Papers by Latife Genç-Kaya
Hybrid Approaches to Scheduling and Clustering
This dissertation consists of four self-contained chapters. The first two chapters concentrate on... more This dissertation consists of four self-contained chapters. The first two chapters concentrate on the circuit constraint. The circuit constraint requires that a sequence of n vertices in a directed graph describe a hamiltonian cycle. The constraint is useful for the succinct formulation of sequencing problems, such as the traveling salesman problem, which it formulates with only one constraint and n variables. In the first chapter, ”The Circuit Polytope”, we analyze the circuit polytope as an alternative to the traveling salesman polytope as a means of obtaining linear relaxations for sequencing problems. We provide a nearly complete characterization of the circuit polytope by showing how to generate, using a greedy algorithm, all facet-defining inequalities that contain at most n − 4 terms. We suggest efficient separation heuristics. Finally, we show that proper choice of the numerical values that index the vertices can allow the resulting relaxation to exploit structure in the obj...
Factory Crane Scheduling by Dynamic Programming
We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovati... more We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovative data structure controls the memory requirements of the state space and enables solution of problems of realistic size. The algorithm finds optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must not interfere with each other, although one may yield to the other. The state space representation permits a wide variety of constraints and objective functions. It is stored in a compressed data structure that uses a cartesian product of intervals of states and an array of two-dimensional circular queues. We also show that only certain types of trajectories need be considered. The algorithm found optimal solutions in less than a minute for medium-sized instances of the problem (160 tasks, spanning...
We study the problem of finding optimal space-time trajectories for two factory cranes or hoists ... more We study the problem of finding optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is a assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must be operated so as not to interfere with each other, although one crane may need to yield to another. The objective is generally to follow a production schedule as closely as possible. We show that only certain types of trajectories need be considered to obtain an optimal solution. This simplifies the operation of the cranes and enhances safety, because the cranes move according to predictable patterns. We present a specialized dynamic programming algorithm that solves the 1 problem. To control the state space size we use an innovative state space representation based on a cartesian product of intervals of states and an array of two-dimensional circular queues. The algorithm solves medium-sized insta...
A filter for the circuit constraint
Abstract. We present an incomplete filtering algorithm for the circuit constraint. The filter rem... more Abstract. We present an incomplete filtering algorithm for the circuit constraint. The filter removes redundant values by eliminating nonhamiltonian edges from the associated graph. We identify nonhamiltonian edges by analyzing a smaller graph with labeled edges that is defined on a separator of the original graph. The complexity of the procedure for each separator S is approximately O(|S | 5). We found that it identified all infeasible instances and eliminated about one-third of the redundant domain elements in feasible instances. The circuit constraint can be written circuit(x1,...,xn) where the domain of each xi is Di ⊂{1,...,n}. The constraint requires that y1,...,yn be a cyclic permutation of 1,...,n,where yi+1 = xyi, i =1,...,n − 1 y1 = xyn Let directed graph G contain an edge (i, j) if and only if j belongs to the domain
J.N.: Factory crane scheduling by dynamic programming
We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovati... more We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovative data structure controls the memory re-quirements of the state space and enables solution of problems of realistic size. The algorithm nds optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specied locations that must be performed within given time windows. The cranes must not in-terfere with each other, although one may yield to the other. The state space representation permits a wide variety of constraints and objective functions. It is stored in a compressed data structure that uses a carte-sian product of intervals of states and an array of two-dimensional circular queues. We also show that only certain types of trajectories need be con-sidered. The algorithm found optimal solutions in less than a minute for medium-sized instances of the problem (160 tasks, spanning...
The Hamiltonian Circuit Polytope
The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constra... more The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems without using 0–1 variables. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these tools to derive several families of facets. The tools include necessary and sufficient conditions for an inequality to be facet defining, and an algorithm for generating all undominated circuits. We use a novel approach to identifying families of facet-defining inequalities, based on the structure of variable indices rather than on subgraphs such as combs or subtours. This leads to our main result, a hierarchy of families of facet-defining inequalities and polynomial-time separation algorithms for them.
The Circuit Polytope
The circuit constraint requires that a sequence of n vertices in a directed graph describe a hami... more The circuit constraint requires that a sequence of n vertices in a directed graph describe a hamiltonian cycle. The constraint is useful for the succinct formulation of sequencing problems, such as the traveling salesman problem. We analyze the circuit polytope as an alternative to the traveling salesman polytope as a means of obtaining linear relaxations for sequencing problems. We provide a nearly complete characterization of the polytope by showing how to generate, using a greedy algorithm, all facet-defining inequalities that contain at most n − 4 terms. We suggest efficient separation heuristics. Finally, we show that proper choice of the numerical values that index the vertices can allow the resulting relaxation to exploit structure in the objective function.
The Circuit Polytope
The circuit constraint requires that a sequence of n vertices in a directed graph describe a hami... more The circuit constraint requires that a sequence of n vertices in a directed graph describe a hamiltonian cycle. The constraint is useful for the succinct formulation of sequencing problems, such as the traveling salesman problem. We analyze the circuit polytope as an alternative to the traveling salesman polytope as a means of obtaining linear relaxations for sequencing problems. We provide a nearly complete characterization of the polytope by showing how to generate, using a greedy algorithm, all facet-defining inequalities that contain at most n− 4 terms. We suggest efficient separation heuristics. Finally, we show that proper choice of the numerical values that index the vertices can allow the resulting relaxation to exploit structure in the objective function. 1 The Circuit Constraint The circuit constraint requires that a sequence of vertices in a directed graph define a hamiltonian circuit. Let G be a directed graph on vertices 1, . . . , n, and let variable xi denote the vert...
