Artur Tomaszewski

A polynomial algorithm for solving a general max-min fairness problem

An iterative algorithm of polynomial complexity is presented that solves a max-min fairness problem which is often encountered while dealing with traffic routing or capacity allocation problems. The algorithm does not depend on any specific traffic routing problem formulation and is sufficiently general to be applied to a broad class of traffic routing and capacity allocation problems. The correctness of the algorithm and its complexity are formally proved.

Fair Optimization and Networks: A Survey

Optimization models related to designing and operating complex systems are mainly focused on some efficiency metrics such as response time, queue length, throughput, and cost. However, in systems which serve many entities there is also a need for respecting fairness: each system entity ought to be provided with an adequate share of the systems services. Still, due to system operations-dependant constraints, fair treatment of the entities does not directly imply that each of them is assigned equal amount of the services. That leads to concepts of fair optimization expressed by the equitable models that represent inequality averse optimization rather than strict inequality minimization; a particular widely applied example of that concept is the so-called lexicographic maximin optimization (max-min fairness). The fair optimization methodology delivers a variety of techniques to generate fair and efficient solutions. This paper reviews fair optimization models and methods applied to systems that are based on some kind of network of connections and dependencies, especially, fair optimization methods for the location problems and for the resource allocation problems in communication networks.

Cooperative multi-provider routing optimization and income distribution

We consider the problem of cooperative distributed routing optimization in multi-domain/multi-provider networks. The main object of our investigation are ASON/G-MPLS transport networks, still the results of our investigations could be extended to any multi-domain network where particular domains have limited mutual visibility of intra-domain resources. This paper refines a distributed decomposition mechanism for reliable cooperative optimization of flow reservation levels introduced, by considering the fundamental issue of fair income distribution. The proposed idea of fair income distribution mechanism has been adopted from the theory of cooperative games (Shapley value). We show the benefits of adopting the proposed income distribution scheme by numerical simulations.

Optimization of OSPF Routing in IP Networks

The Internet is a huge world-wide packet switching network comprised of more than 13,000 distinct subnetworks, referred to as Autonomous Systems (ASs) . They all rely on the Internet Protocol (IP) for transport of packets across the network. And most of them use shortest path routing protocols , such as OSPF or IS-IS, to control the routing of IP packets within an AS. The idea of the routing is extremely simple — every packet is forwarded on IP links along the shortest route between its source and destination nodes of the AS. The AS network administrator can manage the routing of packets in the AS by supplying the so-called administrative weights of IP links, which specify the link lengths that are used by the routing protocols for their shortest path computations. The main advantage of the shortest path routing policy is its simplicity, allowing for little administrative overhead. From the network engineering perspective, however, shortest path routing can pose problems in achieving satisfactory traffic handling efficiency. As all routing paths depend on the same routing metric , it is not possible to configure the routing paths for the communication demands between different pairs of nodes explicitly or individually; the routing can be controlled only indirectly and only as a whole by modifying the routing metric. Thus, one of the main tasks when planning such networks is to find administrative link weights that induce a globally efficient traffic routing configuration of an AS. It turns out that this task leads to very difficult mathematical optimization problems. In this chapter, we discuss and describe exact integer programming models and solution approaches as well as practically efficient smart heuristics for such shortest path routing problems .

Optimization of The Shortest-Path Routing with Equal-Cost Multi-Path Load Balancing

In this paper we address the problem of routing optimization in IP networks. We assume that traffic is routed along the shortest paths computed with respect to administrative link metrics. Metrics are distributed in a network by open shortest path first (OSPF) or a similar routing protocol. If it happens that the shortest path is not unique then equal-cost multi-path (ECMP) load balancing principle is applied. It means that the demand traffic destined to specific node is split among all the shortest paths to that node. The problem considered here is to determine the shortest-path routing pattern satisfying traffic demands, and to find appropriate link metrics while link capacities are not exceeded. Besides that many traffic engineering criterias can be used as objective function of the problem, we assume that the residual capacity volume is maximized. In this paper we formulate the problem as a mixed integer programme (MIP) and propose some combinatorial separation cuts for the problem and give an effective method for deriving such cuts

