The reverse search technique has been recently introduced by the authors for efficient enumeration of vertices of polyhedra and arrangements. In this paper, we develop this idea in a general framework and show its broader applications to various problems in operations research, combinatorics, and geometry. In particular, we propose new algorithms for listing (i) all triangulations of a set of n points in the plane, (ii) all cells in a hyperplane arrangement in R d , (iii) all spanning trees of a graph, (iv) all Euclidean (non-crossing) trees spanning a set of n points in the plane, (v) all connected induced subgraphs of a graph, and (vi) all topological orderings of an acyclic graph. Finally we propose a new algorithm for the 0-1 integer programming problem which can be considered as an alternative to the branch-and-bound algorithm. 1 Introduction The listing of all objects that satisfy a specified property is a fundamental problem in combinatorics, computational geometr...

The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex-type algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the so-called self-dual method, is analyzed. The algorithm is not started at the traditional point (1,..., but points of the form (1, e, e2,...)T, with t sufficiently small, are used. The result is better, in two respects, than those of the previous analyses. First, it is shown that the expected number of steps is bounded between two quadratic functions cl(min(m, n))' and cz(min(m, n)) ' of the smaller dimension of the problem. This should be compared with the previous two major results in the field. Borgwardt proves an upper bound of 0(n4m1'(n-1') under a model that implies that the zero vector satisfies all the constraints, and also the algorithm under his consideration solves only problems from that particular subclass. Smale analyzes the self-dual algorithm starting at (1,..., He shows that for any fixed m there is a constant c(m) such the expected number of steps is less than ~(m)(lnn)"'("+~); Megiddo has shown that, under Smale's model, an upper bound C(m) exists. Thus, for the first time, a polynomial upper bound with no restrictions (except for nondegeneracy) on the problem is proved, and, for the first time, a nontrivial lower bound of precisely the same order of magnitude is established. Both Borgwardt and Smale require the input vectors to be drawn from

Many optimization techniques, including several targeted specifically at embedded systems, depend on the ability to calculate the number of elements that satisfy certain conditions. If these conditions can be represented by linear constraints, then such problems are equivalent to counting the number of integer points in (possibly) parametric polytopes. It is well known that this parametric count can be represented by a set of Ehrhart polynomials. Previously, interpolation was used to obtain these polynomials, but this technique has several disadvantages. Its worst-case computation time for a single Ehrhart polynomial is exponential in the input size, even for fixed dimensions. The worst-case size of such an Ehrhart polynomial (measured in bits needed to represent the polynomial) is also exponential in the input size. Under certain conditions this technique even fails to produce a solution.

We study generalization properties of the area under an ROC curve (AUC), a quantity that has been advocated as an evaluation criterion for bipartite ranking problems. The AUC is a different and more complex term than the error rate used for evaluation in classification problems; consequently, existing generalization bounds for the classification error rate cannot be used to draw conclusions about the AUC. In this paper, we define a precise notion of the expected accuracy of a ranking function (analogous to the expected error rate of a classification function), and derive distribution-free probabilistic bounds on the deviation of the empirical AUC of a ranking function (observed on a finite data sequence) from its expected accuracy. We derive both a large deviation bound, which serves to bound the expected accuracy of a ranking function in terms of its empirical AUC on a test sequence, and a uniform convergence bound, which serves to bound the expected accuracy of a learned ranking function in terms of its empirical AUC on a training sequence. Our uniform convergence bound is expressed in terms of a new set of combinatorial parameters that we term the bipartite rank-shatter coefficients; these play the same role in our result as do the standard shatter coefficients (also known variously as the counting numbers or growth function) in uniform convergence results for the classification error rate. We also compare our result with a recent uniform convergence result derived by Freund et al. (2003) for a quantity closely related to the AUC; as we show, the bound provided by our result is considerably tighter. 1 1

We introduce the convex combinatorial optimization problem, a far-reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.

This paper addresses the problem of noisy Generalized Binary Search (GBS). GBS is a well-known greedy algorithm for determining a binary-valued hypothesis through a sequence of strategically selected queries. At each step, a query is selected that most evenly splits the hypotheses under consideration into two disjoint subsets, a natural generalization of the idea underlying classic binary search. GBS is used in many applications, including fault testing, machine diagnostics, disease diagnosis, job scheduling, image processing, computer vision, and active learning. In most of these cases, the responses to queries can be noisy. Past work has provided a partial characterization of GBS, but existing noise-tolerant versions of GBS are suboptimal in terms of query complexity. This paper presents an optimal algorithm for noisy GBS and demonstrates its application to learning multidimensional threshold functions. 1

The area under the ROC curve (AUC) has been advocated as an evaluation criterion for the bipartite ranking problem. We study uniform convergence properties of the AUC; in particular, we derive a distribution-free uniform convergence bound for the AUC which serves to bound the expected accuracy of a learned ranking function in terms of its empirical AUC on the training sequence from which it is learned. Our bound is expressed in terms of a new set of combinatorial parameters that we term the bipartite rank-shatter coefficients; these play the same role in our result as do the standard VC-dimension related shatter coefficients (also known as the growth function) in uniform convergence results for the classification error rate. A comparison of our result with a recent uniform convergence result derived by Freund et al. [FISS03] for a quantity closely related to the AUC shows that the bound provided by our result can be considerably tighter. 1

Abstract. We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of variables. 1 Introduction. In this paper we study the curvature of the central path of linear programming theory. We establish that for a linear programming problem defined on a compact polytope contained in R n, the total curvature of the central path is less than or

Typechecking consists of statically verifying whether the output of an XML transformation always conforms to an output type for documents satisfying a given input type. In this general setting, both the input and output schema as well as the transformation are part of the input for the problem. However, scenarios where the input or output schema can be considered to be fixed, are quite common in practice. In the present work, we investigate the computational complexity of the typechecking problem in the latter setting. 1