Vladimir Kotov

Semi on-line algorithms for the partition problem

The partition problem is one of the basic NP-complete problems. While an efficient heuristic for the optimization version, which is equivalent to minimizing the makespan on two identical machines, is known with worst-case ratio 1211, no deterministic heuristic for the on-line problem can have a worst-case ratio lower than 32. In this paper we investigate three different semi on-line versions of the partition problem. In the first case, we assume that a buffer of length k is available to maintain k items. In the second case, two parallel processors are available which assign each item independently to the partition sets. The best of the two produced solutions is chosen. Finally, in the third problem the total sum of the items is known in advance. For each version we propose a heuristic and investigate its worst-case ratio. All algorithms have a worst-case ratio of 43 which is shown to be the best possible worst-case ratio.

Algorithms for on-line bin-packing problems with cardinality constraints

The bin-packing problem asks for a packing of a list of items of sizes from (0,1] into the smallest possible number of bins having unit capacity. The k-item bin-packing problem additionally imposes the constraint that at most k items are allowed in one bin. We present two efficient on-line algorithms for this problem. We show that, for increasing values of k, the bound on the asymptotic worst-case performance ratio of the first algorithm tends towards 2 and that the second algorithm has a ratio of 2. Both algorithms considerably improve upon the best known result of an algorithm, which has an asymptotic bound of 2.7 on its ratio. Moreover, we improve known bounds for all values of k by presenting on-line algorithms for k = 2 and 3 with bounds on their ratios close to optimal.

Thek-partitioning problem

Thek-partitioning problem is defined as follows: Given a set of items {I1,I2,...,In} where itemIj is of weightwj ≥ 0, find a partitionS1,S2,...,Sm of this set with ¦Si¦ =k such that the maximum weight of all subsetsSi is minimal,k-partitioning is strongly related to the classical multiprocessor scheduling problem of minimizing the makespan on identical machines. This paper provides suitable tools for the construction of algorithms which solve exactly the problem. Several approximation algorithms are presented for this NP-hard problem. The worst-case behavior of the algorithms is analyzed. The best of these algorithms achieves a performance bound of 4/3.

Semi-on-line multiprocessor scheduling with given total processing time

We are given a set of identical machines and a sequence of jobs, the sum of whose weights is known in advance. The jobs are to be assigned on-line to one of the machines and the objective is to minimize the makespan. An algorithm with performance ratio 1.6 and a lower bound of 1.5 is presented. These results improve on the recent results by Azar and Regev, who proposed an algorithm with performance ratio 1.625 for the less general problem that the optimal makespan is known in advance.

An approximation algorithm with absolute worst-case performance ratio 2 for two-dimensional vector packing

The two-dimensional vector packing problem is the generalization of the classical one-dimensional bin packing problem to two dimensions. While an asymptotic polynomial time approximation scheme has been designed for one-dimensional bin packing, the existence of an asymptotic polynomial time approximation scheme for two dimensions would imply P=NP. The existence of an approximation algorithm for the two-dimensional vector packing problem with an asymptotic performance guarantee 2 was an open problem so far. In this paper we present an O(nlogn) time algorithm for two-dimensional vector packing with absolute performance guarantee 2.

The EOQ problem with decidable warehouse capacity: Analysis, solution approaches and applications

The Economic Order Quantity (EOQ) problem is a fundamental problem in supply and inventory management. In its classical setting, solutions are not affected by the warehouse capacity. We study a type of EOQ problem where the (maximum) warehouse capacity is a decision variable. Furthermore, we assume that the warehouse cost dominates all the other inventory holding costs. We call this the EOQ-Max problem and the D-EOQ-Max problem, if the product is continuously divisible and discrete, respectively. The EOQ-Max problem admits a closed form optimal solution, while the D-EOQ-Max problem does not because its objective function may have several local minima. We present an optimal polynomial time algorithm for the discrete problem. Construction of this algorithm is supported by the fact that continuous relaxation of the D-EOQ-Max problem provides a solution that can be up to 50% worse than the optimal solution, and this worst-case error bound is tight. Applications of the D-EOQ-Max problem include supply and inventory management, logistics and scheduling.

Batch scheduling of step deteriorating jobs

Abstract In this paper we consider the problem of scheduling n jobs on a single machine, where the jobs are processed in batches and the processing time of each job is a step function depending on its waiting time, which is the time between the start of the processing of the batch to which the job belongs and the start of the processing of the job. For job i, if its waiting time is less than a given threshold value D, then it requires a basic processing time ai; otherwise, it requires an extended processing time ai+bi. The objective is to minimize the completion time of the last job. We first show that the problem is NP-hard in the strong sense even if all bi are equal, it is NP-hard even if bi=ai for all i, and it is non-approximable in polynomial time with a constant performance guarantee Δ<3/2, unless

A new algorithm for online uniform-machine scheduling to minimize the makespan

\mathcal {P}=\mathcal{NP}

The Stock Size Problem

. We then present O(nlog n) and O(n3F−1log n/FF) algorithms for the case where all ai are equal and for the case where there are F, F≥2, distinct values of ai, respectively. We further propose an O(n2log n) approximation algorithm with a performance guarantee

A 7/6–Approximation Algorithm For 3-Partitioning And Its Application To Multiprocessor Scheduling

\Delta\le1+\lfloor\frac{m^{*}}{2}\rfloor/m^{*}\le3/2