SWAT 2006: Algorithm Theory – SWAT 2006 pp 400-410

# The Weighted Maximum-Mean Subtree and Other Bicriterion Subtree Problems

• Josiah Carlson
• David Eppstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)

## Abstract

We consider problems where we are given a rooted tree as input, and must find a subtree with the same root, optimizing some objective function of the nodes in the subtree. When the objective is the sum of linear function weights of a parameter, we show how to list all optima for all parameter values in O(nlogn) time. This can be used to solve many bicriterion optimizations problems in which each node has two values x i and y i associated with it, and the objective function is a bivariate function f(∑x i ,∑y i ) of the sums of these two values. When f is the ratio of the two sums, we have the Weighted Maximum-Mean Subtree Problem, or equivalently the Fractional Prize-Collecting Steiner Tree Problem on Trees; we provide a linear time algorithm when all values are positive, improving a previous O(nlogn) solution, and prove NP-completeness when certain negative values are allowed.

## Keywords

Rooted Tree Binary Search Piecewise Linear Function Tree Node Linear Time Algorithm

## References

1. 1.
Agarwal, P.K., Eppstein, D., Guibas, L.J., Henzinger, M.R.: Parametric and kinetic minimum spanning trees. In: Proc. 39th Symp. Foundations of Computer Science, pp. 596–605. IEEE, Los Alamitos (1998)Google Scholar
2. 2.
Ahuja, R., Magnanti, T., Orlin, J.: Network Flows. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
3. 3.
Brown, M., Tarjan, R.: A fast merging algorithm. J. ACM 26(2), 211–226 (1979)
4. 4.
Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001)
5. 5.
Katoh, N.: Parametric combinatorial optimization problems and applications. J. Inst. Electronics, Information and Communication Engineers 74, 949–956 (1991)Google Scholar
6. 6.
Katoh, N.: Bicriteria network optimization problems. IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences E75-A, 321–329 (1992)Google Scholar
7. 7.
Klau, G., Ljubi, I., Mutzel, P., Pferschy, U., Weiskircher, R.: The Fractional Prize-Collecting Steiner Tree Problem on Trees. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 691–702. Springer, Heidelberg (2003)
8. 8.
Shah, R., Farach-Colton, M.: Undiscretized dynamic programming: faster algorithms for facility location and related problems on trees. In: Proc. 13th ACM-SIAM Symp. Discrete Algorithms (SODA 2002), pp. 108–115 (2002)Google Scholar