Selected Publications
A heuristic for cumulative vehicle routing using column generation
Cumulative vehicle routing problems are a simplified model of fuel consumption in vehicle routing problems. Here we computationally study, an inexact approach for constructing solutions to cumulative vehicle routing problems based on rounding solutions to a linear program. The linear program is based on the set cover formulation and is solved using column generation. The pricing subproblem is solved heuristically using dynamic programming. Simulation results show that a simple scalable strategy gives solutions with cost close to the lower bound given by the linear programming relaxation. We also give theoretical bounds on the integrality gap of the set cover formulation.
Discrete Applied Mathematics, In Press Corrected Proof, July 2016
Conflict Resolution in the Scheduling of Television Commercials
We extend a previous model for scheduling commercial advertisements during breaks in television programming. The proposed extension allows differential weighting of conflicts between pairs of commercials. We formulate the problem as a capacitated generalization of the max k-cut problem in which the vertices of a graph correspond to commercial insertions and the edge weights to the conflicts between pairs of insertions. The objective is to partition the vertices into k capacitated sets to maximize the sum of conflict weights across partitions. We note that the problem is NP-hard. We extend a previous local-search procedure to allow for the differential weighting of edge weights. We show that for problems with equal insertion lengths and break durations, the worst-case bound on the performance of the proposed algorithm increases with the number of program breaks and the number of insertions per break, and that it is independent of the number of conflicts between pairs of insertions. Simulation results suggest that the algorithm performs well even if the problem size is small.
Operations Research 57(5): 1098–1105 (2009)
Constant Ratio Approximation Algorithms for the Rectangle Stabbing Problem and the Rectilinear Partitioning Problem
We provide constant ratio approximation algorithms for two NP-hard problems, the rectangle stabbing problem and the rectilinear partitioning problem. In the rectangle stabbing problem, we are given a set of rectangles in two-dimensional space, with the objective of stabbing all rectangles with the minimum number of lines parallel to the x and y axes. We provide a 2-approximation algorithm, while the best known approximation ratio for this problem is O(logn). This algorithm is then extended to a 4-approximation algorithm for the rectilinear partitioning problem, which, given an mx × my array of nonnegative integers and positive integers v, h, asks to find a set of v vertical and h horizontal lines such that the maximum load of a subrectangle (i.e., the sum of the numbers in it) is minimized. The best known approximation ratio for this problem is 27. Our approximation ratio 4 is close to the best possible, as it is known to be NP-hard to approximate within any factor less than 2. The results are then extended to the d-dimensional space for d ≥ 2, where a d-approximation algorithm for the stabbing problem and a dd-approximation algorithm for the partitioning problem are developed.
Journal of Algorithms, 43(1): 138–152 (2002)
On the fractional chromatic number of monotone self-dual Boolean functions
We compute the exact fractional chromatic number for several classes of monotone self-dual Boolean functions. We characterize monotone self-dual Boolean functions in terms of the optimal value of an LP relaxation of a suitable strengthening of the standard IP formulation for the chromatic number. We also show that determining the self-duality of a monotone Boolean function is equivalent to determining the feasibility of a certain point in a polytope defined implicitly.
Disc. Math. 309(4): 867–877 (2009)
Routing vehicles to minimize fuel consumption
We consider a generalization of the capacitated vehicle routing problem known as the cumulative vehicle routing problem in the literature. Cumulative VRPs are known to be a simple model for fuel consumption in VRPs. We examine four variants of the problem, and give constant factor approximation algorithms. Our results are based on a well-known heuristic of partitioning the traveling salesman tours and the use of the averaging argument.
Oper. Res. Lett. 41(6): 576-580 (2013)