Worst-case comparison of valid inequalities for the TSP

We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst-case improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10/9. The corresponding factor for the class of clique tree inequalities is 8/7, while it is 4/3 for the path configuration inequalities.Research supported in part by Air Force contract F49620-92-J-0125, DARPA contract N00014-92-J-1799 and NSF contract 9302476-CCR.     

A minimal cutset of the boolean lattice with almost all members

Two almost explicit constructions are given satisfying the title.This research was done while the authors visited the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, MN 55455.Research supported in part by Hungarian National Foundation for Scientific Research, grant no. 1812.Research supported in part by NSF Grant MDS 87-01475.Research supported in part by NSF grant DMS 86-06225 and Airforce Grant OSR-86-0076.     

A least-squares-based method for a class of nonsmooth minimization problems with applications in plasticity

This paper introduces a globally convergent algorithm for solving a class of nonsmooth optimization problems, involving square roots of quadratic forms. The class includes in particular limit analysis problems in plasticity. The algorithm combines smoothing with successive approximation. The main computational effort in each iteration is solving a linear weighted least-squares problem. The convergence of the algorithm is proved and ana priori error estimate is obtained. Numerical results are presented for two limit analysis problems.The work of the first author was partially supported by NSF Grant DDM-89-96112. Parts of the work was done during his stay at the University of Bayreuth as a guest of the DFG. The work of the second author was supported in part by the Air Force Office of Scientific Research under contract AFOSR-88-0218 and by a National Science Foundation Grant ECS-8802239 at the University of Maryland, Baltimore County Campus.     

Bounds on the size of tetrahedralizations

This paper gives lower and upper bounds on the complexity of triangulating the region between polyhedra. Particular attention is given to the case of convex polyhedra and terrains. The first author was suported in part by NSF Grant CCR-90-02352 and The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc. The second author was supported in part by NSF Grant PHY-90-21984.     

Tight bounds for christofides' traveling salesman heuristic

Christofides [1] proposes a heuristic for the traveling salesman problem that runs in polynomial time. He shows that when the graphG = (V, E) is complete and the distance matrix defines a function onV × V that is metric, then the length of the Hamiltonian cycle produced by the heuristic is always smaller than 3/2 times the length of an optimal Hamiltonian cycle. The purpose of this note is to refine Christofides' worst-case analysis by providing a tight bound for everyn 3, wheren is the number of vertices of the graph. We also show that these bounds are still tight when the metric is restricted to rectilinear distances, or to Euclidean distances for alln 6.This work was supported, in part, by NSF Grant ENG 75-00568 to Cornell University. This work was done when the authors were affiliated with the Center for Operations Research and Econometrics, University of Louvain, Belgium.     

Identifying an optimal basis in linear programming

We propose a sufficient condition that allows an optimal basis to be identified from a central path point in a linear programming problem. This condition can be applied when there is a gap in the sorted list of slack values. Unlike previously known conditions, this condition is valid for real-number data and does not involve the number of bits in the data.This work is supported in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF Grant DMS-8920550. Also supported in part by an NSF Presidential Young Investigator Award with matching funds received from AT&T and the Xerox Corporation. Part of this work was carried out while the author was visiting the Sandia National Laboratories, supported by the U.S. Department of Energy under Contract DE-AC04-76DP00789.The author is supported in part by NSF Grant DDM-9207347. Part of this work was carried out while the author was on a sabbatical leave from the University of Iowa and visiting the Cornell Theory Center, Cornell University, Ithaca, NY 14853, supported in part by the Cornell Center for Applied Mathematics and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center, which receives major funding from the National Science Foundation and the IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.     

Rademacher complexity in Neyman-Pearson classification

Neyman-Pearson（NP） criterion is one of the most important ways in hypothesis testing. It is also a criterion for classification. This paper addresses the problem of bounding the estimation error of NP classification, in terms of Rademacher averages. We investigate the behavior of the global and local Rademacher averages, and present new NP classification error bounds which are based on the localized averages, and indicate how the estimation error can be estimated without a priori knowledge of the class at hand.     

