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A network for the transportation of supplies can be described as a rootedtree with a weight of a degree of congestion for each edge. We take the sum ofroot-leaf distance (SRD) on a rooted tree as the whole degree of congestion ofthe tree. Hence, we consider the SRD interdiction problem on trees withcardinality constraint by upgrading edges (denoted by (SDIPTC) in brief). Itaims to maximize the SRD by upgrading the weights of $N$ critical edges suchthat the total upgrade cost under some measurement is upper-bounded by a givenvalue. The relevant minimum cost problem (MCSDIPTC) aims to minimize the totalupgrade cost on the premise that the SRD is lower-bounded by a given value. Wedevelop two different norms including weighted $l_\infty$ norm and weightedbottleneck Hamming distance to measure the upgrade cost. We propose two binarysearch algorithms within O($n\log n$) time for the problems (SDIPTC) under thetwo norms, respectively. For problems (MCSDIPTC),we propose two binary searchalgorithms within O($N n^2$) and O($n \log n$) under weighted $l_\infty$ normand weighted bottleneck Hamming distance, respectively. These problems aresolved through their subproblems (SDIPT) and (MCSDIPT), in which we ignore thecardinality constraint on the number of upgraded edges. Finally, we designnumerical experiments to show the effectiveness of these algorithms.

The sum of root-leaf distance interdiction problem with cardinality constraint by upgrading edges on trees