In SODA'10, Huang introduced the laminar classified stable matching problem (LCSM for short) that is motivated by academic hiring. This problem is an extension of the wellknown hospitals/residents problem in which a hospital has laminar classes of residents and it sets lower and upper bounds on the number of residents that it would hire in that class. Against the intuition that stable matching problems with lower quotas are difficult in general, Huang proved that this problem can be solved in polynomial time. In this paper, we propose a matroid-based approach to this problem and we obtain the following results. (i) We solve a generalization of the LCSM problem. (ii) We exhibit a polyhedral description for stable assignments of the LCSM problem, which gives a positive answer to Huang's question. (iii) We prove that the set of stable assignments of the LCSM problem has a lattice structure similarly to the ordinary stable matching model.• We solve a generalization of the LCSM problem.• By exhibiting a polyhedral description for stable assignments of the LCSM problem, we give a positive answer to Huang's question in [9].• We prove that similarly to the ordinary stable matchings, the set of stable assignments of the LCSM problem has a natural lattice structure.for any class C of C Ci . Hence, I i , J i ∈ I Ci follows from I, J ∈ I C . Moreover, by the condition in the lemma,for any class C of C Ci on which I i ∩ C is deficient. So, by the induction hypothesis,

Given a directed graph D = (V, A) with a set of d specified vertices S = {s1, . . . , s d } ⊆ V and a function f : S → N where N denotes the set of natural numbers, we present a necessary and sufficient condition such that there existare rooted at si and each Ti,j spans the vertices from which si is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D = (V, A) with a specified vertex s ∈ V , there are k arc-disjoint in-trees rooted at s each of which spans V . Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case.

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In this paper, we consider the quickest flow problem in a network which consists of a directed graph with capacities and transit times on its arcs. We present an O(n log n) time algorithm for the quickest flow problem in a network of grid structure with uniform arc capacity which has a single sink where n is the number of vertices in the network.