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Two-phase scheduling with unknown speeds
Minařík, Josef ; Sgall, Jiří (advisor) ; Eberle, Franziska (referee)
Speed-robust scheduling is a two-stage scheduling problem with a makespan objective. We are given processing times of n jobs, number of machines m and number of bags b. We have to group the jobs into bags that are to be scheduled on machines of currently unknown speed. The goal is to minimize the worst-case ratio of our makespan and makespan of an adversary who does not have to create bags and assigns jobs directly to machines. So far, the problem has been mostly studied for b = m. We generalize previously known results for infinitesimal jobs (called sand) and prove that the best achievable competitive ratio is mb mb−(m−1)b . We present an algorithm for the case of identical jobs (called bricks) with competitive ratio at most 1.6 in the case b = m, improving the best previously known value of 1.8. We introduce a new category called p-pebbles, those are jobs with processing time at most p times the average load of a machine. Pebbles are half way between sand and the general case (called rocks). We present an algorithm for pebbles that has better robustness factor than the best known algorithm for rocks for small values of p (for p less than 2 − e e−1 in the case b = m). 1
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Algorithms for timetabling in sports
Koštejn, Vít ; Sgall, Jiří (advisor) ; Vu, Tung Anh (referee)
This bachelor thesis studies the timetabling problem of the creation of startlists for orienteering events. We focus both on theoretical and practical results. In the theo- retical part, the problem is translated to the scheduling terminology and then analyzed together with its special cases. We are mostly interested in the approximation algorithms and their ratios. In the practical part, we study multiple methods based on the pre- viously analyzed approximation algorithms and constraint programming for generating such startlists. Subsequently, these methods are tested on real data from previous events and compared with the startlists made by humans. 1
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Approximation and Online Algorithms
Tichý, Tomáš ; Sgall, Jiří (advisor) ; Kolman, Petr (referee) ; van Stee, Rob (referee)
This thesis presents results of our research in the area of optimization problems with incomplete information-our research is focused on the online scheduling problems. Our research is based on the worst-case analysis of studied problems and algorithms; thus we use methods of the competitive analysis during our research. Althrough there are many "real-world" industrial and theoretical applications of the online scheduling problems there are still so many open problems with so simple description. Therefore it is important, interesting and also challenging to study the online scheduling problems and their simplified variants as well. In this thesis we have shown the following our results of our research on the online scheduling problems: A 1.58-competitive online algorithm for the problem of randomized scheduling of unit jobs on a single processor, where the jobs are arriving over time and the total weight of processed jobs ismaximized. A lower bound 1.172 on the competitive ratio for the problem of randomized scheduling of 2-uniform unit jobs on a single processor, where the jobs are arriving over time andthe totalweight of processed jobs is maximized. A lower bound 1.25 on the competitive ratio for the problem of randomized scheduling of s-uniform unit jobs on a single processor where s is tending to...
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Algorithms for scheduling with conflicts
Zajíček, Ondřej ; Sgall, Jiří (advisor) ; Čepek, Ondřej (referee)
Scheduling with conflicts supposes graph of conflicts. Vertices of that graph represent machines and edges represent conflicts between them. Every machine can be switched on or switched off . Two conflicting machines cannot be both switched on at the same time. At certain times new tasks arrive to speci c machines and enqueue to its input buff ers. Each machine continuously processes tasks from its input bu ffer whenever it is switched on. An algorithm decides which machines should be switched on or switched o ff at any time, obeying conflict constraints. The objective is to schedule machine switching to minimize the maximum bu er size of all processors. The problem is online, so an algorithm has to make decisions about current con ffiguration without knowledge of future tasks. In this thesis I consider the algorithm based on maximization of scalar product of work vector (vector describing con guration of machines) and vector of bu ffer lengths. I prove that this algorithm is well de ffined, finite on every input and for speciffi c graph (path of length 3) it has competitive ratio of 7=3. Further I consider possibilities of implementation of that algorithm.
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Výpočetní složitost v teorii grafů
Ondráčková, Eva ; Kratochvíl, Jan (advisor) ; Sgall, Jiří (referee)
Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this thesis, we study the computational complexity the problem S(P) for a certain graph property P: given a graph G, determine if G is switching-equivalent to a graph having P. First, we give an overview of known results, including both properties P for which S(P) is polynomial, and those for which S(P) is NP-complete. Then we show the NP-completeness of the following problem for each c (0; 1): determine if a graph G can be switched to contain a clique of size at least cn, where n is the number of vertices of G. We also study the problem if, for a xed graph H, a given graph is switching-equivalent to an H-free graph. We show that for H isomorphic to a claw, the problem is polynomial. Further, we give a characterization of graphs witching-equivalent to a K1;2-free graph by ten forbidden induced subgraphs, each having ve vertices.
