Daya Ram Gaur

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.

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.

Self-Duality of Bounded Monotone Boolean Functions and Related Problems

In this paper we show the equivalence between the problem of determining self-duality of a boolean function in DNF and a special type of satisfiability problem called NAESPI. Eiter and Gottlob [8] use a result from [2] to show that self-duality of monotone boolean functions which have bounded clause sizes (by some constant) can be determined in polynomial time. We show that the self-duality of instances in the class studied by Eiter and Gottlob can be determined in time linear in the number of clauses in the input, thereby strengthening their result. Domingo [7] recently showed that self-duality of boolean functions where each clause is bounded by (√log n) can be solved in polynomial time. Our linear time algorithm for solving the clauses with bounded size infact solves the (√log n) bounded self-duality problem in O(n2 √log n) time, which is better bound then the algorithm of Domingo [7], O(n3). n nAnother class of self-dual functions arising naturally in application domain has the property that every pair of terms in f intersect in at most constant number of variables. The equivalent subclass of NAESPI is the c-bounded NAESPI. We also show that c-bounded NAESPI can be solved in polynomial time when c is some constant. We also give an alternative characterization of almost self-dual functions proposed by Bioch and Ibaraki [5] in terms of NAESPI instances which admit solutions of a particular type.

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.

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.

Constan 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(log n). This algorithm is then extended to a 4-approximation algorithm for the rectilinear partitioning problem, which, given an m × n array of non-negative integers, asks to find a set of vertical and horizontal lines such that the maximum load of a subrectangle (i.e., the sum of the numbers in it) is minimized. This problem arises when a mapping of an m × n array onto an h × v mesh of processors is required such that the largest load assigned to a processor is minimized. The best known approximation ratio for this problem is 27. Our approximation ratio 4 is close to the best possible, as there is evidence that it is NP-hard to approximate within a factor of 2.

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.

Simple Approximation Algorithms for MAXNAESP and Hypergraph 2-colorability

Hypergraph 2-colorability, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph 2-colorability, set splitting and MAXCUT (which is a special case of hypergraph 2-colorability) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of 1/2 + Ɛ. This is an improvement over the ratio of 1/2 attributed to another local improvement heuristic for MAX-CUT [6]. In fact we provide a bound of k/k+1 for this problem, where k ≥ 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques [3] appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both [4] yield a bound of 1/2 for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of 1/2 + Ɛ for this problem.

An approximation algorithm for nonpreemptive scheduling on hypercube parallel task systems

We study a generalization of the nonpreemptive scheduling problem on hypercube parallel task systems. We generalize the problem by limiting the maximum number of tasks that can be simultaneously executed. We describe a simple 3-approximation algorithm for this problem when speedup and execution times are monotone in the number of processors assigned to a task.

Disjunctive constraint satisfaction over reals

Optimal algorithms are given for the disjunctive constraint satisfaction problem when the dimensions are two and three. These algorithms are based on techniques frequently used in computational geometry. The running times of the algorithms are O(n/sup 2/) and O(n/sup 3/) in two and three dimensions, respectively.<<ETX>>