arXiv subjects

Dana Shapira

Publications and source records attributed to Dana Shapira.

4 recordsLinked to original sources

arXiv2025

Asymmetric Number Partitioning with Splitting and Interval Targets

The n-way number partitioning problem, a fundamental challenge in combinatorial optimization, has significant implications for applications such as fair division and machine scheduling. Despite these problems being NP-hard, many approximation techniques exist. We consider three closely related kinds of approximations, and various objectives such as decision, min-max, max-min, and even a generalized objective, in which the bins are not considered identical anymore, but rather asymmetric (used to solve fair division to asymmetric agents or uniform machine scheduling problems). The first two variants optimize the partition such that: in the first variant some fixed number s of items can be split between two or more bins and in the second variant we allow at most a fixed number t of splittings. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number u times the largest item size. When the number of bins n is unbounded, we show that every variant is strongly NP-complete. When the number of bins n is fixed, the running time depends on the fixed parameters s,t,u. For each variant, we give a complete picture of its running time. For n=2, the running time is easy to identify. Our main results consider any fixed n>=3. Using a two-way polynomial-time reduction between the first and the third variant, we show that n-way number-partitioning with s split items can be solved in polynomial time if s>=n-2, and it is NP-complete otherwise. Also, n-way number-partitioning with t splittings can be solved in polynomial time if t>=n-1, and it is NP-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if u>=(n-2)/n, and it is NP-complete otherwise. Our positive results for the optimization problems consider both asymmetric min-max and asymmetric max-min versions.

cs.DS↗

arXiv2021

Weighted Burrows-Wheeler Compression

A weight based dynamic compression method has recently been proposed, which is especially suitable for the encoding of files with locally skewed distributions. Its main idea is to assign larger weights to closer to be encoded symbols by means of an increasing weight function, rather than considering each position in the text evenly. A well known transformation that tends to convert input files into files with a more skewed distribution is the Burrows-Wheeler Transform. This paper employs the weighted approach on Burrows-Wheeler transformed files and provides empirical evidence of the efficiency of this combination.

cs.DS↗

arXiv2020

Weighted Adaptive Coding

Huffman coding is known to be optimal, yet its dynamic version may be even more efficient in practice. A new variant of Huffman encoding has been proposed recently, that provably always performs better than static Huffman coding by at least $m-1$ bits, where $m$ denotes the size of the alphabet, and has a better worst case than the standard dynamic Huffman coding. This paper introduces a new generic coding method, extending the known static and dynamic variants and including them as special cases. In fact, the generalization is applicable to all statistical methods, including arithmetic coding. This leads then to the formalization of a new adaptive coding method, which is provably always at least as good as the best dynamic variant known to date. Moreover, we present empirical results that show improvements over static and dynamic Huffman and arithmetic coding achieved by the proposed method, even when the encoded file includes the model description.

cs.DS↗

arXiv2018

Scheduling with regular performance measures and optional job rejection on a single machine

We address single machine problems with optional jobs - rejection, studied recently in Zhang et al. [21] and Cao et al. [2]. In these papers, the authors focus on minimizing regular performance measures, i.e., functions that are non-decreasing in the jobs completion time, subject to the constraint that the total rejection cost cannot exceed a predefined upper bound. They also prove that the considered problems are ordinary NP-hard and provide pseudo-polynomial-time Dynamic Programming (DP) solutions. In this paper, we focus on three of these problems: makespan with release-dates; total completion times; and total weighted completion, and present enhanced DP solutions demonstrating both theoretical and practical improvements. Moreover, we provide extensive numerical studies verifying their efficiency.

cs.DM↗