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Combinatorial algorithms for online problems: Semi-online scheduling on related machines
Ebenlendr, Tomáš ; Sgall, Jiří (advisor) ; Barták, Roman (referee) ; Epstein, Leah (referee) ; Woeginger, Gerhard (referee)
Mgr. Tomáš Ebenlendr Combinatorial algorithms for online problems: Semi-online scheduling on related machines Abstract of doctoral thesis We construct a framework that gives optimal algorithms for a whole class of scheduling problems. This class covers the most studied semi-online variants of preemptive online scheduling on uniformly related machines with the objective to minimize makespan. The algorithms from our framework are deterministic, yet they are optimal even among all randomized algorithms. In addition, they are optimal for any fixed combination of speeds of the machines, and thus our results subsume all the previous work on various special cases. We provide new lower bound of 2.112 for the original online problem. The (deterministic) upper bound is e ≈ 2.718 as there was known e-competitive randomized algorithm before. Our framework applies to all semi-online variants which are based on some knowledge about the input sequence. I.e., they are restrictions of the set of valid inputs. We use our framework to study restrictions that were studied before, and we derive some new bounds. Namely we study known sum of processing times, known maximal processing time, sorted (decreasing) jobs, tightly grouped processing times, approximately known optimal makespan and few combinations. Based on the analysis...
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On the Hardness of General Caching
Folwarczný, Lukáš ; Sgall, Jiří (advisor) ; Koutecký, Martin (referee)
Caching (also known as paging) is a classical problem concerning page re- placement policies in two-level memory systems. General caching is its vari- ant with pages of different sizes and fault costs. We aim at a better charac- terization of the computational complexity of general caching in the offline version. General caching in the offline version was recently shown to be strongly NP- hard, but the proof needed instances of caching with pages larger than half of the cache size. The primary result of this work addresses this problem as we prove: General caching is strongly NP-hard even when page sizes are limited to {1, 2, 3}. In the structural part of this work, a new simpler proof for the full characterization of work functions by layers for classical caching is given and then extended to caching with variable cache size. We invent two algorithms for restricted instances of general caching building on results around caching with variable cache size.
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Online Algorithms for Packet Scheduling
Veselý, Pavel ; Sgall, Jiří (advisor) ; Stein, Clifford (referee) ; Englert, Matthias (referee)
We study online scheduling policies for buffer management models, in which packets are arriving over time to a buffer of a network switch to be sent through its single output port. However, the bandwidth of the port is limited and some packets need to be dropped, based on their weights. The goal of the scheduler is to maximize the weighted throughput, that is, the total weight of packets transmitted. Due to the natural lack of information about future, an optimal performance cannot be achieved, we thus pursue competitive analysis and its refinements to analyze online algorithms on worst-case inputs. Specifically, in the first part of the thesis, we focus on a simple online scheduling model with unit-size packets and deadlines, called Bounded-Delay Packet Scheduling. We design an optimal φ-competitive deterministic algo- rithm for the problem, where φ ≈ 1.618 is the golden ratio. It is based on a detailed understanding of an optimal schedule of pending packets, called the plan, which may be of independent interest. We also propose a semi-online setting with lookahead that allows the algorithm to see a little bit of future, namely, packets arriving in the next few steps. We provide an algorithm with lookahead for instances in which each packet can be scheduled in at most two consecutive slots and prove lower...
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Online Bin Stretching: Algorithms and Computer Lower Bounds
Böhm, Martin ; Sgall, Jiří (advisor) ; Durr, Christoph (referee) ; Kellerer, Hans (referee)
Online Bin Stretching: Algorithms and Computer Lower Bounds Author: Martin Böhm Abstract: We investigate a problem in semi-online algorithm design, called Online Bin Stretching. The problem can be understood as an online repacking problem: the goal of the algorithm is to repack items of various sizes into m containers of identical size R > 1. The input items arrive one by one and the algorithm must assign an item to a container before the next item arrives. A specialty of this problem is that there is a specific guarantee made to the algorithm: the algorithm learns at the start of the input that there exists a packing of all input items into m containers of capacity 1. Our goal is to design algorithms for this problem which successfully pack the entire incoming sequence one by one while requiring the lowest container capacity R possible. In this thesis, we show several new results about Online Bin Stretching: First, we design an algorithm that is able to pack the entire input into m containers of capacity 1.5 regardless of what the vale of m will be. Second, we show a specialized algorithm for the setting of just 3 containers; this algorithm is able to pack into 3 bins of capacity 1.375. Finally, we design and implement an involved search algorithm which is able to find lower bounds for Online Bin...
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Online scheduling of multiprocessor jobs with preemption
Šimsa, Štěpán ; Sgall, Jiří (advisor) ; Kolman, Petr (referee)
Abstract. The thesis is devoted to the problem of online preemptive scheduling of mul- tiprocessor jobs. It gives a summary of previous work on this problem. For some special variants of the problem, especially if we restrict the sizes of jobs to one and two, new results are given, both in the terms of lower bounds and in the terms of competitive al- gorithms. A previously published lower bound is showed to be computed incorrectly and it is replaced by a correct lower bound in this thesis. An algorithm is presented for the special case of four processors and sizes of jobs one and two that is conjectured to achieve the best possible competitive ratio.
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