Shortest paths algorithms: Theory and experimental evaluation

We conduct an extensive computational study of shortest paths algorithms, including some very recent algorithms. We also suggest new algorithms motivated by the experimental results and prove interesting theoretical results suggested by the experimental data. Our computational study is based on several natural problem classes which identify strengths and weaknesses of various algorithms. These problem classes and algorithm implementations form an environment for testing the performance of shortest paths algorithms. The interaction between the experimental evaluation of algorithm behavior and the theoretical analysis of algorithm performance plays an important role in our research. This work was done while Boris V. Cherkassky was visiting Stanford University Computer Science Department and supported by the NSF and Powell Foundation grants mentioned below. Andrew V. Goldberg was supported in part by ONR Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation. Corresponding author. This work was done while Tomasz Radzik was a Postdoctoral Fellow at SORIE, Cornell University, and supported by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550, and by the Packard Fellowship of éva Tardos.     

Applications of the Dirichlet Principle to finite reversible nearest particle systems

Summary The Dirichlet Principle provides a variational expression for the survival probability of a supercritical finite reversible nearest particle system. We use this expression to derive improved bounds on this survival probability, and to develop techniques for comparing different systems with the same critical value.Research supported in part by NSF Grants MCS 83-00836 and DMS-8601800     

A primal-dual approximation algorithm for generalized steiner network problems

We present the first polynomial-time approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. Ifk is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primal-dual and shows the importance of this technique in designing approximation algorithms.Research supported by an NSF Graduate Fellowship, DARPA contracts N00014-91-J-1698 and N00014-92-J-1799, and AT&T Bell Laboratories.Research supported in part by Air Force contract F49620-92-J-0125 and DARPA contract N00014-92-J-1799.Part of this work was done while the author was visiting AT&T Bell Laboratories and Bellcore.     

Error estimates for approximate solutions to Bellman equations associated with controlled jump-diffusions

We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jump-diffusion processes, which are fully nonlinear integro-partial differential equations. Two main results are obtained: (i) error bounds for a class of monotone approximation schemes, which under some assumptions includes finite difference schemes, and (ii) bounds on the error induced when the original Lévy measure is replaced by a finite measure with compact support, an approximation process that is commonly used when designing numerical schemes for integro-partial differential equations. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen. This work is supported by the European network HYKE, contract HPRN-CT-2002-00282. The research of E. R. Jakobsen is supported by the Research Council of Norway through grant no 151608/432. The research of K. H. Karlsen is supported by an Outstanding Young Investigators Award from the Research Council of Norway. This work was done while C. La Chioma visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.     

On the convergence of the iteration sequence in primal-dual interior-point methods

Recently, numerous research efforts, most of them concerned with superlinear convergence of the duality gap sequence to zero in the Kojima—Mizuno—Yoshise primal-dual interior-point method for linear programming, have as a primary assumption the convergence of the iteration sequence. Yet, except for the case of nondegeneracy (uniqueness of solution), the convergence of the iteration sequence has been an important open question now for some time. In this work we demonstrate that for general problems, under slightly stronger assumptions than those needed for superlinear convergence of the duality gap sequence (except of course the assumption that the iteration sequence converges), the iteration sequence converges. Hence, we have not only established convergence of the iteration sequence for an important class of problems, but have demonstrated that the assumption that the iteration sequence converges is redundant in many of the above mentioned works.This research was supported in part by NSF Coop. Agr. No. CCR-8809615. A part of this research was performed in June, 1991 while the second and the third authors were at Rice University as visiting members of the Center for Research in Parallel Computation.Corresponding author. Research supported in part by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Research supported in part by NSF DMS-9102761 and DOE DE-FG05-91ER25100.Research supported in part by NSF DDM-8922636.     

