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In the popular edge problem, the input is a bipartite graph G = (A ∪ B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A∪ B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M' such that the number of vertices that prefer M' to M is more than the number of vertices that prefer M to M'. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018]. We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings. We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes.

In this paper, we consider reachability oracles and reachability preservers for directed graphs/networks prone to edge/node failures. Let G = (V, E) be a directed graph on n-nodes, and P ⊆ V× V be a set of vertex pairs in G. We present the first non-trivial constructions of single and dual fault-tolerant pairwise reachability oracle with constant query time. Furthermore, we provide extremal bounds for sparse fault-tolerant reachability preservers, resilient to two or more failures. Prior to this work, such oracles and reachability preservers were widely studied for the special scenario of single-source and all-pairs settings. However, for the scenario of arbitrary pairs, no prior (non-trivial) results were known for dual (or more) failures, except those implied from the single-source setting. One of the main questions is whether it is possible to beat the O(n |P|) size bound (derived from the single-source setting) for reachability oracle and preserver for dual failures (or O(2^k n|P|) bound for k failures). We answer this question affirmatively. Below we summarize our contributions. - For an n-vertex directed graph G = (V, E) and P ⊆ V× V, we present a construction of O(n √{|P|}) sized dual fault-tolerant pairwise reachability oracle with constant query time. We further provide a matching (up to the word size) lower bound of Ω(n √{|P|}) on the size (in bits) of the oracle for the dual fault setting, thereby proving that our oracle is (near-)optimal. - Next, we provide a construction of O(n + min{|P|√ n,~n√{|P|}}) sized oracle with O(1) query time, resilient to single node/edge failure. In particular, for |P| bounded by O(√n) this yields an oracle of just O(n) size. We complement the upper bound with a lower bound of Ω(n^{2/3}|P|^{1/2}) (in bits), refuting the possibility of a linear-sized oracle for P of size ω(n^{2/3}). - We also present a construction of O(n^{4/3} |P|^{1/3}) sized pairwise reachability preservers resilient to dual edge/vertex failures. Previously, such preservers were known to exist only under single failure and had O(n+min{|P|√n,~n√ {|P|}}) size [Chakraborty and Choudhary, ICALP'20]. We also show a lower bound of Ω(n √{|P|}) edges on the size of dual fault-tolerant reachability preservers, thereby providing a sharp gap between single and dual fault-tolerant reachability preservers for |P| = o(n). - Finally, we provide a generic pairwise reachability preserver construction that provides a o(2^k n |P|) sized subgraph resilient to k failures, for any k ≥ 1. Before this work, we only knew of an O(2^k n |P|) bound implied from the single-source setting [Baswana, Choudhary, and Roditty, STOC'16].