The subdivision-constrained minimum spanning tree problem

• Authors: Jianping Li;Weidong Li;Tongquan Zhang;Zhongxu Zhang
• Venue: Theoretical Computer Science
• Year: 2009

Motivated by the constrained minimum spanning tree (CST) problem inHassin and Levin [R. Hassin, A. Levin, An efficient polynomial timeapproximation scheme for the constrained minimum spanning treeproblem using matroid intersection, SIAM Journal on Computing 33(2) (2004) 261-268], we study a new combinatorial optimizationproblem in this paper, called the general subdivision-constrainedspanning tree problem (GSCST): given a graphG=(V,E;w,c) with two nonnegative integersw(e) and c(e) for each edge e∈E,two positive integers B and d, the GSCST problem isto first find a spanning tree T=(V,ET) ofG with weightΣe∈ETw(e)≤B andthen to insert some new vertices on some suitable edges in T suchthat each edge in the subdivision tree T' of T hasits weight not beyond d. The objective is to minimize thecostΣe∈ETinsert(e)c(e)of such new vertices inserted on the suitable edges among allspanning trees of G subject to the two precedingconstraints, where a subdivision tree T' of T isconstructed by inserting some new vertices on the suitable edges inT, the value insert(e)=[w(e)/d] - 1 is theleast number of vertices inserted and c(e) is the cost ofeach vertex inserted on the edge e. We obtain the followingmain results: (1) the GSCST problem and its variant are stillNP-hard, by a reduction from the 0-1 knapsack problem,respectively; (2) the GSCST problem as well as its variant ispolynomially equivalent to the CST problem, which implies theexistence of a polynomial time approximation scheme to solve theGSCST problem and its variant; (3) we finally design three stronglypolynomial time algorithms to solve the special versions of theGSCST problem and its variant, respectively.