We consider directed graph algorithms in a streaming setting, focusing on problems concerning orderings of the vertices. This includes such fundamental problems as topological sorting and acyclicity testing. We also study the related problems of finding a minimum feedback arc set (edges whose removal yields an acyclic graph), and finding a sink vertex. We are interested in both adversarially-ordered and randomlyordered streams. For arbitrary input graphs with edges ordered adversarially, we show that most of these problems have high space complexity, precluding sublinear-space solutions. Some lower bounds also apply when the stream is randomly ordered: e.g., in our most technical result we show that testing acyclicity in the p-pass random-order model requires roughly n 1+1/p space. For other problems, random ordering can make a dramatic difference: e.g., it is possible to find a sink in an acyclic tournament in the one-pass randomorder model using polylog(n) space whereas under adversarial ordering roughly n 1/p space is necessary and sufficient given Θ(p) passes. We also design sublinear algorithms for the feedback arc set problem in tournament graphs; for random graphs; and for randomly ordered streams. In some cases, we give lower bounds establishing that our algorithms are essentially space-optimal. Together, our results complement the much maturer body of work on algorithms for undirected graph streams.
In this paper we study oriented bipartite graphs. In particular, we introduce bitransitive graphs and bitournaments. Several characterizations of bitransitive bitournaments are obtained. Next we prove the Caccetta-Häggkvist Conjecture for oriented bipartite graphs for some cases for which it is unsolved in general. We introduce the concept of oriented odd-even graphs and (undirected) odd-even graphs and characterize (oriented) bipartite graphs in terms of them.In fact, we show that any (oriented) bipartite graph can be represented by some (oriented) odd-even graph. We obtain some conditions for connectedness of odd-even graphs. Finally we introduce Goldbach graphs which are special type of odd-even graphs. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other related conjectures are related to various parameters of Goldbach graphs. We study nature of degrees of vertices and independent sets of Goldbach graphs.
A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an n-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most ∆ and aim for colorings using a small number f (∆) of colors.A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, nonrobust, streaming setting, (∆ + 1)-colorings can be obtained while using only O(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(∆ 2 ) colors and that robust O(∆)-coloring requires a linear amount of space, namely Ω(n∆). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust.We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(∆ 2 )-coloring using O(n √ ∆) space and an O(∆ 3 )-coloring using O(n) space.
Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time.This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a dynamic setting, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of (1 + )r 2 and an update time of O(poly(r, log n)), where r denotes the maximum rank of the input across all the updates.We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of (1 + ) that is independent of r, and a similar update time of O(poly(r, log n)). It is the first (1 + )-approximation algorithm even for the special case of weighted simple graphs.To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.
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