Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method

An Improved Multi-objective Algorithm for the Urban Transit Routing Problem

The determination of efficient routes and schedules in public transport systems is complex due to the vast search space and multi- ple constraints involved. In this paper we focus on the Urban Transit Routing Problem concerned with the physical network design of pub- lic transport systems. Historically, route planners have used their local knowledge coupled with simple guidelines to produce network designs. Several major studies have identified the need for automated tools to aid in the design and evaluation of public transport networks. We propose a new construction heuristic used to seed a multi-objective evolutionary al- gorithm. Several problem specific mutation operators are then combined with an NSGAII framework leading to improvements upon previously published results

Stochastic programming approaches to stochastic scheduling

Practical scheduling problems typically require decisions without full information about the outcomes of those decisions. Yields, resource availability, performance, demand, costs, and revenues may all vary. Incorporating these quantities into stochastic scheduling models often produces diffculties in analysis that may be addressed in a variety of ways. In this paper, we present results based on stochastic programming approaches to the hierarchy of decisions in typical stochastic scheduling situations. Our unifying framework allows us to treat all aspects of a decision in a similar framework. We show how views from different levels enable approximations that can overcome nonconvexities and duality gaps that appear in deterministic formulations. In particular, we show that the stochastic program structure leads to a vanishing Lagrangian duality gap in stochastic integer programs as the number of scenarios increases.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44935/1/10898_2004_Article_BF00121682.pd

An Adaptive Memory-Based Approach Based on Partial Enumeration

We propose an iterative memory-based algorithm for solving a class of combinatorial optimization problems. The algorithm generates a sequence of gradually improving solutions by exploiting at each iteration the knowledge gained in previous iterations. At each iteration, the algorithm builds an enumerative tree and stores at each tree level a set of promising partial solutions that will be used to drive the tree exploration in the following iteration. We tested the effectiveness of the proposed method on an hard combinatorial optimization problem arising in the design of telecommunication networks, the Non Bifurcated Network Design Problem, and we report computational results on a set of test problems simulating real life instances

Bounded-degree spanning tree problems: models and new algorithms

Spanning trees, Spanning spiders, Branch vertices, Heuristics, Optical networks,