We consider the problem faced by a company that wants to use viral marketing to introduce a new product into a market where a competing product is already being introduced. We assume that consumers will use only one of the two products and will influence their friends in their decision of which product to use. We propose two models for the spread of influence of competing technologies through a social network and consider the influence maximization problem from the follower's perspective. In particular we assume the follower has a fixed budget available that can be used to target a subset of consumers and show that, although it is NP-hard to select the most influential subset to target, it is possible to give an efficient algorithm that is within 63% of optimal.Our computational experiments show that by using knowledge of the social network and the set of consumers targeted by the competitor, the follower may in fact capture a majority of the market by targeting a relatively small set of the right consumers.

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We consider ranking and clustering problems related to the aggregation of inconsistent information, in particular, rank aggregation, (weighted) feedback arc set in tournaments, consensus and correlation clustering, and hierarchical clustering. Ailon et al. [Ailon, N., M. Charikar, A. Newman. 2005. Aggregating inconsistent information: Ranking and clustering. Proc. 37th Annual ACM Sympos. Theory Comput. (STOC '05), 684-693], Ailon and Charikar [Ailon, N., M. Charikar. 2005. Fitting tree metrics: Hierarchical clustering and phylogeny. Proc. 46th Annual IEEE Sympos. Foundations Comput. Sci. (FOCS '05), 73-82], and Ailon [Ailon, N. 2007. Aggregation of partial rankings, p-ratings and top-m lists. Proc. 18th Annual ACM-SIAM Sympos. Discrete Algorithms (SODA '07), [415][416][417][418][419][420][421][422][423][424] proposed randomized constant factor approximation algorithms for these problems, which recursively generate a solution by choosing a random vertex as "pivot" and dividing the remaining vertices into two groups based on the pivot vertex.In this paper, we answer an open question in these works by giving deterministic approximation algorithms for these problems. The analysis of our algorithms is simpler than the analysis of the randomized algorithms. In addition, we consider the problem of finding minimum-cost rankings and clusterings that must obey certain constraints (e.g., an input partial order in the case of ranking problems), which were introduced by Hegde and Jain [Hegde, R., K. Jain. 2006. Personal communication]. We show that the first type of algorithms we propose can also handle these constrained problems. In addition, we show that in the case of a rank aggregation or consensus clustering problem, if the input rankings or clusterings obey the constraints, then we can always ensure that the output of any algorithm obeys the constraints without increasing the objective value of the solution.

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Abstract. In this work we study the problem of Bipartite Correlation Clustering (BCC), a natural bipartite counterpart of the well studied Correlation Clustering (CC) problem. Given a bipartite graph, the objective of BCC is to generate a set of vertex-disjoint bi-cliques (clusters) which minimizes the symmetric difference to it. The best known approximation algorithm for BCC due to Amit (2004) guarantees an 11-approximation ratio. 4 In this paper we present two algorithms. The first is an improved 4-approximation algorithm. However, like the previous approximation algorithm, it requires solving a large convex problem which becomes prohibitive even for modestly sized tasks. The second algorithm, and our main contribution, is a simple randomized combinatorial algorithm. It also achieves an expected 4-approximation factor, it is trivial to implement and highly scalable. The analysis extends a method developed by Ailon, Charikar and Newman in 2008, where a randomized pivoting algorithm was analyzed for obtaining a 3-approximation algorithm for CC. For analyzing our algorithm for BCC, considerably more sophisticated arguments are required in order to take advantage of the bipartite structure. Whether it is possible to achieve (or beat) the 4-approximation factor using a scalable and deterministic algorithm remains an open problem.