Faculty of Management Studies

We give the first non-trivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ...

Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. Virtual private networks (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must be connected, bandwidth in the underlying network is reserved for communication within the group, forming a virtual "sub-network." Provisioning a virtual private network over a set of terminals gives rise to the following general network design problem. We have bounds on the cumulative amount of traffic each terminal can send and receive; we must choose a path for each pair of terminals, and a bandwidth allocation for each edge of the network, so that any traffic matrix consistent with the given upper bounds can be feasibly routed. Thus, we are seeking to design a network that can support a continuum of possible traffic scenarios. We provide optimal and approximate algorithms for several variants of this problem, depending on whether the traffic matrix is required to be symmetric, and on whether the designed network is required to be a tree (a natural constraint in a number of basic applications). We also establish a relation between this collection of network design problems and a variant of the facility location problem introduced by Karger and Minkoff; we extend their results by providing a stronger approximation algorithm for this latter problem.

Several studies have demonstrated the effectiveness of the wavelet decomposition as a tool for reducing large amounts of data down to compact wavelet synopses that can be used to obtain fast, accurate approximate query answers. Conventional wavelet synopses that greedily minimize the overall root-mean-squared (i.e., L 2 -norm) error in the data approximation can suffer from important problems, including severe bias and wide variance in the quality of the data reconstruction, and lack of nontrivial guarantees for individual approximate answers. Thus, probabilistic thresholding schemes have been recently proposed as a means of building wavelet synopses that try to probabilistically control maximum approximation-error metrics (e.g., maximum relative error).A key open problem is whether it is possible to design efficient deterministic wavelet-thresholding algorithms for minimizing general, non-L 2 error metrics that are relevant to approximate query processing systems, such as maximum r...

We propose the first known solution to the problem of correlating, in small space, continuous streams of XML data through approximate (structure and content) matching, as defined by a general tree-edit distance metric. The key element of our solution is a novel algorithm for obliviously embedding tree-edit distance metrics into an L 1 vector space while guaranteeing a (worst-case) upper bound of O (log 2 n log* n ) on the distance distortion between any data trees with at most n nodes. We demonstrate how our embedding algorithm can be applied in conjunction with known random sketching techniques to (1) build a compact synopsis of a massive, streaming XML data tree that can be used as a concise surrogate for the full tree in approximate tree-edit distance computations; and (2) approximate the result of tree-edit-distance similarity joins over continuous XML document streams. Experimental results from an empirical study with both synthetic and real-life XML data trees validate our app...

We present approximation algorithms for the unsplittable flow problem (UFP) in undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the non-uniform capacity case in which the edge capacities can vary arbitrarily over the graph. Our results are:

Virtual Private Networks (VPNs) provide customers with predictable and secure network connections over a shared network. The recently proposed hose model for VPNs allows for greater flexibility since it permits traffic to and from a hose endpoint to be arbitrarily distributed to other endpoints. In this paper, we develop novel algorithms for provisioning VPNs in the hose model. We connect VPN endpoints using a tree structure and our algorithms attempt to optimize the total bandwidth reserved on edges of the VPN tree. We show that even for the simple scenario in which network links are assumed to have infinite capacity, the general problem of computing the optimal VPN tree is NP hard. Fortunately, for the special case when the ingress and egress bandwidths for each VPN endpoint are equal, we can devise an algorithm for computing the optimal tree whose time complexity is O ( mn ), where m and n are the number of links and nodes in the network, respectively. We present a novel integer ...

In the Multiway Cut problem, we are given an undirected edge-weighted graph G = (V; E) with c e denoting the cost (weight) of edge e. We are also given a subset S of V , of size k, called the terminals. The objective is to nd a minimum cost set of edges whose removal ensures that the terminals are disconnected. In this paper, we study a bidirected linear programming relaxation of Multiway Cut. We resolve an open problem posed by Vazirani 10], and show that the integrality gap of this relaxation is no better than that for a geometric linear programming relaxation given by C alinescu et al. 2], and may be strictly worse on some instances.

