Comparing and Aggregating Partial Orders with Kendall Tau Distances

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order of all candidates. Recently, this approach was generalized to bucket orders, which allow ties. In this work we further generalize and consider total, bucket, interval and partial orders. The disagreement between two orders is measured by the nearest neighbor Spearman footrule distance, which has not been studied so far. For two bucket orders and for a total and an interval order the nearest neighbor Spearman footrule distance is shown to be computable in linear time, whereas for a total and a partial order the computation is NP-hard, 4-approximable and fixed-parameter tractable. Moreover, in contrast to the well-known efficient solution of the rank aggregation problem for total orders, we prove the NP-completeness for bucket orders and establish a 4-approximation.

Discovering a bucket order B from a collection of possibly noisy full rankings is a fundamental problem that relates to various applications involving rankings. Informally, a bucket order is a total order that allows ties between items in a bucket. A bucket order B can be viewed as a representative that summarizes a given set of full rankings {T1, T2, ..., Tm}, or conversely B can be an approximation of some ground truth G where the rankings {T1, T2, ..., Tm} are simply the extensions of G.Current work of finding bucket orders such as the dynamic programming algorithm is mainly developed from the representative perspective, which maximizes items' intra-bucket similarity when forming a bucket. The underlying idea of maximizing intra-bucket similarity is realized via minimizing the sum of the deviations of median ranks within a bucket. In contrast, from the approximation perspective, since each observed full ranking Ti is simply a linear extension of the given ground truth bucket order G, items in a big bucket b in G are forced to have different median ranks, and as a result b will have a big sum of deviations. Thus, minimizing the sum of deviations may result in an undesirable scenario that big buckets are mostly decomposed into small ones.In this paper, we propose a novel heuristic called Abnormal Rank Gap to capture the inter-bucket dissimilarity for better bucket forming. In addition, we propose to use the closeness on multiple quantile ranks to determine if two items should be put into the same bucket. We develop a novel bucket order discovering method termed the Bucket Gap algorithm. Our extensive experiments demonstrate that the Bucket Gap algorithm significantly outperforms the major related work, i.e., the Bucket Pivot algorithm. In particular, the error distance of the generated bucket order can be reduced by about 30% on a real paleontological dataset and the noise tolerance can be increased from 30% to 50% in the synthetic dataset.

Comparing genomes of different species is a crucial problem in comparative genomics. Different measures have been proposed to compare two genomes: number of common intervals, number of adjacencies, number of reversals, etc. These measures are classically used between two totally ordered genomes. However, genetic mapping techniques often give rise to different maps with some unordered genes. Starting from a partial order between genes of a genome, one method to find a total order consists in optimizing a given measure between a linear extension of this partial order and a given total order of a close and well-known genome. However, for most common measures, the problem turns out to be NP-hard. In this paper, we propose a (0,1)-linear programming approach to compute a linear extension of one genome that maximizes the number of common intervals (resp. the number of adjacencies) between this linear extension and a given total order. Next, we propose an algorithm to find linear extensions of two partial orders that maximize the number of adjacencies.KeywordsPartial OrderTotal OrderLinear ExtensionBoolean VariableReduction RuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The flow set with partial order is a mixed-integer set described by a budget on total flow and a partial order on the arcs that may carry positive flow. This set is a common substructure of resource allocation and scheduling problems with precedence constraints and robust network flow problems under demand/capacity uncertainty. We give a polyhedral analysis of the convex hull of the flow set with partial order. Unlike for the flow set without partial order, cover-type inequalities based on partial order structure are a function of a lifting sequence. We study the lifting sequences and describe structural results on the lifting coefficients for general and simpler special cases. We show that all lifting coefficients can be computed in polynomial time by solving maximum weight closure problems in general. For the special case of induced-minimal covers, we give a sequence-dependent characterization of the lifting coefficients. We prove, however, if the partial order is defined by an arborescence, then lifting is sequence-independent and all lifting coefficients can be computed in linear time. Moreover, if the partial order is defined by a path (total order), then the coefficients can be expressed explicitly. We also give a complete polyhedral description of the flow set with partial order for the polynomially-solvable total order case. We show that finding an optimal lifting order for a given induced-minimal cover and a given fractional solution is a submodular optimization problem, which is solved greedily. Finally, we present preliminary computational results with a cutting-plane algorithm based on the lifting and separation results.

On a Class of Lattice Ordered Inverse Semigroups

The total order assumption (TOA) is the assumption that all execution sequences of observable actions or events are totally ordered by precedence. As long as in some cases TOA is to the point, total order and partial order semantics are both legitimate but lead to different theories. I argue that TOA is a simplifying assumption, and present an example of a total order theory without interleaving, and a partial order theory with interleaving.KeywordsPartial OrderTotal OrderOrder TheoryParallel CompositionExecution SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let \({(X, m, \leq)}\) be a partially ordered metric space, that is, a metric space (X, m) equipped with a partial order \({\leq}\) on X. We say that a T 0-quasi-metric d on X is m-splitting provided that \({d\vee d^{-1}=m}\). Furthermore d is said to be \({(X, m, \leq)}\)-producing provided that d is m-splitting and the specialization partial preorder of d is equal to \({\leq}\). It is known and easy to see that if \({(X, m, \leq)}\) is a partially ordered metric space that is produced by a T 0-quasi-metric and \({\leq}\) is a total order, then there exists exactly one producing T 0-quasi-metric on X. We first will give an example that shows that a partially ordered metric space can be uniquely produced by a T 0-quasi-metric although \({\leq}\) is not total.

