Pseudo Boolean Programming for Partially Ordered Genomes

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 or a bucket order of all candidates, where bucket orders allow ties. In this work we consider the generalization of total and bucket orders to partial orders and compare them by the nearest neighbor and the Hausdorff Kendall tau distances. For total and bucket orders these distances can be computed in $\mathcal{O}(n \log n)$ time. We show that the computation of the nearest neighbor Kendall tau distance is NP-hard, 2-approximable and fixed-parameter tractable for a total and a partial order. The computation of the Hausdorff Kendall tau distance for a total and a partial order is shown to be coNP-hard. The rank aggregation problem is known to be NP-complete for total and bucket orders, even for four voters and solvable in $\mathcal{O}(n\log n)$ for two voters. It is NP-complete for two partial orders and the nearest neighbor Kendall tau distance. For the Hausdorff Kendall tau distance it is in $\mathbf{\Sigma_2^p}$ , but not in NP or coNP unless $\ensuremath{\mathbf{NP}} = \ensuremath{\mathbf{coNP}} $ , even for four voters.

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.

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.

This study intends to demonstrate the value of using the partial order set theory comparing different but intertwined sets of indicators or indexes. We illustrate this approach by analysing the relative positions (partial order) of a set of countries with consideration for environmental and development indicators. Using data from 2013, the analysis mainly covers the countries with economies having a strong impact on climate change—China, the USA, the European Union (member States), India, Russian Federation, Japan, Brazil, Canada, and Mexico. The concepts of total and partial orders, linear extension or comparability are introduced and used in the analysis. The inclusion of three integrative environmental indicators and two development indicators (human development index and GDP per capita) shows that in 2013 the BRICS were the worst positioned countries. In contrast, several countries in Northern Europe (Denmark, followed by Germany, Ireland, the Netherlands, Sweden and the UK) were associated with the best overall indicators. Canada is not comparable to any other country, the values of its indicators being sometimes higher and sometimes lower than those associated to any other country considered in this study. The USA, comparable to a single country, shows a similar behaviour for the same reasons.

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.

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.

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

Partial order theory (POT) is an attractive and operationally simple method that allows ordering of compounds, based on selected structural and/or electronic descriptors (modeled order), or based on their end points, e.g., solubility (experimental order). If the modeled order resembles the experimental order, compounds that are not experimentally investigated can be assigned a position in the model that eventually might lead to a prediction of an end-point value. However, in the application of POT in quantitative structure-activity relationship modeling, only the compounds directly comparable to the noninvestigated compounds are applied. To explore the possibilities of improving the methodology, the theory is extended by application of the so-called linear extensions of the model order. The study show that partial ordering combined with linear extensions appears as a promising tool providing probability distribution curves in the range of possible end-point values for compounds not being experimentally investigated.

E. Szpilrajn [2] showed that any partial order r on a set A can be extended to a linear order on A. He also proved that any partial order is the intersection of its linear extensions. The following generalization of Szpilrajn's classical result can be found in [1]. Given a poset (A, r) and an acyclic r-monotone function f : A-~A, then r can be extended to a linear order R with xRy=~f(x)RJ(y) for all x, yCA. Such linear extensions are called monotonicity preserving or simply f-linear extensions. The triple (A,f , R) can be considered as a linearly ordered extension of the acyclic partially ordered 1-unary algebra (A,f, r). The main aim of this note is to describe the intersection of all f-linear extensions of a given acyclic partially ordered algebra (A,f, r). In what follows we present an improved version of a theorem first a n n o u n c e d in [1].

Risk stratification is an invaluable tool for modern healthcare systems. By separating patients into subgroups with distinct disease severity and prognosis, it allows better clinical decision making due to targeted care thus ultimately fosters healthier patient populations. For example, the American Joint Committee on Cancer staging is an indispensible first step for any cancer care. Based on the stages, appropriate treatments are given to optimize clinical outcomes such as recurrence free survival or overall survival. Consequently, it enhances communication, engagement, and research focus. This paper presents a new algorithm entitled 'Ordering Poset Elements by Recursive Amalgamation' (OPERA) for patient stratification using many risk factors. Health risk factors frequently exhibit total or partial ordering and when considered jointly, they form a partially ordered set or a poset. Compared with the usual regression models, OPERA can explore high order interactions similar to the well-known tree method. On the other hand, by explicitly exploring the poset structure, OPERA allows flexible staging patterns and faster pruning for better performance. OPERA is evaluated in extensive simulation studies and cancer staging data to demonstrate its ability in performing risk stratification using ordered risk factors.

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.