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The conditions upon which young adults enter the labor market have been demonstrated to affect various later-life employment and family formation outcomes. After the 2008–2009 Global Financial Crisis, a thick strand of the literature has shown that precarious initial employment leads to postponed childbearing and higher ultimate childlessness. However, it is not only individual conditions that matter. Broader macroeconomic conditions upon entry also matter. The “scarring” literature has illustrated the consequences of entering the labor market during a recession on later life outcomes. Speaking to both strands and using detailed employment and birth histories of labor market entrants in Germany, this paper examines the effects of initial conditions, operationalized using fixed-term employment and recession year entry, on subsequent fertility behavior. To partly address bias from endogenous selection into initial conditions, we employ a two-step estimation strategy combining a non-parametric optimal full matching step and a parametric event history modeling step using the matched data. Results suggest that entering the labor market with a fixed-term contract is persistently and negatively associated with first births up to a decade after entry, and this pattern is pronounced only among women, whereas entering during a recession has persistent negative associations only among men.

Cluster randomization trials with relatively few clusters have been widely used in recent years for evaluation of health-care strategies. On average, randomized treatment assignment achieves balance in both known and unknown confounding factors between treatment groups, however, in practice investigators can only introduce a small amount of stratification and cannot balance on all the important variables simultaneously. The limitation arises especially when there are many confounding variables in small studies. Such is the case in the INSTINCT trial designed to investigate the effectiveness of an education program in enhancing the tPA use in stroke patients. In this article, we introduce a new randomization design, the balance match weighted (BMW) design, which applies the optimal matching with constraints technique to a prospective randomized design and aims to minimize the mean squared error (MSE) of the treatment effect estimator. A simulation study shows that, under various confounding scenarios, the BMW design can yield substantial reductions in the MSE for the treatment effect estimator compared to a completely randomized or matched-pair design. The BMW design is also compared with a model-based approach adjusting for the estimated propensity score and Robins-Mark-Newey E-estimation procedure in terms of efficiency and robustness of the treatment effect estimator. These investigations suggest that the BMW design is more robust and usually, although not always, more efficient than either of the approaches. The design is also seen to be robust against heterogeneous error. We illustrate these methods in proposing a design for the INSTINCT trial.

This Monte Carlo simulation study compares methods to estimate the effects of programs with multiple versions when assignment of individuals to program version is not random. These methods use generalized propensity scores, which are predicted probabilities of receiving a particular level of the treatment conditional on covariates, to remove selection bias. The results indicate that inverse probability of treatment weighting (IPTW) removes the most bias, followed by optimal full matching (OFM), and marginal mean weighting through stratification (MMWTS). The study also compared standard error estimation with Taylor series linearization, bootstrapping and the jackknife across propensity score methods. With IPTW, these standard error estimation methods performed adequately, but standard errors estimates were biased in most conditions with OFM and MMWTS.

Given a bipartite graph \\[G = (A B,E)\\] with strict preference lists and given an edge \\[e^* ın E\\], we ask if there exists a popular matching in G that contains \\[e^*\\]. We call this the popular edge problem. A matching M is popular if there is no matching \\[M'\\] such that the vertices that prefer \\[M'\\] to M outnumber those that prefer M to \\[M'\\]. It is known that every stable matching is popular; however G may have no stable matching with the edge \\[e^*\\]. In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge \\[e^*\\], then there is either a stable matching that contains \\[e^*\\] or a dominant matching that contains \\[e^*\\]. This allows us to design a linear time algorithm for identifying the set of popular edges. When preference lists are complete, we show an \\[O(n^3)\\] algorithm to find a popular matching containing a given set of edges or report that none exists, where \\[n = |A| + |B|\\].

An edge-weighted, vertex-capacitated graph G is called stable if the value of a maximum-weight capacity-matching equals the value of a maximum-weight fractional capacity-matching. Stable graphs play a key role in characterizing the existence of stable solutions for popular combinatorial games that involve the structure of matchings in graphs, such as network bargaining games and cooperative matching games. The vertex-stabilizer problem asks to compute a minimum number of players to block (i.e., vertices of G to remove) in order to ensure stability for such games. The problem has been shown to be solvable in polynomial-time, for unit-capacity graphs. This stays true also if we impose the restriction that the set of players to block must not intersect with a given specified maximum matching of G . In this work, we investigate these algorithmic problems in the more general setting of arbitrary capacities. We show that the vertex-stabilizer problem with the additional restriction of avoiding a given maximum matching remains polynomial-time solvable. Differently, without this restriction, the vertex-stabilizer problem becomes NP-hard and even hard to approximate, in contrast to the unit-capacity case.

