Refinements and Randomised Versions of Some Tournament Solutions

We consider voting rules that are based on the majority graph. Such rules typically output large sets of winners. Our goal is to investigate a general method which leads to refinements of such rules. In particular, we use the idea of parallel universes, where each universe is connected with a permutation over alternatives. The permutation allows us to construct resolute voting rules (i.e. rules that always choose unique winners). Such resolute rules can be constructed in a variety of ways: we consider using binary voting trees to select a single alternative. In turn this permits the construction of neutral rules that output the set the possible winners of every parallel universe. The question of which rules can be constructed in this way has already been partially studied under the heading of agenda implementability. We further propose a randomised version in which the probability of being the winner is the ratio of universes in which the alternative wins. We also investigate (typically novel) rules that elect the alternatives that have maximal winning probability. These rules typically output small sets of winners, thus provide refinements of known tournament solutions

Single-Elimination Brackets Fail to Approximate Copeland Winner

Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature

A memetic algorithm for the university course timetabling problem

This article is posted here with permission from IEEE - Copyright @ 2008 IEEEThe design of course timetables for academic institutions is a very hectic job due to the exponential number of possible feasible timetables with respect to the problem size. This process involves lots of constraints that must be respected and a huge search space to be explored, even if the size of the problem input is not significantly large. On the other hand, the problem itself does not have a widely approved definition, since different institutions face different variations of the problem. This paper presents a memetic algorithm that integrates two local search methods into the genetic algorithm for solving the university course timetabling problem (UCTP). These two local search methods use their exploitive search ability to improve the explorative search ability of genetic algorithms. The experimental results indicate that the proposed memetic algorithm is efficient for solving the UCTP

Pipelined genetic propagation

© 2015 IEEE.Genetic Algorithms (GAs) are a class of numerical and combinatorial optimisers which are especially useful for solving complex non-linear and non-convex problems. However, the required execution time often limits their application to small-scale or latency-insensitive problems, so techniques to increase the computational efficiency of GAs are needed. FPGA-based acceleration has significant potential for speeding up genetic algorithms, but existing FPGA GAs are limited by the generational approaches inherited from software GAs. Many parts of the generational approach do not map well to hardware, such as the large shared population memory and intrinsic loop-carried dependency. To address this problem, this paper proposes a new hardware-oriented approach to GAs, called Pipelined Genetic Propagation (PGP), which is intrinsically distributed and pipelined. PGP represents a GA solver as a graph of loosely coupled genetic operators, which allows the solution to be scaled to the available resources, and also to dynamically change topology at run-time to explore different solution strategies. Experiments show that pipelined genetic propagation is effective in solving seven different applications. Our PGP design is 5 times faster than a recent FPGA-based GA system, and 90 times faster than a CPU-based GA system

Level-Based Analysis of Genetic Algorithms and Other Search Processes

The fitness-level technique is a simple and old way to derive upper bounds for the expected runtime of simple elitist evolutionary algorithms (EAs). Recently, the technique has been adapted to deduce the runtime of algorithms with non-elitist populations and unary variation operators [2,8]. In this paper, we show that the restriction to unary variation operators can be removed. This gives rise to a much more general analytical tool which is applicable to a wide range of search processes. As introductory examples, we provide simple runtime analyses of many variants of the Genetic Algorithm on well-known benchmark functions, such as OneMax, LeadingOnes, and the sorting problem

Nested Balanced Incomplete Block Designs

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b1;r;k1) are each partitioned into sub-blocks of size k2, and the b2 =b1k1=k2 sub-blocks themselves constitute a BIBD with parameters (v; b2;r;k2), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k1 =2k2= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, they are almost unknown in the combinatorial literature, except in the literature of tournaments, and detailed combinatorial studies of them have been lacking. The present paper therefore reviews and extends mathematical knowledge of NBIBDs. Isomorphism and automorphisms are defined for NBIBDs, and methods of construction are outlined. Some special types of NBIBD are de4ned and illustrated. A first-ever detailed table of NBIBDs with v⩽16, r⩽30 is provided; this table contains many newly discovered NBIBDs. © 2001 Elsevier Science B.V. All rights reserved

CSM429: Abstract Geometric Crossover for the Permutation Representation

Abstract crossover and abstract mutation are representation-independent operators that are well-defined once a notion of distance over the solution space is defined. They were obtained as generalization of genetic operators for binary strings and real vectors. In this paper we explore how the abstract geometric framework applies to the permutation representation. This representation is challenging for various reasons: because of the inherent difference between permutations and the representations that inspired the abstraction; because the whole notion of geometry over permutation spaces radically departs from traditional geometries and it is almost unexplored mathematical territory; because the many notions of distance available and their subtle interconnections make it hard to see the right distance to use, if any; because the various available interpretations of permutations make ambiguous what a permutation represents, hence, how to treat it; because of the existence of various permutation-like representations that are incorrectly confused with permutations; and finally because of the existence of many mutation and recombination operators and their many variations for the same representation. This article shows that the application of our geometric framework naturally clarifies and unifies an important domain,the permutation representation and the related operators, in which there was little or no hope to find order. In addition the abstract geometric framework is used to improve the design of crossover operators for well-known problems naturally connected with the permutation representation

A genetic approach to statistical disclosure control

Statistical disclosure control is the collective name for a range of tools used by data providers such as government departments to protect the confidentiality of individuals or organizations. When the published tables contain magnitude data such as turnover or health statistics, the preferred method is to suppress the values of certain cells. Assigning a cost to the information lost by suppressing any given cell creates the cell suppression problem. This consists of finding the minimum cost solution which meets the confidentiality constraints. Solving this problem simultaneously for all of the sensitive cells in a table is NP-hard and not possible for medium to large sized tables. In this paper, we describe the development of a heuristic tool for this problem which hybridizes linear programming (to solve a relaxed version for a single sensitive cell) with a genetic algorithm (to seek an order for considering the sensitive cells which minimizes the final cost). Considering a range of real-world and representative artificial datasets, we show that the method is able to provide relatively low cost solutions for far larger tables than is possible for the optimal approach to tackle. We show that our genetic approach is able to significantly improve on the initial solutions provided by existing heuristics for cell ordering, and outperforms local search. This approach is then extended and applied to large statistical tables with over 200000 cells. © 2012 IEEE