Ordering and ranking items of different types (observations, web pages, etc.) are important tasks in various applications, such as query processing and scientific data mining. We consider different problems of inferring total or partial orders from data, with special emphasis on applications to the seriation problem in paleontology. Seriation can be viewed as the task of ordering rows of a 0-1 matrix so that certain conditions hold. We review different approaches to this task, including spectral ordering methods, techniques for finding partial orders, and probabilistic models using MCMC methods.Joint work with Antti Ukkonen, Aris Gionis, Mikael Fortelius, Kai Puolamäki, and Jukka Jernvall.KeywordsArtificial IntelligenceLecture NoteData MiningProbabilistic ModelPartial OrderThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A unified framework for explanation-based generalization of partially ordered and partially instantiated plans

In this paper, we introduce the weighted poset game, which is defined as an extension of the poset (partially ordered set) game by adding a weight on every element of the poset. Each player has their own non-negative number of lives, and loses as many lives as the sum of the element weights they took. The player whose lives become negative first is the loser. We consider winning ways of this problem. First, for the problem with {0, 1}-weights, we find that (1) if the number of lives are different, then the player who has the large number of lives is the winner, (2) if the number of lives are the same and all maximal elements have positive weights, then the second player is a winner, and (3) otherwise, the game is reduced to an (unweighted) poset game. Next, for general weights, we consider the case where the partial order is a total order, and derive a polynomial-time algorithm for calculating who is the winner and the winning way for the winner.

To prove really difficult theorems, resolution principle programs need to make better inferences and to make them faster. An approach is presented for taking advantage of the structure of some special theories. These are theories with simplifiers, commutativity, and associativity, which are valuable concepts to build in, since they so frequently occur in important theories, for example, number theory (plus and times) and set theory (union and intersection). The object of the approach is to build in such concepts in a (refutation) complete, valid, efficient (in time) manner by means of a “natural” notation and/or new inference rules. Some of the many simplifiers that can be built in are axioms for (left and right) identities, inverses, and multiplication by zero. As for results, commutativity is built in by a straightforward modification to the unification (matching) algorithm. The results for simplifiers and associativity are more complicated. These theoretical results can be combined with one another and/or extended to either C -linear refutation completeness or theories with partial ordering, total ordering, or sets. How these results can serve as the basis of practical computer programs is discussed.

In this chapter, we describe some preference structures used more or less frequently in literature concerned with preferences: tournaments, total orders, weak orders, interval orders, semiorders, partial orders, quasi orders.

This paper proposes a new methodology for performing comparative validation between two specifications of a system at different levels of abstraction. The methodology consists of two steps : extracting a high-level simulation from a low-level simulation by recursively recognizing and naming patterns of events, and compare the extracted simulation to a high-level simulation. This paper introduces a new semantics for high-level simulation (partial order with duration events), describes a new algorithm to compare an extracted simulation (total order with duration events) and a high-level simulation, and lists performance results of the comparison algorithm.

Let (S, *) be a ring with involution *. The involution is positive definite if, for all finite subsets {r,-} of S, £ w * = 0 implies all the r{ are zero. Then the set of self-adjoint elements of S, denoted S, possesses a natural partial ordering, with positive cone consisting of elements of the form Z l w * ; with this ordering, S is a directed partially ordered abelian group. Let Sb denote the set of bounded elements, that is, the set of elements s such that ss* is less than an integral multiple of 1 in this ordering. Then Sb is a *-subalgebra of S whenever S is an algebra over the rationals. We will be studying the objects Sb9 S , and (Sb) , from the point of view of their ordered structures. For instance, suppose S is a field, and * is the identity. Then S is formally real, and Sb must be Prufer domain, all of whose residue fields are themselves formally real (and in fact, are embeddable in the reals). Viewing Sb as a partially ordered abelian group with order unit 1 (indeed, Sb is the convex subgroup of S generated by 1), Sb has the Riesz decomposition property, and its normalized extremal states are precisely the ring homomorphisms into the reals. There is a natural mapping from the collection of total orderings of S to the set of extremal states of Sb9 and this in turn maps to Spec Sb (the prime ideal space of Sb); when S is even a real algebra much more can be said. If either S is a field and * is not the identity, or S is a quaternionic division algebra with the natural involution, essentially the same properties hold, with the appropriate modifications. A useful tool here is an involutory version of the Artin Schreier Theorem, about the existence of sufficiently many total orderings finer than the natural ordering. Studies are made of several specific bounded subrings. For instance, if S is the rational function field in one variable over the reals, then Sb is a Dedekind domain with class group of order 2, with spectrum the unit circle (in the point-open topology), and all of its maximal ideals are not principal. Expanding the scope of S somewhat, we next allow S to be a division

Duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on different solution concepts