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We consider the setting of asynchronous opinion diffusion with majority threshold: given a social network with each agent assigned to one opinion, an agent will update its opinion if more than half of its neighbors agree on a different opinion. The stabilized final outcome highly depends on the sequence in which agents update their opinion. We are interested in optimistic sequences---sequences that maximize the spread of a chosen opinion. We complement known results for two opinions where optimistic sequences can be computed in time and length linear in the number of agents. We analyze upper and lower bounds on the length of optimistic sequences, showing quadratic bounds in the general and linear bounds in the acyclic case. Moreover, we show that in networks with more than two opinions determining a spread-maximizing sequence becomes intractable; surprisingly, already with three opinions the intractability results hold in highly restricted cases, e.g., when each agent has at most three neighbors, when looking for a short sequence, or when we aim for approximate solutions.

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Given a matrix A ∈ R n×n , we consider the problem of maximizing x T Ax subject to the constraint x ∈ {−1, 1} n . This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an Ω(1/ lg n) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an Ω(1) approximation when A corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is Õ(n 1.5 • min{N, n 1.5 }), where N is the number of nonzero entries in A and Õ ignores polylogarithmic factors.In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where A is sparse (i.e., has O(n) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that:-Unit MaxQP, where A ∈ {−1, 0, 1} n×n , admits an (1/3d)-approximation in O(n 1.5 ) time, when the corresponding graph has no isolated vertices and at most dn edges. -MaxQP admits an Ω(1/ lg amax)-approximation in O(n 1.5 lg amax) time, where amax is the maximum absolute value in A, when the corresponding graph is d-degenerate. -Unit MaxQP admits a (1 − ε)-approximation in O(n 2 ) time when the corresponding graph is H-minor free. -MaxQP admits a (1 − ε)-approximation in O(n) time when the corresponding graph and each of its minors have bounded local treewidth.

Betweenness centrality-measuring how many shortest paths pass through a vertex-is one of the most important network analysis concepts for assessing the relative importance of a vertex. The well-known algorithm of Brandes [J. Math. Sociol. '01] computes, on an $n$-vertex and $m$-edge graph, the betweenness centrality of all vertices in $O(nm)$ worst-case time. In later work, significant empirical speedups were achieved by preprocessing degree-one vertices and by graph partitioning based on cut vertices. We contribute an algorithmic treatment of degree-two vertices, which turns out to be much richer in mathematical structure than the case of degree-one vertices. Based on these three algorithmic ingredients, we provide a strengthened worst-case running time analysis for betweenness centrality algorithms. More specifically, we prove an adaptive running time bound $O(kn)$, where $k < m$ is the size of a minimum feedback edge set of the input graph.