We study the multicommodity rent-or-buy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can berented, with cost incurred on a per-unit of capacity basis, orbought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way

Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints ''communicated.'' We will call such a graph a temporal network. To model the notion that information in such a network ''flows'' only on paths whose labels respect the ordering of time, we call a path time-respecting if the time labels on its edges are non-decreasing. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the traditional setting, but very rich in its own right. We provide results on two types of problems for temporal networks. First, we consider connectivity problems, in which we seek disjoint time-respecting paths between pairs of nodes. The natural analogue of Menger's Theorem for node-disjoint paths fails in general for time-respecting paths; we give a non-trivial characterization of those graphs for which the theorem does hold in terms of an excluded subdivision theorem, and provide a polynomial-time algorithm for connectivity on this class of graphs. (The problem on general graphs is NP-complete.) View metadata, citation and similar papers at core.ac.uk We then define and study the class of inference problems, in which we seek to reconstruct a partially specified time labeling of a network in a manner consistent with an observed history of information flow.

We present constant-factor approximation algorithms for several widely-studied NP-hard optimization problems in network design, including the multicommodity rent-or-buy, virtual private network design, and single-sink buy-at-bulk problems. Our algorithms are simple and their approximation ratios improve over those previously known, in some cases by orders of magnitude. We develop a general analysis framework to bound the approximation ratios of our algorithms. This framework is based on a novel connection between random sampling and game-theoretic cost sharing.

A new packet routing model proposed by the Internet Engineering Task Force is MultiProtocol Label Switching, or MPLS [B. Davie and Y. Rekhter, MPLS: Technology and Applications, Morgan Kaufmann (Elsevier), New York, 2000]. Instead of each router's parsing the packet network layer header and doing its lookups based on that analysis (as in much of conventional packet routing), MPLS ensures that the analysis of the header is performed just once. The packet is then assigned a stack of labels, where the labels are usually much smaller than the packet headers themselves. When a router receives a packet, it examines the label at the top of the label stack and makes the decision of where the packet is forwarded based solely on that label. It can pop the top label off the stack if it so desires, and can also push some new labels onto the stack, before forwarding the packet. This scheme has several advantages over conventional routing protocols, the two primary ones being (a) reduced amount of header analysis at intermediate routers, which allows for faster switching times, and (b) better traffic engineering capabilities and hence easier handling of quality of service issues. However, essentially nothing is known at a theoretical level about the performance one can achieve with this protocol, or about the intrinsic trade-offs in its use of resources. This paper initiates a theoretical study of MPLS protocols, and routing algorithms and lower bounds are given for a variety of situations. We first study the routing problem on the line, a case which is already nontrivial, and give routing protocols whose trade-offs are close to optimality. We then extend our results for paths to trees, and thence onto more general graphs. These routing algorithms on general graphs are obtained by finding a tree cover of a graph, i.e., a small family of subtrees of the graph such that, for each pair of vertices, one of the trees in the family contains an (almost-)shortest path between them. Our results show tree covers of logarithmic size for planar graphs and graphs with bounded separators, which may be of independent interest.

The Border Gateway Protocol (BGP) is the standard protocol for exchanging routing information between border routers of Autonomous Systems (ASes) in today's Internet. Within an AS, border routers exchange externally-learned BGP route advertisements via Internal-BGP (I-BGP) peerings. Naive solutions for these I-BGP peering sessions (e.g., based on full-mesh topologies) simply cannot scale to the sizes of modern AS networks. Carefully designed route-reflector configurations can drastically reduce the total number and connection cost of the required I-BGP sessions. Nevertheless, no principled algorithmic approaches exist for designing such configurations, and current practice relies on manual reflector selection using simple, ad-hoc rules. In this paper, we address the novel and challenging optimization problems involved in designing effective BGP route-reflector configurations for AS networks. More specifically, we consider the problems of selecting route reflectors in an AS topology to minimize: (1) the total connection cost of all I-BGP peering sessions, and (2) the average distance traversed by route advertisements within the AS. We present N P-hardness results that establish the intractability of these problems, and propose several polynomial-time approximation algorithms (based on LP-rounding and combinatorial techniques) with guaranteed (constant-factor or logarithmic) bounds on the quality of the approximate solution. Our simulation results validate our approach, demonstrating the effectiveness of our configuration algorithms over a wide range of network topologies.

