Reasoning about optimal stable matchings under partial information

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.

Possibilistic preference networks

IntroductionThe “stable marriage” algorithm underlying the National Residency Match Program (NRMP) has been shown to create optimal outcomes when students submit true preference lists. Previous research has shown students may allow external information to affect their rank lists. The objective of this study was to determine whether medical students consistently make rank lists that reflect their true preferences.MethodsA voluntary online survey was sent to third-year students at a single midwestern medical school. Students were given hypothetical scenarios that either should or should not affect their true residency preferences and rated the importance of six factors to their final rank list. The survey was edited by a group of education scholars and revised based on feedback from a pilot with current postgraduate year 1 residents.ResultsOf 175 students surveyed, 140 (80%) responded; 63% (88/140) reported that their “perceived competitiveness” would influence their rank list at least a “moderate amount. Of 135 students, 31 (23%) moved a program lower on their list if they learned they were ranked “low” by that program, while 6% (8/135) of respondents moved a program higher if they learned they were ranked “at the top of the list.” Participants responded similarly (κ = 0.71) when presented with scenarios asking what they would do vs what a classmate should do.ConclusionStudents’ hypothetical rank lists did not consistently match their true residency preferences. These results may stem from a misunderstanding of the Match algorithm. Medical schools should consider augmenting explicit education related to the NRMP Match algorithm to ensure optimal outcomes for students.

Results are presented concerning the main problem of algebraic geometry over partially ordered sets from a computational point of view, namely, the solvability problem for systems of equations over a partial order. This problem is solvable in polynomial time if the directed graph corresponding to the partial order is a adjusted interval digraph, and is NP-complete if the base of the directed graph corresponding to the partial order is a cycle of length at least 4. We also present a result characterizing the possibility of transition from infinite systems of equations over partial orders to finite systems. Algebraic systems with this property are called equationally Noetherian. A partially ordered set is equationally Noetherian if and only if any of its upper and lower cones with base are finitely defined.

The paper proposes an algorithm to compute the full set of many-to-many stable matchings when agents have substitutable preferences. The algorithm starts by calculating the two optimal stable matchings using the deferred-acceptance algorithm. Then, it computes each remaining stable matching as the firm-optimal stable matching corresponding to a new preference profile, which is obtained after modifying the preferences of a previously identified sequence of firms.

Entropy and Sorting

The Blocking Lemma for a many-to-one matching model

The theory of social choice is concerned with the problem of aggregating the preferences of several persons into a single preference order. The problem has the same structure as that of aggregating the preference orders of a single person with regard to several aspects of alternatives into a single preference order on the set of alternatives. For this reason, the theory of social choice comes logically not under the topic of collective action (as one might suppose) but rather under the topic of multi-objective decisions. This is so because the several participants in a situation defined as a problem of social choice do not really select courses of action among several available ones when they present their preference orders on a set of alternatives. Therefore they are not ‘actors’, as actors are defined in the theory of decision. To be sure, there are situations where voters do choose strategies, that is, choose among different preference orders to present, which may or may not represent their true preferences but which they believe will be more likely to lead to a more desirable aggregated order. These situations can be considered as n-person decision problems, and we will examine them in the context of non-cooperative n-person games (see Chapter 14. At this time, however, we will suppose that each voter acts ‘sincerely’, that is, presents his ‘actual’ preference order. The decision is now up to some agency, which must combine all these submitted preference orders into a ‘social’ preference order.KeywordsSocial ChoicePreference OrderPareto OptimalitySocial Welfare FunctionSimple MajorityThese 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.

Stable marriage and indifference

The k-secretary problem deals with online selection of at most k numerically scored applicants, where selection decisions are immediate upon their arrival and irrevocable, with the goal of maximizing total score. There is, however, wide prevalence of bias in evaluations of applicants from different demographic groups (e.g., gender, age, race), and the assumption of an algorithm observing their true score is unreasonable in practice. In this work, we propose the poset secretary problem, where selection decisions must be made by observing a partial order over the applicants. This partial order is constructed to account for uncertainty and biases in applicant scores. We assume that each applicant has a fixed score, which is not visible to the algorithm and is consistent with the partial order. Using a random partitioning technique from the matroid secretary literature, we provide order-optimal competitive algorithms for the poset secretary problem and provide matching lower bounds. Further, we develop the theory of thresholding in posets to provide a tight, adaptive-thresholding algorithm under regimes where k grows quickly enough, thus matching the adaptiveness to k shown for the classical k-secretary problem. We then study a special case, in which applicants belong to g disjoint demographic groups and the bias is group specific. We provide competitive algorithms for adversarial and stochastic variants of this special case, including a framework, Gap, for parallelizing any vanilla k-secretary algorithm to the group setting. Finally, we perform a case study on real-world data to demonstrate the responsiveness of our algorithms to data and their impact on selection rates. This paper was accepted by Hamid Nazerzadeh, data science. Funding: This work was supported by the National Science Foundation [Grant 2112533]. S. Gupta’s research was supported in part by NSF CAREER Award 2239824. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2023.4926 .