We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This problem, which we call maximum submodular-function matching (MSM), is a natural generalization of maximum weight matching (MWM), which is in turn a generalization of maximum cardinality matching. We give two incomparable algorithms for this problem with space usage falling in the semi-streaming range—they store only O ( n ) edges, using O ( n log n ) working memory—that achieve approximation ratios of 7.75 in a single pass and ( 3 + ε ) in O ( ε - 3 ) passes respectively. The operations of these algorithms mimic those of Zelke’s and McGregor’s respective algorithms for MWM; the novelty lies in the analysis for the MSM setting. In fact we identify a general framework for MWM algorithms that allows this kind of adaptation to the broader setting of MSM. Our framework is not specific to matchings. Rather, we identify a general pattern for algorithms that maximize linear weight functions over “independent sets” and prove that such algorithms can be adapted to maximize a submodular function. The notion of independence here is very general; in particular, appealing to known weight-maximization algorithms, we obtain results for submodular maximization over hypermatchings in hypergraphs as well as independent sets in the intersection of multiple matroids.

We introduce a new two-sided stable matching problem that describes the summer internship matching practice of an Australian university. The model is a case between two models of Kamada and Kojima on matchings with distributional constraints. We study three solution concepts, the strong and weak stability concepts proposed by Kamada and Kojima, and a new one in between the two, called cutoff stability. Kamada and Kojima showed that a strongly stable matching may not exist in their most restricted model with disjoint regional quotas. Our first result is that checking its existence is NP-hard. We then show that a cutoff stable matching exists not just for the summer internship problem but also for the general matching model with arbitrary heredity constraints. We present an algorithm to compute a cutoff stable matching and show that it runs in polynomial time in our special case of summer internship model. However, we also show that finding a maximum size cutoff stable matching is NP-hard, but we provide a Mixed Integer Linear Program formulation for this optimisation problem.

We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless P = NP , there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph. Conditioned on P ≠ NP , this disproves a conjecture by Ito et al. (SIAM J Discrete Math 36(2):1102–1123, 2022). Assuming the Exponential Time Hypothesis we prove the stronger result that there exists no polynomial-time algorithm computing a path of length at most 1 4 - o ( 1 ) log N / log log N between two vertices at distance two of the perfect matching polytope of an N -vertex bipartite graph. These results remain true if the bipartite graph is restricted to be of maximum degree three. The above has the following interesting implication for the performance of pivot rules for the simplex algorithm on simply-structured combinatorial polytopes: If P ≠ NP , then for every simplex pivot rule executable in polynomial time and every constant k ∈ N there exists a linear program on a perfect matching polytope and a starting vertex of the polytope such that the optimal solution can be reached in two monotone non-degenerate steps from the starting vertex, yet the pivot rule will require at least k non-degenerate steps to reach the optimal solution. This result remains true in the more general setting of pivot rules for so-called circuit-augmentation algorithms .

Given an undirected graph, are there k matchings whose union covers all of its nodes, that is, a matching-k-cover ? When k = 1 , the problem is equivalent to the existence of a perfect matching for which Tutte’s celebrated matching theorem (J. Lon. Math. Soc., 1947) provides a ‘good’ characterization. We prove here, when k is greater than one, a ‘good’ characterization à la Kőnig : for k ≥ 2 , there exist k matchings covering every node if and only if for every stable set S , we have | S | ≤ k · | N ( S ) | . Moreover, somewhat surprisingly, we use only techniques from bipartite matching in the proof, through a simple, polynomial algorithm. A different approach to matching-k-covers has been previously suggested by Wang et al. (Math. Prog., 2014), relying on general matching and using matroid union for matching-matroids, or the Edmonds-Gallai structure theorem. Our approach provides a simpler polynomial algorithm together with an elegant certificate of non-existence when appropriate. Further results, generalizations and interconnections between several problems are then deduced as consequences of the new minimax theorem, with surprisingly simple proofs (again using only the level of difficulty of bipartite matchings). One of the equivalent formulations leads to a solution of weighted minimization for non-negative edge-weights, while the edge-cardinality maximization of matching-2-covers turns out to be already NP-hard. We have arrived at this problem as the line graph special case of a model arising for manufacturing integrated circuits with the technology called ‘Directed Self Assembly’.

We study the i.i.d. online bipartite matching problem, a dynamic version of the classical model where one side of the bipartition is fixed and known in advance, while nodes from the other side appear one at a time as i.i.d. realizations of a uniform distribution, and must immediately be matched or discarded. We consider various relaxations of the polyhedral set of achievable matching probabilities, introduce valid inequalities, and discuss when they are facet-defining. We also show how several of these relaxations correspond to ranking policies and their time-dependent generalizations. We finally present a computational study of these relaxations and policies to determine their empirical performance.