Backward error analysis for linear systems associated with inverses ofH-matrices

In this paper, bounds on the growth factors resulting from Gaussian elimination applied to inverses ofH-matrices are developed and investigated. These bounds are then used in the error analysis for solving linear systemsAx =b whose coefficient matricesA are of this type. For each such system our results show that the Gaussian elimination without pivoting can proceed safely provided that the elements of the inverse of a certainM-matrix (associated with the coefficient matrixA) are not excessively large. We exhibit a particularly satisfactory situation for the special case whenA itself is an inverse of anM-matrix. Part of the first section of this paper is devoted to a discussion on some results of de Boor and Pinkus for the stability of triangular factorizations of systemsAx =b, whereA is a nonsingular totally nonnegative matrix, and to the explanation of why the analysis of de Boor and Pinkus is not applicable to the case when the coefficient matrixA is an inverse of anM-matrix.Research supported in part by NSF Grant MCS-8102114.Research supported in part by the U.S. Army Research Office under contract No. DAAG-29-81-K-0132 and in part by NSF Grant MCS-8219500.     

A real algorithm for the Hermitian eigenvalue decomposition

Modifying complex plane rotations, we derive a new Jacobi-type algorithm for the Hermitian eigendecomposition, which uses only real arithmetic. When the fast-scaled rotations are incorporated, the new algorithm brings a substantial reduction in computational costs. The new method has the same convergence properties and parallelism as the symmetric Jacobi algorithm. Computational test results show that it produces accurate eigenvalues and eigenvectors and achieves great reduction in computational time.The work of this author was supported in part by the National Science Foundation grant CCR-8813493 and by the University of Minnesota Army High Performance Computing Research Center contract DAAL 03-89-C-0038.The work of this author was supported in part by the University of Minnesota Army High Performance Computing Research Center contract DAAL 03-89-C-0038.     

Polynomial systems with few real zeroes

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ℤⁿ, this bound is 2n+1, while the Khovanskii bound is exponential in n². The bound 2n+1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic. Part of work done at MSRI was supported by NSF grant DMS-9810361. Work of Sottile is supported by the Clay Mathematical Institute. Sottile and Bihan were supported in part by NSF CAREER grant DMS-0134860. Bertrand is supported by the European research network IHP-RAAG contract HPRN-CT-2001-00271.     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

Mean lattice point discrepancy bounds,II: Convex domains in the plane

We consider planar curved strictly convex domains with very weak (or no) smoothness assumptions on the boundary and prove sharp bounds for square-functions associated to the lattice point discrepancy. Research supported in part by NSF grants.     

On Ricci flat 3-fold

In this paper we study a sub-class of compact Ricci flat manifolds from certain standard constructions. Research supported in part by NSF grant #DMS84-09447 and ONR contract #N-00014-85-K-0367.     

L-functions of second-order cusp forms

We discuss equivalent definitions of holomorphic second-order cusp forms and prove bounds on their Fourier coefficients. We also introduce their associated L-functions, prove functional equations for twisted versions of these L-functions and establish a criterion for a Dirichlet series to originate from a second order form. In the last section we investigate the effect of adding an assumption of periodicity to this criterion. 2000 Mathematics Subject Classification Primary—11F12, 11F66 G. Mason: Research supported in part by NSF Grant DMS 0245225. C. O’Sullivan: Research supported in part by PSC CUNY Research Award No. 65453-00 34.     

Worst-case analyses, linear programming and the bin-packing problem

In this paper we consider the familiar bin-packing problem and its associated set-partitioning formulation. We show that the optimal solution to the bin-packing problem can be no larger than 4/3 ⌈Z _LP⌉, whereZ _LP is the optimal solution value of the linear programming relaxation of the set-partitioning formulation. An example is provided to show that the bound is tight. A by-product of our analysis is a new worst-case bound on the performance of the well studied First Fit Decreasing and Best Fit Decreasing heuristics. This research was supported in part by ONR Contracts N00014-90-J-1649 and N00014-95-1-0232, and NSF Contracts DDM-8922712 and DDM-9322828.