We introduce three different approaches for decision making under uncertainty, if (I) there is only partial (both cardinal and ordinal) information on an agent’s preferences and (II) the uncertainty about the states of nature is described by a credal set. Particularly, (I) is modeled by a pair of relations, one specifying the partial rank order of the alternatives and the other modeling partial information on the strength of preference. Our first approach relies on criteria that construct complete rankings of the acts based on generalized expectation intervals. Subsequently, we introduce different concepts of global admissibility that construct partial orders by comparing all acts simultaneously. Finally, we define criteria induced by suitable binary relations on the set of acts and, therefore, can be understood as concepts of local admissibility. Whenever suitable, we provide linear programming based algorithms for checking optimality/admissibility of acts.

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.

Numerous AI problems in planning, robot motion, distributed systems, cooperating agents, and intelligence gathering have domains with sub-collections of events or actions over time which are measured on incomparable or unsynchronized time scales from one to another. In such situations, a temporal model providing only a partial order on time moments is appropriate. Unlike a branching-time model, no sense of a common past and divergent futures occurs; unlike a ’’parallel worlds‘‘ model, check points or intercommunication between the sub-collections of events may exist, providing a true, rich partially ordered set of temporal information. In applications for which temporal intervals and their relations are appropriate, constraint propagation is a recognized reasoning tool. We discuss several temporal interval models and their relationship to one another but particularly focus on the general partial order model. In each model the emphasis is on the atomic relations, so we amplify on the meaning of atomic and show that what is atomic in one model may not be so in another. Utilizing results established earlier about the lattice properties of the models, the paper describes the closure of the atomic relations under composition and conjunction, relating the structures of relations in linear time, partially ordered time, and relativistic time. Lattice and algebraic isomorphisms explain what appeared to be only coincidental similarities between different models. The results are shown to provide especially efficient representations for constraint propagation algorithms.

Given a weighted graph, a distance oracle takes as an input a pair of vertices and returns the distance between them. The labeling approach to distance oracle design is to precompute a label for every vertex so that the distances can be computed from the corresponding labels, without looking at the graph. In the hub labeling algorithm (HL), a vertex label consists of a set of other vertices (hubs) with distances to the hubs. We survey theoretical and experimental results on HL.Although computing optimal hub labels is hard, in polynomial time one can approximate them up to a factor of O(logn). This can be done for the total label size (i.e., memory required to store the labels), the maximum label size (which determines the worst-case query time), and in general for an L p norm of the vector induced by the vertex label sizes. One can also simultaneously approximate L p and L q norms.Hierarchical labels are a special class of HL for which the relationship ”v is a hub of w” is a partial order. For networks with a small highway dimension, one can compute provably small hierarchical labels in polynomial time.While some graphs admit small labels, there are graphs for which the labels are large. Furthermore, one can prove that for some graphs hierarchical labels are significantly larger than the general ones.A heuristic for computing hierarchical labels leads to the fastest known implementation of a distance oracle for road networks. One can use label compression to trade off time for space, making the algorithm practical for a wider range of applications. We give experimental results showing that the heuristic hierarchical labels are efficient on road networks as well as on some other graph classes, but not on all graphs.We also discuss efficient implementations of the provably good approximation algorithms. Although not as fast as the hierarchical heuristic, the algorithms can solve moderate-size problems. For some graph classes, the theoretically justified algorithms compute significantly smaller labels, although for many graphs the label size difference is very small.

Since the existing terrestrial fifth generation (5G) network has limited coverage, it is difficult to meet the growing demand for seamless network connection. Meanwhile, current network resource allocation methods mainly research on how to improve system performance only from the perspective of resource utilization, but rarely take users’ specific needs for network content into consideration. This brings severe challenges to efficient network service and flexible resource allocation. Therefore, we construct the content service-oriented resource allocation model for space–air–ground integrated sixth generation networks (SAGIN 6G), and formulate the three-sided matching issue among the space–air–ground integrated network equipment (SAGINE), content sources, and users. In this model, users request to establish connection with SAGIN which forwards users’ request to content service provider (CSP). CSP manages the creation of content data, and finally returns the requested content to users through SAGIN. For the content service-oriented resource allocation in SAGIN, finding the optimal stable three-sided matching with the largest cardinality is an NP-complete problem. Therefore, to efficiently solve the above issue, we design some reasonable restrictions and convert it to a restricted three-sided matching problem with size and cycle preferences. We further develop the content-oriented resource allocation algorithm (COR2A) and the user-oriented resource allocation algorithm (UOR2A) in a distributed manner. Extensive simulations verify our approach outperforms traditional benchmark resource allocation schemes in terms of system throughput, CSP revenue, and user experience.