Pareto optimality in many-to-many matching problems

Consider a many-to-many matching market that involves two finite disjoint sets, a set A of applicants and a set C of courses. Each applicant has preferences on the different sets of courses she can attend, while each course has a quota of applicants that it can admit. In this paper, we examine Pareto optimal matchings (briefly POM) in the context of such markets, that can also incorporate additional constraints, e.g., each course bearing some cost and each applicant having a limited budget available. We provide necessary and sufficient conditions for a many-to-many matching to be Pareto optimal and show that checking whether a given matching is Pareto optimal can be accomplished in 0(1 A 12 I C 12) time. Moreover, we provide a generalized version of serial dictatorship, which can be used to obtain any many-to-many POM. We also study some structural questions related to POM. We show that, unlike in the one-to-one case, finding a maximum cardinality POM is NP-hard for many-to-many markets. (C) 2014 Elsevier B.V. All rights reserved

Rotation-based formulation for stable matching

We introduce new CP models for the many-to-many stable matching problem. We use the notion of rotation to give a novel encoding that is linear in the input size of the problem. We give extra filtering rules to maintain arc consistency in quadratic time. Our experimental study on hard instances of sex-equal and balanced stable matching shows the efficiency of one of our propositions as compared with the state-of-the-art constraint programming approach

A complexity perspective on entailment of parameterized linear constraints

Extending linear constraints by admitting parameters allows for more abstract problem modeling and reasoning. A lot of focus has been given to conducting research that demonstrates the usefulness of parameterized linear constraints and implementing tools that utilize their modeling strength. However, there is no approach that considers basic theoretical tools related to such constraints that allow for reasoning over them. Hence, in this paper we introduce satisfiability with respect to polyhedral sets and entailment for the class of parameterized linear constraints. In order to study the computational complexities of these problems, we relate them to classes of quantified linear implications. The problem of satisfiability with respect to polyhedral sets is then shown to be co- NP hard. The entailment problem is also shown to be co- NP hard in its general form. Nevertheless, we characterize some subclasses for which this problem is in ℙ. Furthermore, we examine a weakening and a strengthening extension of the entailment problem. The weak entailment problem is proved to be NP complete. On the other hand, the strong entailment problem is shown to be co- NP hard

Computational complexity of inclusion queries over polyhedral sets

In this paper we discuss the computational complexities of procedures for inclusion queries over polyhedral sets. The polyhedral sets that we consider occur in a wide range of applications, ranging from logistics to program verification. The goal of our study is to establish boundaries between hard and easy problems in this context

On Quantified Linear Implications

A Quantified Linear Implication (QLI) is an inclusion query over two polyhedral sets, with a quantifier string that specifies which variables are existentially quantified and which are universally quantified. Equivalently, it can be viewed as a quantified implication of two systems of linear inequalities. In this paper, we provide a 2-person game semantics for the QLI problem, which allows us to explore the computational complexities of several of its classes. More specifically, we prove that the decision problem for QLIs with an arbitrary number of quantifier alternations is PSPACE-hard. Furthermore, we explore the computational complexities of several classes of 0, 1, and 2-quantifier alternation QLIs. We observed that some classes are decidable in polynomial time, some are NP-complete, some are coNP-hard and some are P2-hard.We also establish the hardness of QLIs with 2 or more quantifier alternations with respect to the first quantifier in the quantifier string and the number of quantifier alternations. All the proofs that we provide for polynomially solvable problems are constructive, i.e., polynomial-time decision algorithms are devised that utilize well-known procedures. QLIs can be utilized as powerful modelling tools for real-life applications. Such applications include reactive systems, real-time schedulers, and static program analyzers

On the complexity of quantified linear systems

In this paper, we explore the computational complexity of the conjunctive fragment of the first-order theory of linear arithmetic. Quantified propositional formulas of linear inequalities with $(k-1)$ quantifier alternations are log-space complete in $\Sigma_k^P$ or $\Pi_k^P$ depending on the initial quantifier. We show that when we restrict ourselves to quantified conjunctions of linear inequalities, i.e., quantified linear systems, the complexity classes collapse to polynomial time. In other words, the presence of universal quantifiers does not alter the complexity of the linear programming problem, which is known to be in P. Our result reinforces the importance of sentence formats from the perspective of computational complexity