April 7, 2008
Partitioning Graphs of Supply and Demand
-- Generalization of Knapsack Problem --
Takehiro Ito
Tohoku University
Suppose that each vertex of a graph $G$ is either a supply vertex or a demand vertex and is assigned a positive real number, called the supply or the demand. Each demand vertex can receive ``power'' from at most one supply vertex through edges in $G$. One thus wishes to partition $G$ into connected components so that each component $C$ either has no supply vertex or has exactly one supply vertex whose supply is at least the sum of demands in $C$, and one wishes to maximize the fulfillment, that is, the sum of demands in all components with supply vertices. This maximization problem is a generalization of the multiple knapsack problem. In this talk, we give some results on the hardness and approximability of the problem. The problem is NP-hard even for trees having exactly one supply vertex and is strongly NP-hard for general graphs. Furthermore, the problem is APX-hard and hence there is no polynomial-time approximation scheme (PTAS) for general graphs unless P=NP. However, there is a fully polynomial-time approximation scheme (FPTAS) for trees. The FPTAS can be extended for series-parallel graphs and partial $k$-trees, that is, graphs with bounded treewidth, if there is exactly one supply vertex in the graph.

Joint work with Professors Takao Nishizeki and Xiao Zhou of Tohoku University, Japan and with Professor Erik D. Demaine of MIT, USA.