There are many applications in which it is desirable to order rather than classify instances. Here we consider the problem of learning how to order, given feedback in the form of preference judgments, i.e., statements to the effect that one instance should be ranked ahead of another. We outline a two-stage approach in which one first learns by conventional means a preference function, of the form PREF(u; v), which indicates whether it is advisable to rank u before v. New instances are then ordered so as to maximize agreements with the learned preference function. We show that the problem of finding the ordering that agrees best with a preference function is NP-complete, even under very restrictive assumptions. Nevertheless, we describe a simple greedy algorithm that is guaranteed to find a good approximation. We then discuss an on-line learning algorithm, based on the "Hedge" algorithm, for finding a good linear combination of ranking "experts." We use the ordering algorith...

This paper is concerned with probabilistic approximation of metric spaces. In previous work we introduced the method of ecient approximation of metrics by more simple families of metrics in a probabilistic fashion. In particular we study probabilistic approximations of arbitrary metric spaces by \hierarchically wellseparated tree" metric spaces. This has proved as a useful technique for simplifying the solutions to various problems.

In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

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We present a novel divide-and-conquer paradigm for approximating NP-hard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divide-and-conquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns rational lengths to either edges or vertices of the input graph, such that all subgraphs on which the optimization problem is non-trivial have large diameters. In addition, the spreading metric provides a lower bound, ø , on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is O (minflog ø log log ø; log k log log kg), where k denotes the number of "interesting" vertices in the problem instance, and is at most the number of vertices. We present seven problems that can be formulated to fit the paradigm. For all these problems our algorithm improves ...

This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (fvs) problem, and the weighted feedback edge set problem (fes). In the fvs (resp. fes) problem, one is given a directed graph with weights (each of which is at least 1) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-Hard problems and have many applications. We also consider a generalization of these problems: subset-fvs and subset-fes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-Hard even when |X| = 2. We present approximation algorithms for the subset-fvs and subset-fes problems. The first algorithm we present achieves an approximation factor of O(log2 |X|). The second algorithm achieves an approximation factor of O(min(log tau log log tau; log n log log n)), where tau is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subset-fes and subset-fvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1 + epsilon) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.

Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his

We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, , ? and 6= are considered. While Min RVLS with equations was known to be NP-hard in [27], we established in [2, 5] that Min ULR with equalities and inequalities are NP-hard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in [8]. In this paper we determine strong bou...

Most optimization problems on an undirected graph reduce in complexity when restricted to instances on a tree. A recent result [3] for probabilistically approximating graph metrics by trees such that no edge stretches (in an expected sense) by more than a factor of O(log 2 n) has resulted in several approximation algorithms which exploit the ease of solving problems on trees. The tree construction in [3] is inherently randomized and a natural question to ask is whether approximation algorithms which use this construction can be derandomized. We present a general framework for derandomizing approximation algorithms which use the above tree construction as a primitive. Let \Pi be a graph optimization problem which can be expressed as an integer program with 0-1 variables ¯ x(e) for each edge and with an objective function expressible as...

... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an n-vertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds is listed as one of the long-standing open questions in the area of low-distortion embeddings of metrics (Matousek 2002). We significantly reduce this gap by constructing a spanning tree in G of average stretch O((log n log log n) 2). Moreover, we show that this tree can be constructed in time O(m log 2 n) in general, and in time O(m log n) if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for diagonally dominant linear systems of Spielman and Teng from to m2 (O( √ log n log log n)) log(1/ɛ) m log O(1) n log(1/ɛ), and to O(n(log n log log n) 2 log(1/ɛ)) when the system is planar. Applying a recent reduction of Boman, Hendrickson and Vavasis, this provides an O(n(log n log log n) 2 log(1/ɛ)) time algorithm for solving the linear systems that arise when applying the finite element method to solve twodimensional elliptic partial differential equations. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.