We consider the stochastic Steiner forest problem: suppose we were given a collection of Steiner forest instances, and were guaranteed that a random one of these instances would appear tomorrow; moreover, the cost of edges tomorrow will be λ times the cost of edges today. Which edges should we buy today so that we can extend it to a solution for the instance arriving tomorrow, to minimize the expected total cost? While very general results have been developed for many problems in stochastic discrete optimization over the past years, the approximation status of the stochastic Steiner Forest problem has remained open, with previous works yielding constant-factor approximations only for special cases. We resolve the status of this problem by giving a constant-factor primal-dual based approximation algorithm.

We give simple and easy-to-analyze randomized approximation algorithms for several well-studied NP-hard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following. • We give a randomized 3.55-approximation algorithm for the connected facility location problem. The algorithm requires three lines to state, one page to analyze, and improves the best-known performance guarantee for the problem. • We give a 5.55-approximation algorithm for virtual private network design. Previously, constant-factor approximation algorithms were known only for special cases of this problem. • We give a simple constant-factor approximation algorithm for the single-sink buy-at-bulk network design problem. Our performance guarantee improves over what was previously known, and is an order of magnitude improvement over previous combinatorial approximation algorithms for the problem.

In the online Steiner tree problem, a sequence of points is revealed one-by-one: when a point arrives, we only have time to add a single edge connecting this point to the previous ones, and we want to minimize the total length of edges added. Here, a tight bound has been known for two decades: the greedy algorithm maintains a tree whose cost is O(log n) times the Steiner tree cost, and this is best possible. But suppose, in addition to the new edge we add, we have time to change a single edge from the previous set of edges: can we do much better? Can we, e.g., maintain a tree that is constant-competitive? We answer this question in the affirmative. We give a primal-dual algorithm that makes only a single swap per step (in addition to adding the edge connecting the new point to the previous ones), and such that the tree's cost is only a constant times the optimal cost. Our dual-based analysis is quite different from previous primal-only analyses. In particular, we give a correspondence between radii of dual balls and lengths of tree edges; since dual balls are associated with points and hence do not move around (in contrast to edges), we can closely monitor the edge lengths based on the dual radii. Showing that these dual radii cannot change too rapidly is the technical heart of the paper, and allows us to give a hard bound on the number of swaps per arrival, while maintaining a constant-competitive tree at all times. Previous results for this problem gave an algorithm that performed an amortized constant number of swaps: for each n, the number of swaps in the first n steps was O(n). We also give a simpler tight analysis for this amortized case.

Consider the following edge-orientation problem: edges of a graph appear online one-by-one and they to be directedgiven an "orientation". We want to ensure that the indegree of each vertex remains low. (This is a simple case of scheduling unit-sized jobs on machines, where each job can only go on one of two machines.) If the edge-orientations we assign are irrevocable, we suffer a significant loss in quality due to online decision-making (as compared to the offline performance). For instance the best online competitive ratio achievable is Θ(log m) for even this toy problem. But what if the decisions are not irrevocable-what if we allow a limited number of reassignments? Can we do much better? We show that indeed we can. For instance, for edge-orientation, we can achieve a constant-competitive load while doing only a constant number of reorientations per edge (in an amortized sense). For more substantial problems, our results are as follows: • For online matching, where the left vertices arrive online and must be matched to the right vertices, we give an algorithm that reassigns the left vertices an (amortized) constant number of times, and maintains a constant factor to the optimal load on the right vertices. We extend this to restricted machine scheduling with arbitrary sized jobs and give an algorithm that maintains load which is O(log log mn) times the optimum, and reassigns each job only an (amortized) constant number of times. • Consider a digraph with a single source, where sinks arrive online and want to send unit flow to the source. The goal is to minimize the congestion on the edges. Suppose there is an offline flow such that the total length of the flow paths is F *. We give an algorithm that reroutes flow along O(F *) edges and achieves a O(1)-approximation to the congestion.

