Abstract
AbstractLet
$T=(V,E)$
be a tree in which each edge is assigned a cost; let
$\mathcal{P}$
be a set of source–sink pairs of vertices in V in which each source–sink pair produces a profit. Given a lower bound K for the profit, the K-prize-collecting multicut problem in trees with submodular penalties is to determine a partial multicut
$M\subseteq E$
such that the total profit of the disconnected pairs after removing M from T is at least K, and the total cost of edges in M plus the penalty of the set of still-connected pairs is minimized, where the penalty is determined by a nondecreasing submodular function. Based on the primal-dual scheme, we present a combinatorial polynomial-time algorithm by carefully increasing the penalty. In the theoretical analysis, we prove that the approximation factor of the proposed algorithm is
$(\frac{8}{3}+\frac{4}{3}\kappa+\varepsilon)$
, where
$\kappa$
is the total curvature of the submodular function and
$\varepsilon$
is any fixed positive number. Experiments reveal that the objective value of the solutions generated by the proposed algorithm is less than 130% compared with that of the optimal value in most cases.
Publisher
Cambridge University Press (CUP)
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