Restricted assignment scheduling with resource constraints

We consider parallel machine scheduling with job assignment restrictions, i.e., each job can only be processed on a certain subset of the machines. Moreover, each job requires a set of renewable resources. Any resource can be used by only one job at any time. The objective is to minimize the makespan. We present approximation algorithms with constant worst-case bound in the case that each job requires only a fixed number of resources. For some special cases optimal algorithms with polynomial running time are given. If any job requires at most one resource and the number of machines is fixed, we give a PTAS. On the other hand we prove that the problem is APX-hard, even when there are just three machines and the input is restricted to unit-time jobs. (C) 2018 Published by Elsevier B.V

Using weight decision for decreasing the price of anarchy in selfish bin packing games

A selfish bin packing game is a variant of the classical bin packing problem in a game theoretic setting. In our model the items have not only a size but also a nonnegative weight. Each item plays the role of a selfish agent, and any agent/item pays some cost for being in a bin. The cost of a bin is 1, and this cost is shared among the items being in the bin, proportionally to their weight. A packing of the items into bins is called a Nash equilibrium if no item can decrease its cost by moving to another bin. In this paper we present two different settings for the weights which provide better values for the price of anarchy (PoA) than previous settings investigated so far. The improved PoA is not bigger than 16/11 approximate to 1.4545. Moreover, we give a general lower bound for the price of anarchy which holds for all possible choices of the weights. (C) 2019 Elsevier B.V. All rights reserved

Bounds for online bin packing with cardinality constraints

Abstract We study a bin packing problem in which a bin can contain at most k items of total size at most 1, where k ≥ 2 is a given parameter. Items are presented one by one in an online fashion. We analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k ≥ 4 . Additionally, we present bounds for relatively small values of k with respect to the asymptotic competitive ratio and the absolute competitive ratio. In particular, we provide tight bounds on the absolute competitive ratio of First Fit for k = 2 , 3 , 4 , and improve the known lower bounds on asymptotic competitive ratios for multiple values of k. Our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general, and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing

Lehet-e a vizuális mentális kép kétértelmű? A képesség és a gyakorlás hatása = Can visual mental image be ambiguous? Effects of ability and practice

A kétértelmű ábrák észlelésével és mentális átalakíthatóságával kapcsolatban, két irányban végeztünk kutatásokat. Kimutattuk a mentális forgatás gyakorlásának és a teszttel mért képzeleti tárgymanipulációs képesség hatását a kétértelmű ábrák képzeleti újraértelmezésére. Mentális forgatással, fixációs váltással, illetve a referenciakeret átszervezéssel megoldható feladatokban egyaránt találtunk sikeres újraértelmezési kivitelezéseket, s részben alátámasztottuk a mentális forgatás kiemelt részfolyamatának szerepét (1). Más adataink szerint a téri szerkezet képzeletben átalakítható. A síkbeli értelmezést dimenzióváltással térbeli értelmezésre való átváltási feladatot a személyek többsége csak a lerajzolt mentális kép alapján tudta megoldani. A tudás szerepe sem elhanyagolható (2). A kutatás másik vonalán kétértelmű ábrák észlelési megfordulásának fejlődését vizsgálva rámutattunk, hogy a megfordulás komplex, észlelés-alapúnak tartható folyamatához, megfelelő fejlődési szinten a lehetséges kétféle értelmezés megértésén túl, a központi végrehajtó működéséhez köthető szelektív figyelmi, gátlási, figyelemváltási és mentális képzeleti képességek is hozzájárulnak. A spontán megfordulások előfordulása, a kétnyelvű gyerekek "fordított világ" feladat eredményei, valamint a firkarejtvény feladat teljesítési szintje és a gyenge korrelációja az észlelési megfordulásokkal alapján az eredeti naív tudatelméleti értelmezés nem kielégítő (3- 4). | The studies of perception and re-interpretation of ambiguous figures has two directions. We try to find the role of practice of mental operations that contribute to the reinterpretation of previously shown ambiguous figures in mental imagery tasks. There are people who are successful in re-interpreting figures, whereas it becomes obvious that the practice of mental rotation enhances performance. This effect cannot explicitly be explained re-interpreting rotated figures. Moreover, performance increases in specific groups. This suggests the unequal transfer of the sub-components of mental rotation. Secondly, two experiments were done with children (3.5-9 yrs). 1. We assessed the proportion of the spontaneous reversal of images, and the relationship between the reversal of the informed condition ambiguous figures, and the results of theory of mind tests, and those of tests assessing interpretation differences (False Belief, Droodle task). Results: 5-9 year old children are able to experience spontaneous reversal, whereas informations about ambiguity facilitated the perception reversals of 4-5 year-old children. Weak correlation was found between the scores of the Droodle task and perception of reversal. 2. Our findings from monolingual and bilingual children indicated that selective attention, inhibition, attentional switch and mental imagery abilities were connected to the central executive function also contribute to the complex perception-based process of reversal

The convergence time for selfish bin packing

In classic bin packing, the objective is to partition a set of n items with positive rational sizes in (0, 1] into a minimum number of subsets called bins, such that the total size of the items of each bin at most 1. We study a bin packing game where the cost of each bin is 1, and given a valid packing of the items, each item has a cost associated with it, such that the items that are packed into a bin share its cost equally. We find tight bounds on the exact worst-case number of steps in processes of convergence to pure Nash equilibria. Those are processes that are given an arbitrary packing as an initial packing. As long as there exists an item that can reduce its cost by moving from its bin to another bin, in each step, a controller selects such an item and instructs it to perform such a beneficial move. The process converges when no further beneficial moves exist. The tight function of n that we find is in Θ(n 3/2 ). This improves the previous bound of Ma et al. [14], who showed an upper bound of O(n 2)

A New Lower Bound for Classic Online Bin Packing

We improve the lower bound on the asymptotic competitive ratio of any online algorithm for bin packing to above 1.54278. We demonstrate for the first time the advantage of branching and the applicability of full adaptivity in the design of lower bounds for the classic online bin packing problem. We apply a new method for weight based analysis, which is usually applied only in proofs of upper bounds. The values of previous lower bounds were approximately 1.5401 and 1.5403. © 2020, Springer Nature Switzerland AG