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Let G = (V , E) be a graph. A double Roman dominating function (DRDF) of G is a function f : V → { 0 , 1 , 2 , 3 } such that (i) each vertex v with f (v) = 0 is adjacent to either a vertex u with f (u) = 3 or two vertices u 1 and u 2 with f (u 1) = f (u 2) = 2 , and (ii) each vertex v with f (v) = 1 is adjacent to a vertex u with f (u) > 1. The double Roman domination number of G is the minimum weight of a DRDF along all DRDFs on G , where the weight of a DRDF f on G is f (V) = ∑ v ∈ V f (v). In this paper, we first propose an algorithm to compute the double Roman domination number of an interval graph G = (V , E) in O (| V | + | E |) time, answering a problem posed in Banerjee et al. (2020) [2]. Next, we show that the decision problem associated with the double Roman domination is NP-complete for split graphs. Finally, we show that the computational complexities of the Roman domination problem and the double Roman domination problem are different. [ABSTRACT FROM AUTHOR]

The Minimum Vertex Cover P roblem is the optimization problem of finding a vertex cover V c of minimal cardinality in a given graph. It is a classic NP-hard problem, and various algorithms have been suggested for it. In this paper, we start with a basic algorithm for solving the problem. Using a probabilistic idea, we use it to develop an improved algorithm. The algorithm is greedy; at each step it adds to the cover a vertex such that the expected cover size, if we continue randomly after this step, is minimal. We study the new algorithm theoretically and empirically, and run simulations to compare its performance to that of some algorithms of a similar nature. [ABSTRACT FROM AUTHOR]

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree T = (W ⊎ B , E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. It was conjectured by Chaudhuri et al. [6] that the total length of all shortest augmenting paths found is O (n log ⁡ n). In this paper we prove a tight O (n log ⁡ n) upper bound for the total length of shortest augmenting paths for trees improving over O (n log 2 ⁡ n) bound [3]. [ABSTRACT FROM AUTHOR]

• In this paper, we provide a formal definition of a causal broadcast abstraction in the presence of Byzantine processes. • We present a simple causal broadcast algorithm that implements this specification in the presence of Byzantine processes. • We prove our algorithm to be correct under the specification we have provided. • One difficulty consists in proving that an adversary cannot destroy the causality between broadcasts of correct processes. • We illustrate how our algorithm can help implement a money transfer object, with potential application to cryptocurrencies. Causal broadcast is a communication abstraction built on top of point-to-point send/receive networks that ensures that any two messages whose broadcasts are causally related (as captured by Lamport's "happened before" relation) are delivered in their sending order. Several causal broadcast algorithms have been designed for failure-free and crash-prone asynchronous message-passing systems. This article first gives a formal definition of a causal broadcast abstraction in the presence of Byzantine processes, in the form of two equivalent characterizations, and then presents a simple causal broadcast algorithm that implements it. The main difficulty in the design and the proof of this algorithm comes from the very nature of Byzantine faults: Byzantine processes may have arbitrary behavior, and the algorithm must ensure that correct processes (i) maintain a coherent view of causality and (ii) are never prevented from communicating between themselves. To this end, the algorithm is built modularly, namely it works on top of any Byzantine-tolerant reliable broadcast algorithm. Due to this modularity, the proposed algorithm is easy to understand and inherits the computability assumptions (notably the maximal number of processes that may be Byzantine) and the message/time complexities of the underlying reliable broadcast on top of which it is built. [ABSTRACT FROM AUTHOR]

Popular matchings have been intensively studied recently as a relaxed concept of stable matchings. By applying the concept of popular matchings to branchings in directed graphs, Kavitha et al. introduced popular branchings. Let G = (V G , E G) be a directed graph, where each vertex has preferences over its incoming arcs, and B and B ′ be branchings in G. A vertex v ∈ V G prefers B to B ′ if v prefers its incoming arc of B to that of B ′ , where having an arbitrary incoming arc is preferred to having no incoming arc. We say that B is more popular than B ′ if the number of vertices preferring B to B ′ is greater than the number of vertices preferring B to B ′. A branching B is called a popular branching if there is no branching more popular than B. Kavitha et al. proposed an algorithm for finding a popular branching when the preferences of each vertex are given by a strict partial order. The correctness of this algorithm is proved by utilizing classical theorems on the duality of weighted arborescences. In this paper, we generalize popular branchings to weighted popular branchings in vertex-weighted directed graphs, in the same manner as weighted popular matchings by Mestre. We give an algorithm for finding a weighted popular branching for the case where the preferences of each vertex are given by a total preorder and the weights satisfy certain conditions. Our algorithm is based on Kavitha et al.'s algorithm, while it includes elaborated procedures resulting from the vertex-weights. Its correctness also builds upon the duality of weighted arborescences. [ABSTRACT FROM AUTHOR]