Быстрая согласованность по Кемени на основе поиска по стандартным матрицам с минимальным расстоянием до усредненного экспертного ранжирования

Проблематика. Розглядається задача ранжування скінченної множини об’єктів. Мета дослідження. Розробка алгоритму, який дав би змогу пришвидшити пошук узгодженості за Кемені поряд з обґрунтуванням метрики для порівняння ранжувань. Методика реалізації. Пропонується й обґрунтовується підхід щодо об’єднання експертних ранжувань. Також пропонується й обґрунтовується метрика для порівняння ранжувань. Результати дослідження. Розроблений алгоритм знаходить множину ранжувань Кемені значно швидше, ніж класичний прямий пошук. Також ця множина часто містить єдину узгодженість за Кемені, що не вдається за прямого пошуку. Крім цього, єдина узгодженість за Кемені визначається відразу, якщо усереднене експертне ранжування виявляється ациклічним. Так розв’язується задача вибору єдиної узгодженості за Кемені. Висновки. Для 10 і більше об’єктів, де більшість відомих підходів стають незастосовними, алгоритм є реалізовним завдяки пошуку по тільки тих стандартних матрицях, чия відстань до першого ранжування відрізняється від відстані між цим ранжуванням та усередненим експертним ранжуванням на мінімальну величину.Background. The problem of ranking a finite set of objects is considered. Objective. The goal is to develop an algorithm that would let speed up the search of the Kemeny consensus along with substantiation of a metric to compare rankings. Methods. An approach for aggregating experts’ rankings is suggested and substantiated. Also a metric to compare rankings is suggested and substantiated. Results. The developed algorithm finds a set of Kemeny rankings much faster than the classical straightforward search. Also this set often contains a single Kemeny consensus, what fails by the straightforward search. Besides, a single Kemeny consensus is determined at one stroke if the averaged expert ranking turns out acyclic. Thus the problem of selecting a single Kemeny consensus is solved. Conclusions. For 10 objects and more, where most known approaches become intractable, the algorithm still is tractable due to searching over only those standard matrices whose distance to the first ranking differs minimally from the distance between this ranking and the averaged expert ranking.Проблематика. Рассматривается задача ранжирования конечного множества объектов. Цель исследования. Разработка алгоритма, который позволил бы ускорить поиск согласованности по Кемени вместе с обоснованием метрики для сравнения ранжирований. Методика реализации. Предлагается и обосновывается подход относительно объединения экспертных ранжирований. Также предлагается и обосновывается метрика для сравнения ранжирований. Результаты исследования. Разработанный алгоритм находит множество ранжирований Кемени гораздо быстрее, чем классический прямой поиск. Также это множество часто содержит единственную согласованность по Кемени, что не удается при прямом поиске. Кроме этого, единственная согласованность по Кемени определяется сразу, если усредненное экспертное ранжирование оказывается ациклическим. Так решается задача выбора единственной согласованности по Кемени. Выводы. Для 10 и более объектов, где большинство известных подходов становятся неисполнимыми, алгоритм является осуществимым благодаря поиску по только тем стандартным матрицам, чье расстояние к первому ранжированию отличается от расстояния между этим ранжированием и усредненным экспертным ранжированием на минимальную величину

09171 Abstracts Collection -- Adaptive, Output Sensitive, Online and Parameterized Algorithms

From 19.01. to 24.04.2009, the Dagstuhl Seminar 09171 ``Adaptive, Output Sensitive, Online and Parameterized Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Parameterizing by the Number of Numbers

The usefulness of parameterized algorithmics has often depended on what Niedermeier has called, "the art of problem parameterization". In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as Subset Sum, Partition, 3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with Target Sums, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We rely on an FPT result for ILPF to show that all the above-mentioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W[1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable

Egalitarianism in the rank aggregation problem: a new dimension for democracy

Winner selection by majority, in an election between two candidates, is the only rule compatible with democratic principles. Instead, when the candidates are three or more and the voters rank candidates in order of preference, there are no univocal criteria for the selection of the winning (consensus) ranking and the outcome is known to depend sensibly on the adopted rule. Building upon XVIII century Condorcet theory, whose idea was to maximize total voter satisfaction, we propose here the addition of a new basic principle (dimension) to guide the selection: satisfaction should be distributed among voters as equally as possible. With this new criterion we identify an optimal set of rankings. They range from the Condorcet solution to the one which is the most egalitarian with respect to the voters. We show that highly egalitarian rankings have the important property to be more stable with respect to fluctuations and that classical consensus rankings (Copeland, Tideman, Schulze) often turn out to be non optimal. The new dimension we have introduced provides, when used together with that of Condorcet, a clear classification of all the possible rankings. By increasing awareness in selecting a consensus ranking our method may lead to social choices which are more egalitarian compared to those achieved by presently available voting systems.Comment: 18 pages, 14 page appendix, RateIt Web Tool: http://www.sapienzaapps.it/rateit.php, RankIt Android mobile application: https://play.google.com/store/apps/details?id=sapienza.informatica.rankit. Appears in Quality & Quantity, 10 Apr 2015, Online Firs

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context

Towards a Dichotomy for the Possible Winner Problem in Elections Based on Scoring Rules

To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, whether a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2,1,...,1,0), while it is solvable in polynomial time for plurality and veto.Comment: minor changes and updates; accepted for publication in JCSS, online version available

Order-Related Problems Parameterized by Width

In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints. While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory. Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory. In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin

Fixed-Parameter Algorithms for Kemeny Rankings

The computation of Kemeny rankings is central to many applications in the context of rank aggregation. Given a set of permutations (votes) over a set of candidates, one searches for a “consensus permutation” that is “closest” to the given set of permutations. Unfortunately, the problem is NP-hard. We provide a broad study of the parameterized complexity for computing optimal Kemeny rankings. Beside the three obvious parameters “number of votes”, “number of candidates”, and solution size (called Kemeny score), we consider further structural parameterizations. More specifically, we show that the Kemeny score (and a corresponding Kemeny ranking) of an election can be computed efficiently whenever the average pairwise distance between two input votes is not too large. In other words, Kemeny Score is fixedparameter tractable with respect to the parameter “average pairwise Kendall-Tau distance da”. We describe a fixed-parameter algorithm with running time 16 ⌈da ⌉ · poly. Moreover, we extend our studies to the parameters “maximum range ” and “average range ” of positions a candidate takes in the input votes. Whereas Kemen