Multiprotocol Label Switching or MPLS technology is being increasingly deployed by several of the largest Internet service providers to solve problems such as traffic engineering and to offer IP services like Virtual Private Networks (VPNs). In MPLS, the analysis of the packet (network layer) header is performed just once, and each packet is assigned a stack of labels, which is examined by subsequent routers when making forwarding decisions. Despite the fact that MPLS is becoming widespread on the Internet, we know essentially very little about the performance one can achieve with it, and about the intrinsic trade-offs in its use of resources. In this paper, we undertake a comprehensive study of the label size versus stack depth trade-off for MPLS routing protocols on lines and trees. We show that in addition to LSP tunneling, label stacks can also be used to dramatically reduce the number of labels required for setting up MPLS LSPs in a network. Based on this observation, we develop routing algorithms and prove lower bounds for two basic problems: (1) FIXED LABEL ROUTING: Given a fixed number of labels, we want to minimize the stack depth, and (2) FIXED STACK ROUTING: Given a bound on the stack depth, we want to minimize the number of labels used. Our simulation results validate our approach, demonstrating that our novel protocols enable MPLS routing on large trees with few labels and small stack sizes. Thus, our MPLS routing algorithms are applicable to a number of practical scenarios involving the provisioning of VPNs and multicast trees.

We study the Steiner Tree problem in the model of two-stage stochastic optimization with non-uniform inflation factors, and give a poly-logarithmic approximation factor for this problem. In this problem, we are given a graph G = (V, E), with each edge having two costs cM and cT (the costs for Monday and Tuesday, respectively). We are also given a probability distribution π : 2 V → [0, 1] over subsets of V , and will be given a client set S drawn from this distribution on Tuesday. The algorithm has to buy a set of edges EM on Monday, and after the client set S is revealed on Tuesday, it has to buy a (possibly empty) set of edges ET (S) so that the edges in EM ∪ ET (S) connect all the nodes in S. The goal is to minimize the cM (EM) + ES←π[ cT (ET (S)) ]. We give the first poly-logarithmic approximation algorithm for this problem. Our algorithm builds on the recent techniques developed by Chekuri et al. (FOCS 2006) for multi-commodity Cost-Distance. Previously, the problem had been studied for the cases when cT = σ × cM for some constant σ ≥ 1 (i.e., the uniform case), or for the case when the goal was to find a tree spanning all the vertices but Tuesday's costs were drawn from a given distribution π (the so-called "stochastic MST case"). We complement our results by showing that our problem is at least as hard as the single-sink Cost-Distance problem (which is known to be Ω(log log n) hard). Moreover, the requirement that Tuesday's costs are fixed seems essential: if we allow Tuesday's costs to dependent on the scenario as in stochastic MST, the problem becomes as hard as Label Cover (which is Ω(2 log 1−ε n)-hard). As an aside, we also give an LP-rounding algorithm for the multi-commodity Cost-Distance problem, matching the O(log 4 n) approximation guarantee given by Chekuri et al. (FOCS 2006). Tuesday, the actual demand set S arrives (drawn from the distribution π), and we must complete a Steiner tree on the set S, but any edges E T bought on Tuesday cost c T. How can we minimize our expected cost c M (E M) + E S←π [ c T (E T (S)) ] ? The Stochastic Steiner tree problem has been studied before in the special case when Tuesday's cost function c T is a scaled-up version of Monday's costs c M (i.e., there is an constant inflation factor σ > 1 such that c T (e) = σ × c M (e)); for this case, constantfactor approximations are known [9, 10, 12]. While these results can be generalized in some directions (see Section 1.1 for a detailed discussion), it has been an open question whether we could handle the case when the two costs c M and c T are unrelated. (We will refer to this case as the non-uniform inflation case, as opposed to the uniform inflation case when the costs c M and c T are just scaled versions of each other.) This gap in our understanding was made more apparent by the fact that many other problems such as Facility Location, Vertex Cover and Set Cover, were all shown to admit good approximations in the non-uniform inflation model [22, 24]: in fact, the results for these problems could be obtained even when the edge cost could depend on the day as well as on the demand set appearing on Tuesday.