Classified Stable Matching

We introduce the {\sc classified stable matching} problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper/lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of ``order types.'' We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching. In the second part, we investigate the problem using a polyhedral approach. Suppose that all classifications are laminar families and there is no lower bound. We propose a set of linear inequalities to describe stable matching polytope and prove that it is integral. This integrality allows us to find various optimal stable matchings using Ellipsoid algorithm. A further ramification of our result is the description of the stable matching polytope for the many-to-many (unclassified) stable matching problem. This answers an open question posed by Sethuraman, Teo and Qian

A Fully Polynomial-Time Approximation Scheme for Speed Scaling with Sleep State

We study classical deadline-based preemptive scheduling of tasks in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each task is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of tasks in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. [16], its exact computational complexity has been repeatedly posed as an open question (see e.g. [2,8,15]). The currently best known upper and lower bounds are a 4/3-approximation algorithm and NP-hardness due to [2] and [2,17], respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules

Just Get Me to the Church: Assessing Policies to Promote Marriage among Fragile Families

This article examines alternative approaches to encourage family formation among fragile families, including higher cash benefits, more liberal acceptance of welfare applications, more effective child support enforcement, and efforts to increase education and employment of low-income parents. We examine these approaches by refining and expanding previous work on a generalized logit model of the mothers’ actual family formation outcomes, in a hierarchy that includes father absence, father involvement, cohabitation, and marriage. Refinements involve measurements of family formation that make our results more comparable to other studies and new controls for previous fertility with the father of the focal child and with another partner (multiple partner fertility). We estimate these models using interim data from the Fragile Families and Child Well-Being 12 month follow-up Survey. The results indicate that, unlike their effects on mature families, cash benefits increase the odds of family formation (short of marriage) among fragile families and effective child support enforcement increases the odds of marriage. However, the father’s employment status outweighs the effects of these traditional income security policies on family formation, because it affects outcomes all along the hierarchy, including marriage, and its effects are larger. Unlike previous research, our data on previous fertility enables us to separate the effects of previous children in common from multiple partner fertility on family formation. Both significantly affect family formation (though in opposite directions), but even after including these variables, blacks, who are more likely to bring children from previous unions into a new union, have substantially lower odds of cohabitation and marriage than non-Hispanic whites.

The M Word: The Rise and Fall of Interracial Coalitions On Fathers And Welfare Reform

Buoyed by the success of the 1996 Personal Responsibility and Work Opportunities Reconciliation Act (PRWORA), whose time limits and work requirements played a large role in the reduction of the welfare rolls, conservative advocates of welfare reform are now moving to ensure that our welfare system reflects traditional family values as well. Responding to this sentiment, the Bush Administration is encouraging states to use TANF to support marriage promotion efforts and the Administration's 2002 budget includes $100 million in support of demonstration projects to promote marriage (source). By contrast, the$60 million President Bush had committed to support efforts to promote responsible fatherhood, not restricted to marriage, has been pared back to $20 million, along with cutbacks in other domestic initiatives that are needed to pay for the "war against terrorism."

How hard is it to cheat in the Gale-Shapley Stable Matching Algorithm

We study strategy issues surrounding the stable marriage problem. Under the Gale-Shapley algorithm (with men proposing), a classical theorem says that it is impossible for every liar to get a better partner. We try to challenge this theorem. First, observing a loophole in the statement of the theorem, we devise a coalition strategy in which a non-empty subset of the liars gets a better partner and no man is worse off than before. This strategy is restricted in that not everyone has the incentive to cheat. We attack the classical theorem further by means of randomization. However, this theorem shows surprising robustness: it is impossible that every liar has the chance to improve while no one gets hurt. Hence, this impossibility result indicates that it is always hard to induce some people to falsify their lists. Finally, to overcome the problem of lacking motivation, we exhibit another randomized lying strategy in which every liar can expect to get a better partner, though with a chance of getting a worse one

Semi-Streaming Algorithms for Submodular Function Maximization Under b-Matching Constraint

We consider the problem of maximizing a submodular function under the b-matching constraint, in the semi-streaming model. Our main results can be summarized as follows. - When the function is linear, i.e. for the maximum weight b-matching problem, we obtain a 2+? approximation. This improves the previous best bound of 3+? [Roie Levin and David Wajc, 2021]. - When the function is a non-negative monotone submodular function, we obtain a 3 + 2 ?2 ? 5.828 approximation. This matches the currently best ratio [Roie Levin and David Wajc, 2021]. - When the function is a non-negative non-monotone submodular function, we obtain a 4 + 2 ?3 ? 7.464 approximation. This ratio is also achieved in [Roie Levin and David Wajc, 2021], but only under the simple matching constraint, while we can deal with the more general b-matching constraint. We also consider a generalized problem, where a k-uniform hypergraph is given with an extra matroid constraint imposed on the edges, with the same goal of finding a b-matching that maximizes a submodular function. We extend our technique to this case to obtain an algorithm with an approximation of 8/3k+O(1). Our algorithms build on the ideas of the recent works of Levin and Wajc [Roie Levin and David Wajc, 2021] and of Garg, Jordan, and Svensson [Paritosh Garg et al., 2021]. Our main technical innovation is to introduce a data structure and associate it with each vertex and the matroid, to record the extra information of the stored edges. After the streaming phase, these data structures guide the greedy algorithm to make better choices

Cheating to Get Better Roommates in a Random Stable Matching

This paper addresses strategies for the stable roommates problem, assuming that a stable matching is chosen at random. We investigate how a cheating man should permute his preference list so that he has a higher-ranking roommate probabilistically. In the first part of the paper, we identify a necessary condition for creating a new stable roommate for the cheating man. This condition precludes any possibility of his getting a new roommate ranking higher than all his stable roommates when everyone is truthful. Generalizing to the case that multiple men collude, we derive another impossibility result: given any stable matching in which a subset of men get their best possible roommates, they cannot cheat to create a new stable matching in which they all get strictly better roommates than in the given matching. Our impossibility result, considered in the context of the stable marriage problem, easily re-establishes the celebrated Dubins-Freedman Theorem. The more generalized Demange-Gale-Sotomayor Theorem states that a coalition of men and women cannot cheat to create a stable matching in which everyone of them gets a strictly better partner than in the Gale-Shapley algorithm (with men proposing). We give a sharper result: a coalition of men and women cannot cheat together so that, in a newly-created stable matching, every man in the coalition gets a strictly better partner than in the Gale-Shapley algorithm while none of the women in the coalition is worse off. In the second part of the paper, we present two cheating strategies that guarantee that the cheating man\u27s new probability distribution over stable roommates majorizes the original one. These two strategies do not require the knowledge of the probability distribution of the cheating man. This is important because the problem of counting stable matchings is \#P-complete. Our strategies only require knowing the set of stable roommates that the cheating man has and can be formulated in polynomial time. Our second cheating strategy has an interesting corollary in the context of stable marriage with the Gale-Shapley algorithm. Any woman-optimal strategy will ensure that every woman, cheating or otherwise, ends up with a partner at least as good as when everyone is truthful