F. Bruce Shepherd

The stable paths problem and interdomain routing

Dynamic routing protocols such as RIP and OSPF essentially implement distributed algorithms for solving the shortest paths problem. The border gateway protocol (BGP) is currently the only interdomain routing protocol deployed in the Internet. BGP does not solve a shortest paths problem since any interdomain protocol is required to allow policy-based metrics to override distance-based metrics and enable autonomous systems to independently define their routing policies with little or no global coordination. It is then natural to ask if BGP can be viewed as a distributed algorithm for solving some fundamental problem. We introduce the stable paths problem and show that BGP can be viewed as a distributed algorithm for solving this problem. Unlike a shortest path tree, such a solution does not represent a global optimum, but rather an equilibrium point in which each node is assigned its local optimum.We study the stable paths problem using a derived structure called a dispute wheel, representing conflicting routing policies at various nodes. We show that if no dispute wheel can be constructed, then there exists a unique solution for the stable paths problem. We define the simple path-vector protocol (SPVP), a distributed algorithm for solving the stable paths problem. SPVP is intended to capture the dynamic behavior of BGP at an abstract level. If SPVP converges, then the resulting state corresponds to a stable paths solution. If there is no solution, then SPVP always diverges. In fact, SPVP can even diverge when a solution exists. We show that SPVP will converge to the unique solution of an instance of the stable paths problem if no dispute wheel exists.

Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems

We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (si; ti), 1 i k, and the goal is to nd a largest subset of source-sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any > 0, EDP is NP-hard to approximate within m 1=2 . We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any > 0, bounded length EDP is hard to approximate within m 1=2 even in undirected networks, and give an O( p m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. nd the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1=2 , for any > 0.

Domination in graphs with minimum degree two

The domination number γ(G) of a graph G = (V, E) is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in the set. We show that if a connected graph G has minimum degree two and is not one of seven exceptional graphs, then γ(G)γ 2/5|V|. We also characterize those connected graphs with γ(G)γ 2/5|V|.

Route oscillations in I-BGP with route reflection

We study the route oscillation problem [16, 19] in the Internal Border Gateway Protocol (I-BGP)[18] when route reflection is used. We propose a formal model of I-BGP and use it to show that even deciding whether an I-BGP configuration with route reflection can converge is an NP-Complete problem. We then propose a modification to I-BGP and show that route reflection cannot cause the modified protocol to diverge. Moreover, we show that the modified protocol converges to the same stable routing configuration regardless of the order in which messages are sent or received.

Multicommodity demand flow in a tree and packing integer programs

We consider requests for capacity in a given tree network T = (V, E) where each edge e of the tree has some integer capacity ue. Each request f is a node pair with an integer demand df and a profit wf which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provides a maximum profit. This generalizes well-known problems, including the knapsack and b-matching problems. When all demands are 1, we have the integer multicommodity flow problem. Garg et al. [1997] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with arbitrary demands. When the maximum demand dmax is at most the minimum edge capacity umin, we show that the integrality gap of the LP is at most 48. This result is obtained by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed.

Multicommodity flow, well-linked terminals, and routing problems

We study multicommodity routing problems in both edge and node capacitated undirected graphs. The input to each problem is a capacitated graph G=(V,E) and a set Τ of node pairs. In the simplest setting, the goal is to route a unit of flow for as many pairs as possible subject to the edge (node) capacity constraints. If the flow for a routed pair is required to be along a single path, it is the well-studied disjoint paths problem. If we allow fractional routings of the flow, it is known as the all-or-nothing flow problem. The nodes in Τ are referred to as terminals.In recent work [8,9], the authors obtained the first poly-logarithmic approximation algorithms for some edge routing problems. A key idea in these algorithms is to decompose an instance into a collection of instances in which the terminals are well-linked. Informally speaking, a set of nodes is well-linked in a graph if it does not have small separators. A decomposition into well-linked instances was previously achieved in [8] via räckes hierarchical graph decomposition for oblivious routing [32]. In this paper, we design a simple new decomposition algorithm that is based on computing sparse cuts in a graph. Our new algorithm improves the earlier results for edge routing problems. Another important advantage of the algorithm is that it also applies to node-capacitated problems. We note that for oblivious routing with node capacities, an Ω√n) lower bound is known on the congestion [18], and hence the oblivious routing approach cannot yield poly-logarithmic bounds for well-linked decompositions. Using the new decomposition, we obtain a poly-logarithmic approximation for the node capacitated all-or-nothing flow problem in general graphs and node-disjoint path problem in planar graphs with O(1) congestion. We also show that the flow-cut gap for product multicommodity flows in node capacitated planar graphs is O(1), improving upon the O(log n) bound from [28].

