Abstract
Motivated by the constrained minimum spanning tree (CST) problem in Hassin and Levin [R. Hassin, A. Levin, An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection, SIAM Journal on Computing 33 (2) (2004) 261–268], we study a new combinatorial optimization problem in this paper, called the general subdivision-constrained spanning tree problem (GSCST): given a graph G = ( V , E ; w , c ) with two nonnegative integers w ( e ) and c ( e ) for each edge e ∈ E , two positive integers B and d , the GSCST problem is to first find a spanning tree T = ( V , E T ) of G with weight ∑ e ∈ E T w ( e ) ≤ B and then to insert some new vertices on some suitable edges in T such that each edge in the subdivision tree T ′ of T has its weight not beyond d . The objective is to minimize the cost ∑ e ∈ E T i n s e r t ( e ) c ( e ) of such new vertices inserted on the suitable edges among all spanning trees of G subject to the two preceding constraints, where a subdivision tree T ′ of T is constructed by inserting some new vertices on the suitable edges in T , the value i n s e r t ( e ) = ⌈ w ( e ) d ⌉ − 1 is the least number of vertices inserted and c ( e ) is the cost of each vertex inserted on the edge e . We obtain the following main results: (1) the GSCST problem and its variant are still NP-hard, by a reduction from the 0–1 knapsack problem, respectively; (2) the GSCST problem as well as its variant is polynomially equivalent to the CST problem, which implies the existence of a polynomial time approximation scheme to solve the GSCST problem and its variant; (3) we finally design three strongly polynomial time algorithms to solve the special versions of the GSCST problem and its variant, respectively.
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