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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.
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