The all-or-nothing multicommodity flow problem

We consider the all-or-nothing multicommodity flow problem in general graphs. We are given a capacitated undirected graph <i>G</i>=(<i>V,E,u</i>) and set of <i>k</i> pairs <i>s</i><inf>1</inf> <i>t</i><inf>1</inf>, <i>s</i><inf>2</inf><i>t</i><inf>2</inf>, …, <i>s<inf>k</inf>t<inf>k</inf></i>. Each pair has a unit demand. The objective is to find a largest subset <i>S</i> of 1,2,…,<i>k</i> such that for every <i>i</i> in <i>S</i> we can send a flow of one unit between <i>s<inf>i</inf></i> and <i>t<inf>i</inf></i>. Note that this differs from the <i>edge-disjoint path</i> problem (<sc>EDP</sc>) in that we do not insist on integral flows for the pairs. This problem is NP-hard, and APX-hard, even on trees. For trees, a 2--approximation is known for the cardinality case and a 4--approximation for the weighted case. In this paper we build on a recent result of Räcke on low congestion oblivious routing in undirected graphs to obtain a poly-logarithmic approximation for the all-or-nothing problem in <i>general</i> undirected graphs. The best previous known approximation for all-or-nothing flow problem was <i>O</i>(min(<i>n</i><^2/3, √<i>m</i>)), the same as that for <sc>EDP</sc>. Our algorithm extends to the case where each pair <i>s<inf>i</inf>t<inf>i</inf></i> has a demand <i>dⁱ</i> associated with it and we need to completely route <i>dⁱ</i> to get credit for pair <i>i</i>. We also consider the <i>online admission control</i> version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it or not. We obtain a randomized algorithm with a competitive ratio that is similar to the approximation ratio for the offline algorithm.

Clustering and server selection using passive monitoring

We consider the problem of client assignment in a distributed system of content servers. We present a system called Webmapper for clustering IP addresses and assigning each cluster to an optimal content server. The system is passive in that the only information it uses comes from monitoring the TCP connections between the clients and the servers. It is also flexible in that it makes no a priori assumptions about network topology and server placement and it can react quickly to changing network conditions. We present experimental results to evaluate the performance of Webmapper.

An O(sqrt(n)) Approximation and Integrality Gap for Disjoint Paths and Unsplittable Flow.

We consider the maximization version of the edge-disjoint path problem (EDP). In undirected graphs and directed acyclic graphs, we obtain an O( p n) upper bound on the approximation ratio where n is the number of nodes in the graph. We show this by es- tablishing the upper bound on the integrality gap of the natural relaxation based on mul- ticommodity flows. Our upper bound matches within a constant factor a lower bound of Ω( p n) that is known for both undirected and directed acyclic graphs. The best previous upper bounds on the integrality gaps were O(min{n 2/3 , p m}) for undirected graphs and O(min{ p n log n, p m}) for directed acyclic graphs; here m is the number of edges in the graph. These bounds are also the best known approximation ratios for these problems. Our bound also extends to the unsplittable flow problem (UFP) when the maximum demand is at most the minimum capacity.

An upper bound for the k-domination number of a graph

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k, then γk(G) ≤ kp/(k + 1).