Optimal Website Design with the Constrained Subtree Selection Problem

Abstract

We introduce the Constrained Subtree Selection (CSS) problem as a model for the optimal design of websites. Given a hierarchy of topics represented as a DAG G and a probability distribution over the topics, we select a subtree of the transitive closure of G which minimizes the expected path cost. We define path cost as the sum of the page costs along a path from the root to a leaf. Page cost, γ, is a function of the number of links on a page. We give a sufficient condition for γ which makes CSS NP-Complete. This result holds even for the uniform probability distribution. We give a polynomial time algorithm for instances of CSS where G does not constrain the choice of subtrees and γ favors pages with at most k links. We show that CSS remains NP-Hard for constant degree DAGs, but also provide an O(log(k)γ(d+1)) approximation for any G with maximum degree d, provided that γ favors pages with at most k links. We also give a complete characterization of the optimal trees for two special cases: (1) linear degree cost in unconstrained graphs and uniform probability distributions, and (2) logarithmic degree cost in arbitrary DAGs and uniform probability distributions.