Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Interval Routing Schemes for Circular-Arc Graphs

Interval routing is a space efficient method to realize a distributed routing function. In this paper we show that every circular-arc graph allows a shortest path strict 2-interval routing scheme, i.e., by introducing a global order on the vertices and assigning at most two (strict) intervals in this order to the ends of every edge allows to depict a routing function that implies exclusively shortest paths. Since circular-arc graphs do not allow shortest path 1-interval routing schemes in general, the result implies that the class of circular-arc graphs has strict compactness 2, which was a hitherto open question. Additionally, we show that the constructed 2-interval routing scheme is a 1-interval routing scheme with at most one additional interval assigned at each vertex and we an outline algorithm to calculate the routing scheme for circular-arc graphs in O(n^2) time, where n is the number of vertices.Comment: 17 pages, to appear in "International Journal of Foundations of Computer Science

The complexity of the characterization of networks supporting shortest-path interval routing

AbstractInterval Routing is a routing method that was proposed in order to reduce the size of the routing tables by using intervals and was extensively studied and implemented. Some variants of the original method were also defined and studied. The question of characterizing networks which support optimal (i.e., shortest path) Interval Routing has been thoroughly investigated for each of the variants and under different models, with only partial answers, both positive and negative, given so far. In this paper, we study the characterization problem under the most basic model (the one unit cost), and with the most restrictive memory requirements (one interval per edge). We prove that this problem is NP-hard (even for the restricted class of graphs of diameter at most 3). Our result holds for all variants of Interval Routing. It significantly extends some related NP-hardness result, and implies that, unless P=NP, partial characterization results of some classes of networks which support shortest path Interval Routing, cannot be pushed further to lead to efficient characterizations for these classes

EGOIST: Overlay Routing Using Selfish Neighbor Selection

A foundational issue underlying many overlay network applications ranging from routing to P2P file sharing is that of connectivity management, i.e., folding new arrivals into an existing overlay, and re-wiring to cope with changing network conditions. Previous work has considered the problem from two perspectives: devising practical heuristics for specific applications designed to work well in real deployments, and providing abstractions for the underlying problem that are analytically tractable, especially via game-theoretic analysis. In this paper, we unify these two thrusts by using insights gleaned from novel, realistic theoretic models in the design of Egoist – a prototype overlay routing system that we implemented, deployed, and evaluated on PlanetLab. Using measurements on PlanetLab and trace-based simulations, we demonstrate that Egoist's neighbor selection primitives significantly outperform existing heuristics on a variety of performance metrics, including delay, available bandwidth, and node utilization. Moreover, we demonstrate that Egoist is competitive with an optimal, but unscalable full-mesh approach, remains highly effective under significant churn, is robust to cheating, and incurs minimal overhead. Finally, we discuss some of the potential benefits Egoist may offer to applications.National Science Foundation (CISE/CSR 0720604, ENG/EFRI 0735974, CISE/CNS 0524477, CNS/NeTS 0520166, CNS/ITR 0205294; CISE/EIA RI 0202067; CAREER 04446522); European Commission (RIDS-011923

On Efficient Distributed Construction of Near Optimal Routing Schemes

Given a distributed network represented by a weighted undirected graph $G=(V,E)$ on $n$ vertices, and a parameter $k$, we devise a distributed algorithm that computes a routing scheme in $(n^{1/2+1/k}+D)\cdot n^{o(1)}$ rounds, where $D$ is the hop-diameter of the network. The running time matches the lower bound of $\tilde{\Omega}(n^{1/2}+D)$ rounds (which holds for any scheme with polynomial stretch), up to lower order terms. The routing tables are of size $\tilde{O}(n^{1/k})$, the labels are of size $O(k\log^2n)$, and every packet is routed on a path suffering stretch at most $4k-5+o(1)$. Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by \cite[STOC 2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but suffers from substantially larger routing tables of size $O(n^{1/2+1/k})$, while the latter has sub-optimal running time of $\tilde{O}(\min\{(nD)^{1/2}\cdot n^{1/k},n^{2/3+2/(3k)}+D\})$

Throughput optimization in MPR-capable multi-hop wireless networks

Recent advances in the physical layer have enabled the simultaneous reception of multiple packets by a node in wireless networks. This capability has the potential of improving the performance of multi-hop wireless networks by a logarithmic factor with respect to current technologies. However, to fully exploit multiple packet reception (MPR) capability, new routing and scheduling schemes must be designed. These schemes need to reformulate a historically underlying assumption in wireless networks which states that any concurrent transmission of two or more packets results in a collision and failure of all packet receptions. In this work, we present a generalized model for the throughput optimization problem in MPR-capable multi-hop wireless networks. The formulation incorporates not only the MPR protocol model to quantify interference, but also the multi-access channel. The former is related with the MAC and routing layers, and considers a packet as the unit of transmission. The latter accounts for the achievable capacity of links used by simultaneous packet transmissions. The problem is modeled as a joint routing and scheduling problem. The scheduling subproblem deals with finding the optimal schedulable sets, which are defined as subsets of links that can be scheduled or activated simultaneously. Among other results, we demonstrate that any solution of the scheduling subproblem can be built with |E| + 1 or fewer schedulable sets, where |E| is the number of links of the network. This result contrasts with a conjecture that states that a solution of the scheduling subproblem, in general, is composed of an exponential number of schedulable sets. The model can be applied to a wide range of networks, such as half and full duplex systems, networks with directional and omni-directional antennas with one or multiple transmit antennas per node. Due to the hardness of the problem, we propose several polynomial time schemes based on a combination of linear programming, approximation algorithm and greedy paradigms. We illustrate the use of the proposed schemes to study the impact of several design parameters such as decoding capability and number of transmit antennas on the performance of MPR-capable networks