Using data from an ongoing study of 93 single Black mothers of preschoolers who had been welfare recipients, but were employed in low-wage jobs at baseline, this study tests a model of how maternal education, economic conditions (earnings and financial strain), and the availability of instrumental support influence maternal psychological functioning, parenting, and child development. Results indicate that maternal educational attainment was positively associated with earnings, which, together with instrumental support, were negatively associated with financial strain. Financial strain, in turn, was implicated in elevated levels of depressive symptoms, which were directly and negatively implicated in parenting quality. The quality of parenting was associated with children's behavior problems and preschool ability. Specifically, mothers with higher scores on the HOME scale, our measure of involved, supportive parenting, had children with fewer behavior problems and better preschool ability.

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Abstract. The input is a bipartite graph G = (A ∪ B,E) where each vertex u ∈ A ∪ B ranks its neighbors in a strict order of preference. This is the same as an instance of the stable marriage problem with incomplete lists. A matching M * is said to be popular if there is no matching M such that more vertices are better off in M than in M * . Any stable matching of G is popular, however such a matching is a minimum cardinality popular matching. We consider the problem of computing a maximum cardinality popular matching in G. It has very recently been shown that when preference lists have ties, the problem of determining if a given instance admits a popular matching or not is NP-complete. When preference lists are strict, popular matchings always exist, however the complexity of computing a maximum cardinality popular matching was unknown. In this paper we give a simple characterization of popular matchings when preference lists are strict and a sufficient condition for a maximum cardinality popular matching. We then show an O(mn 0 ) algorithm for computing a maximum cardinality popular matching, where m = |E| and n 0 = min(|A|, |B|).

We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1 + o(1))-approximationÕ(n)-time algorithms [LP15, Nan14], which are matched withΩ(n)-time lower bounds [Nan14, LPS13, FHW12] 1 . No ω(n) lower bound or o(m) upper bound were known for exact computation.In this paper, we present anÕ(n 5/4 )-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric (a.k.a. the directed case where communication is bidirectional). Our techniques also yield anÕ(n 3/4 k 1/2 + n)-time algorithm for the k-source shortest paths problem where we want every node to know distances from k sources; this improves Elkin's recent bound [Elk17b] when k =ω(n 1/4 ).We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show anÕ(n √ r)-time algorithm for this problem. Another new algorithm is called short range extension, where we show that inÕ(n √ h) time the knowledge about distances can be "extended" for additional h hops. For this, we use weight rounding to introduce small additive errors which can be later fixed.Remark: Independently from our result, Elkin recently observed in [Elk17b] that the same techniques from an earlier version of the same paper (https://arxiv.org/abs/ 1703.01939v1) led to an O(n 5/3 log 2/3 n)-time algorithm. 1Θ,Õ andΩ hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios.2 See also [PRS17, Elk17a] for recent results. 3 For the maximum flow algorithm, there is an extra n o(1) term in the time complexity.

Abstract. This paper addresses strategies for the stable marriage problem. For the Gale-Shapley algorithm with men proposing, a classical theorem states that it is impossible for every cheating man to get a better partner than the one he gets if everyone is truthful. We study how to circumvent this theorem and incite men to cheat. First we devise coalitions in which a non-empty subset of the liars get better partners and no man is worse off than before. This strategy is limited in that not everyone in the coalition has the incentive to falsify his list. In an attempt to rectify this situation we introduce the element of randomness, but the theorem shows surprising robustness: it is impossible that every liar has a chance to improve the rank of his partner while no one gets hurt. To overcome the problem that some men lack the motivation to lie, we exhibit another randomized lying strategy in which every liar can expect to get a better partner on average, though with a chance of getting a worse one. Finally, we consider a variant scenario: instead of using the Gale-Shapley algorithm, suppose the stable matching is chosen at random. We present a modified form of the coalition strategy ensuring that every man in the coalition has a new probability distribution over partners which majorizes the original one.