We study the problem of finding optimal space-time trajectories for two factory cranes or hoists ... more We study the problem of finding optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is a assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must be operated so as not to interfere with each other, although one crane may need to yield to another. The objective is generally to follow a production schedule as closely as possible. We show that only certain types of trajectories need be considered to obtain an optimal solution. This simplifies the operation of the cranes and enhances safety, because the cranes move according to predictable patterns. We present a specialized dynamic programming algorithm that solves the
School of Business 9-2008 The Circuit Polytope
The circuit constraint requires that a sequence of n vertices in a directed graph describe a hami... more The circuit constraint requires that a sequence of n vertices in a directed graph describe a hamiltonian cycle. The constraint is useful for the succinct formulation of sequencing problems, such as the traveling salesman problem. We analyze the circuit polytope as an alternative to the traveling salesman polytope as a means of obtaining linear relaxations for sequencing problems. We provide a nearly complete characterization of the polytope by showing how to generate, using a greedy algorithm, all facet-defining inequalities that contain at most n − 4 terms. We suggest efficient separation heuristics. Finally, we show that proper choice of the numerical values that index the vertices can allow the resulting relaxation to exploit structure in the objective function. 1 The Circuit Constraint The circuit constraint requires that a sequence of vertices in a directed graph define a hamiltonian circuit. Let G be a directed graph on vertices 1, . . . , n, and let variable xi denote the ver...
We study the problem of finding optimal space-time trajectories for two factory cranes or hoists ... more We study the problem of finding optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must be operated so as not to interfere with each other, although one crane may need to yield to another. The objective is generally to follow a production schedule as closely as possible. We show that only certain types of trajectories need be considered to obtain an optimal solution. This simplifies the operation of the cranes and enhances safety, because the cranes move according to predictable patterns. We present a specialized dynamic programming algorithm that solves the problem and accommodates a wide variety of constraints that typically arise in such problems. To control the state space size we use an innovative state space representation based on a cartesian product of intervals of states and an ...
We present an incomplete filtering algorithm for the circuit constraint. The filter removes redun... more We present an incomplete filtering algorithm for the circuit constraint. The filter removes redundant values by eliminating nonhamiltonian edges from the associated graph. We identify nonhamiltonian edges by analyzing a smaller graph with labeled edges that is defined on a separator of the original graph. The complexity of the procedure for each separator S is approximately O(|S| 5). We found that it identified all infeasible instances and eliminated about one-third of the redundant domain elements in feasible instances.
Domain Reduction for the Circuit Constraint
Lecture Notes in Computer Science, 2005
We study the problem of finding optimal space-time trajectories for two factory cranes or hoists ... more We study the problem of finding optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must be operated so as not to interfere with each other, although one crane may need to yield to another. The objective is generally to follow a production schedule as closely as possible. We show that only certain types of trajectories need be considered to obtain an optimal solution. This simplifies the operation of the cranes and enhances safety, because the cranes move according to predictable patterns. We present a specialized dynamic programming algorithm that solves the problem and accommodates a wide variety of constraints that typically arise in such problems. To control the state space size we use an innovative state space representation based on a cartesian product of intervals of states and an array of two-dimensional circular queues. The algorithm solves medium-sized instances of the problem to optimality and can be used to create benchmarks for tuning heuristic algorithms that solve larger instances.
12th INFORMS Computing Society Conference, 2011
We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovati... more We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovative data structure controls the memory requirements of the state space and enables solution of problems of realistic size. The algorithm finds optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must not interfere with each other, although one may yield to the other. The state space representation permits a wide variety of constraints and objective functions. It is stored in a compressed data structure that uses a cartesian product of intervals of states and an array of two-dimensional circular queues. We also show that only certain types of trajectories need be considered. The algorithm found optimal solutions in less than a minute for medium-sized instances of the problem (160 tasks, spanning four hours). It can also be used to create benchmarks for tuning heuristic algorithms that solve larger instances.
12th INFORMS Computing Society Conference, 2011
We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovati... more We describe a specialized dynamic programming algorithm for factory crane scheduling. An innovative data structure controls the memory requirements of the state space and enables solution of problems of realistic size. The algorithm finds optimal space-time trajectories for two factory cranes or hoists that move along a single overhead track. Each crane is assigned a sequence of pickups and deliveries at specified locations that must be performed within given time windows. The cranes must not interfere with each other, although one may yield to the other. The state space representation permits a wide variety of constraints and objective functions. It is stored in a compressed data structure that uses a cartesian product of intervals of states and an array of two-dimensional circular queues. We also show that only certain types of trajectories need be considered. The algorithm found optimal solutions in less than a minute for medium-sized instances of the problem (160 tasks, spanning four hours). It can also be used to create benchmarks for tuning heuristic algorithms that solve larger instances.
An FPTAS for minimizing the product of two non-negative linear cost functions
Mathematical Programming, 2011
We consider a quadratic programming (QP) problem ( ) of the form minxTCx subject toAx b where C2 ... more We consider a quadratic programming (QP) problem ( ) of the form minxTCx subject toAx b where C2 Rn◊n + ;rank(C) = 1 and A2 Rm ◊n;b2 Rm. We present an FPTAS for this problem by reformulating the QP ( ) as a parametrized LP and "rounding" the optimal solution. Furthermore, our algorithm returns an extreme point solution of the
The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constra... more The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these tools to derive several families of facets. The tools include necessary and sufficient conditions for an inequality to be facet defining, and an algorithm for generating all undominated circuits. We use a novel approach to identifying families of facetdefining inequalities, based on the structure of variable indices rather than on subgraphs such as combs or subtours. This leads to our main result, a hierarchy of families of facet-defining inequalities and polynomial-time separation algorithms for them. *
Uploads
Papers by Latife Genç-Kaya