Integer programming approaches for minimum stabbing problems

The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associated to set of points in the plane. The complexity of the mstr remains open whilst the other two are known to be . This paper presents integer programming (ip) formulations for these three problems, that allowed us to solve them to optimality through ip branch-and-bound (b&b) or branch-and-cut (b&c) algorithms. Moreover, these models are the basis for the development of Lagrangian heuristics. Computational tests were conducted with instances taken from the literature where the performance of the Lagrangian heuristics were compared with that of the exact b&b and b&c algorithms. The results reveal that the Lagrangian heuristics yield solutions with minute, and often null, duality gaps for instances with several hundreds of points in small computation times. To our knowledge, this is the first computational study ever reported in which these three stabbing problems are considered and where provably optimal solutions are given. © 2014 EDP Sciences, ROADEF, SMAI.The problem of finding structures with minimum stabbing number has received considerable attention from researchers. Particularly, [10] study the minimum stabbing number of perfect matchings (mspm), spanning trees (msst) and triangulations (mstr) associat482211233CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO301732/2007-8; 473867/2010-9; 147619/2010-607/52015-

Approximation algorithms for rectangle stabbing and interval stabbing problems.

Inthe weighted rectangle stabbing problem we are given a grid in R2 consisting of columns and rows each having a positive integral weight, and a set of closed axis-parallel rectangles each having a positive integral demand. The rectangles are placed arbitrarily in the grid with the only assumption that each rectangle is intersected by at least one column and at least one row. The objective is to find a minimum-weight (multi)set of columns and rows of the grid so that for each rectangle the total multiplicity of selected columns and rows stabbing it is at least its demand. A special case of this problem arises when each rectangle is intersected by exactly one row. We describe two algorithms, called STAB and ROUND, that are shown to be constant-factor approximation algorithms for different variants of this stabbing problem.Research; Approximation; Algorithms; Problems; Demand;

Stabbing Planes

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting a lifting argument in communication complexity

The Complexity of Some Geometric Proof Systems

In this Thesis we investigate proof systems based on Integer Linear Programming. These methods inspect the solution space of an unsatisfiable propositional formula and prove that this space contains no integral points. We begin by proving some size and depth lower bounds for a recent proof system, Stabbing Planes, and along the way introduce some novel methods for doing so. We then turn to the complexity of propositional contradictions generated uniformly from first order sentences, in Stabbing Planes and Sum-Of-Squares. We finish by investigating the complexity-theoretic impact of the choice of method of generating these propositional contradictions in Sherali-Adams

Capacitated max-Batching with Interval Graph Compatibilities

We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7, 4, 5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.

On Partial Covering For Geometric Set Systems

We study a generalization of the Set Cover problem called the Partial Set Cover in the context of geometric set systems. The input to this problem is a set system (X, R), where X is a set of elements and R is a collection of subsets of X, and an integer k <= |X|. Each set in R has a non-negative weight associated with it. The goal is to cover at least k elements of X by using a minimum-weight collection of sets from R. The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for a certain projection of the set system (X, R). As a corollary of this result, we get improved approximation guarantees for the Partial Set Cover problem for a large class of geometric set systems

Two-stage Robust Optimization Approach for Enhanced Community Resilience Under Tornado Hazards

Catastrophic tornadoes cause severe damage and are a threat to human wellbeing, making it critical to determine mitigation strategies to reduce their impact. One such strategy, following recent research, is to retrofit existing structures. To this end, in this article we propose a model that considers a decision-maker (a government agency or a public-private consortium) who seeks to allocate resources to retrofit and recover wood-frame residential structures, to minimize the population dislocation due to an uncertain tornado. In the first stage the decision-maker selects the retrofitting strategies, and in the second stage the recovery decisions are made after observing the tornado. As tornado paths cannot be forecast reliably, we take a worst-case approach to uncertainty where paths are modeled as arbitrary line segments on the plane. Under the assumption that an area is affected if it is sufficiently close to the tornado path, the problem is framed as a two-stage robust optimization problem with a mixed-integer non-linear uncertainty set. We solve this problem by using a decomposition column-and-constraint generation algorithm that solves a two-level integer problem at each iteration. This problem, in turn, is solved by a decomposition branch-and-cut method that exploits the geometry of the uncertainty set. To illustrate the model's applicability, we present a case study based on Joplin, Missouri. Our results show that there can be up to 20 percent reductions in worst-case population dislocation by investing 15 million dollars in retrofitting and recovery; that our approach outperforms other retrofitting policies, and that the model is not over-conservative

Fast Path Planning Through Large Collections of Safe Boxes

We present a fast algorithm for the design of smooth paths (or trajectories) that are constrained to lie in a collection of axis-aligned boxes. We consider the case where the number of these safe boxes is large, and basic preprocessing of them (such as finding their intersections) can be done offline. At runtime we quickly generate a smooth path between given initial and terminal positions. Our algorithm designs trajectories that are guaranteed to be safe at all times, and it detects infeasibility whenever such a trajectory does not exist. Our algorithm is based on two subproblems that we can solve very efficiently: finding a shortest path in a weighted graph, and solving (multiple) convex optimal control problems. We demonstrate the proposed path planner on large-scale numerical examples, and we provide an efficient open-source software implementation, fastpathplanning