Given a task T, a pool of individuals X with different skills, and a social network G that captures the compatibility among these individuals, we study the problem of finding X, a subset of X, to perform the task. We call this the TEAM FORMATION problem. We require that members of X' not only meet the skill requirements of the task, but can also work effectively together as a team. We measure effectiveness using the communication cost incurred by the subgraph in G that only involves X'. We study two variants of the problem for two different communication-cost functions, and show that both variants are NP-hard. We explore their connections with existing combinatorial problems and give novel algorithms for their solution. To the best of our knowledge, this is the first work to consider the TEAM FORMATION problem in the presence of a social network of individuals. Experiments on the DBLP dataset show that our framework works well in practice and gives useful and intuitive results.

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Finding a team of experts in social networks

We obtain the first non-trivial approximation algorithms for the Steiner Tree problem and the Generalized Steiner Tree problem in general directed graphs. Essentially no approximation algorithms were known for these problems. For the Directed Steiner Tree problem, we design a family of algorithms which achieve an approximation ratio of O(k^\epsilon) in time O(kn^{1/\epsilon}) for any fixed (\epsilon < 0), where k is the number of terminals to be connected. For the Directed Generalized Steiner Tree Problem, we give an algorithm which achieves an approximation ratio of O(k^{2/3}\log^{1/3} k), where k is the number of pairs to be connected. Related problems including the Group Steiner tree problem, the Node Weighted Steiner tree problem and several others can be reduced in an approximation preserving fashion to the problems we solve, giving the first non-trivial approximations to those as well.

Approximation algorithms for directed Steiner problems

Chapter 15 – Geometric Shortest Paths and Network Optimization

A Nearly best-possible approximation algorithm for node-weighted Steiner trees

Approximation Algorithms for Directed Steiner Problems

We consider a generalized version of Steiner's problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with groups of required vertices and Steiner vertices, find a shortest connected subgraph containing at least one required vertex of each group We propose two efficient approximation algorithms computing different approximate solutions in time O(|E| + |V|log|V|) and in time O(g · (|E| + |V|log|V|)), respectively, where |E| is the number of edges in the given graph, |V| is the number of vertices, and g is the number of groups The latter algorithm propagates a set of wavefronts with different distances simultaneously through the graph; it is interesting in its own right

Beyond Steiner's Problem: A VLSI Oriented Generalization

Class Steiner trees and VLSI-design

We are given a set $P$ of points in the plane, together with a partition of $P$ into {\em classes\/} of points; i.e., each point of $P$ belongs to exactly one class. For a given network optimization problem, such as finding a minimum spanning tree or finding a minimum diameter spanning tree, we study the problem of choosing a subset $P''$ of $P$ that contains at least one point of each class and solving the network optimization problem for $P''$, such that the solution is optimal among all possible choices for $P''$. We show that solving the minimum spanning tree problem for classes of points is defined by any of the Minkowski metrics $L_p$, $1\leq p\leq\infty$. In contrast, a class solution for the minimum diameter spanning tree problem can be computed in time $O({|P|}^3)$. By proving the NP-completeness of the minimum spanning tree class problem we also get some results for distance graphs. Here, computing a class solutioin for the minimum spanning tree problem is NP-complete, even under several restrictions, e.g., if the graph is part of a unit grid and is a tree, where the vertex degree and the numver of vertices per class are both bounded by three. This is true even if the graph is a minimum spanning tree for its set of vertices.

Gabriele Reich

Papers

Beyond Steiner's Problem: A VLSI Oriented Generalization

Class Steiner trees and VLSI-design

On Shortest Networks for Classes of Points in the Plane