deadline. The traveling time between two sites is in general dependent on the starting time.For any ε>0, we provide a (2+ε)-approximation algorithm for the time-dependent orienteering problem which runs in polynomial time if the ratio between the maximum and minimum traveling time between any two sites is constant. No prior upper approximation bounds were known for this time-dependent problem.     

An efficient approximation for the Generalized Assignment Problem

We present a simple family of algorithms for solving the Generalized Assignment Problem (GAP). Our technique is based on a novel combinatorial translation of any algorithm for the knapsack problem into an approximation algorithm for GAP. If the approximation ratio of the knapsack algorithm is α and its running time is O(f(N)), our algorithm guarantees a (1+α)-approximation ratio, and it runs in O(M⋅f(N)+M⋅N), where N is the number of items and M is the number of bins. Not only does our technique comprise a general interesting framework for the GAP problem; it also matches the best combinatorial approximation for this problem, with a much simpler algorithm and a better running time.     

Improved approximation algorithms for minimum AND-circuits problem via k-set cover

Arpe and Manthey [J. Arpe, B. Manthey, Approximability of minimum AND-circuits, Algorithmica 53 (3) (2009) 337-357] recently studied the minimum AND-circuit problem, which is a circuit minimization problem, and showed some results including approximation algorithms, APX-hardness and fixed parameter tractability of the problem. In this note, we show that algorithms via the k-set cover problem yield improved approximation ratios for the minimum AND-circuit problem with maximum degree three. In particular, we obtain an approximation ratio of 1.199 for the problem with maximum degree three and unbounded multiplicity.     

On the Largest Empty Axis-Parallel Box Amidst n Points

We give the first efficient (1?ε)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in ? ^d containing n points, compute a maximum-volume empty axis-parallel d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order $\Theta (\frac {1}{n} )$ . Our algorithm finds an empty axis-aligned box whose volume is at least (1?ε) of the maximum in O((8edε ^?2) ^d ?nlog ^d n) time. No previous efficient exact or approximation algorithms were known for this problem for d≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely. We also present a (1?ε)-approximation algorithm that, given an axis-parallel d-dimensional cube R in ? ^d containing n points, computes a maximum-volume empty axis-parallel hypercube contained in R. The minimum of this quantity over all such point sets is also shown to be of the order $\Theta (\frac{1}{n} )$ . A faster (1?ε)-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.