Complexity of a classical flow restoration problem

In this article, we revisit a classical optimization problem occurring in designing survivable multicommodity flow networks. The problem, referred to as FR, assumes flow restoration that takes advantage of the so-called stub release. As no compact linear programming (LP) formulation of FR is known and at the same time all known noncompact LP formulations of FR exhibit NP-hard dual separation, the problem itself is believed to be NP-hard, although without a proof. In this article, we study a restriction of FR (RFR) that assumes only elementary (cycle-free) admissible paths—an important case virtually not considered in the literature. The two problems have the same noncompact LP formulations as they differ only in the definition of admissible paths: all paths (also those including cycles) are allowed in FR, while only elementary paths are allowed in RFR. Because of that, RFR is in general computationally more complex than FR. The purpose of this article, is three-fold. First, the article reveals an interesting special case of RFR—the case with only one failing link—for which a natural noncompact LP formulation obtained by reducing the general RFR formulation still exhibits NP-hard dual separation, but nevertheless this special case of RFR is polynomial. The constructed example of a polynomial multicommodity flow problem with difficult dual separation is of interest since, to our knowledge, no example of this kind has been known. In this article, we also examine a second special case of RFR, this time assuming two failing links instead of one, which turns out to be NP-hard. This implies that problem RFR is NP-hard in general (more precisely, for two or more failure states). This new result is the second contribution of the article. Finally, we discuss the complexity of FR in the light of our new findings, emphasizing the differences between RFR and FR. (Less)

Distributed Inter-Domain Link Capacity Optimization for Inter-Domain IP/MPLS Routing

Our goal is to present a mathematical model useful for distributed optimization of traffic routing in multi-domain Internet networks. In such environments each domain is operated autonomously and has access to limited information about the rest of the network. Our model reflects this through an appropriate problem decomposition with respect to individual domains. The decomposition aims at supporting a distributed process of routing optimization that could be run in the control plane of the network using existing EGP routing protocols. The usefulness of the decomposition is investigated and illustrated with numerical examples.

Comparison of IP-based and explicit paths for one-to-one fast reroute in MPLS networks

Primary and backup paths in MPLS fast reroute (FRR) may be established as shortest paths according to the administrative link costs of the IP control plane, or as explicitly calculated arbitrary paths. In both cases, the path layout can be optimized so that the maximum link utilization for a specific traffic matrix and for a set of considered failure scenarios is minimized. In this paper, we propose a linear program for the optimization of the path layout for explicitly calculated paths, which can either produce single paths and route entire traffic along those paths, or generate multiple paths and spread the traffic among those paths providing load balancing. We compare the resulting lowest maximum link utilization in both cases with the lowest maximum link utilization that can be obtained by optimizing unique IP-based paths. Our results quantify the gain in resource efficiency usage provided by optimized explicit multiple paths or explicit single paths as compared to optimized IP-based paths. Furthermore, we investigate if explicit path layouts cause an increased configuration effort compared to IP-based layouts and if yes, to what extend.

Optimized IP-based vs. explicit paths for one-to-one backup in MPLS fast reroute

Primary and backup paths in MPLS fast reroute (FRR) may be established as shortest paths according to the administrative link costs of the IP control plane, or as explicitly calculated arbitrary paths. In both cases, the path layout can be optimized so that the maximum link utilization for a set of considered failure scenarios is minimized. In this paper, we propose a linear program for the optimization of the path layout for explicitly calculated paths, which can either produce single paths and route entire traffic along those paths, or generate multiple paths and spread the traffic among those paths providing load balancing. We compare the resulting lowest maximum link utilization in both cases with the lowest maximum link utilization that can be obtained by optimizing unique IP-based paths. Our results quantify the gain in resource efficiency usage provided by optimized explicit multiple paths or explicit single paths as compared to optimized IP-based paths.

Complexity of resilient network optimisation

Path restoration (PR) is one of the basic mechanisms for securing telecommunication networks against failures. In the paper, we discuss the complexity of certain variants of a multi-commodity flow network optimisation problem in directed graphs related to state-independent path restoration mechanisms. We demonstrate that most variants of the considered problem are NP-hard. Depending on the variant, we show how the problem can be reduced either from the partition problem or from the problem of finding an arc-disjoint pair of paths that connect two distinct pairs of nodes. We also demonstrate that at the same time the considered problem is difficult to approximate. The complexity results of the paper are important as they can help to devise proper algorithms for resilient network design tools. Copyright (C) 2009 John Wiley & Soils, Ltd. (Less)