The Density Maximization Problem in Graphs

Abstract

Abstract. We consider a framework for bi-objective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G = (V, E) with edge weights we ∈ Z and edge lengths e ∈ N for e ∈ E we define the density of a pattern subgraph H = (V , E ) ⊆ G as the ratio (H) = e∈E we/ e∈E e. We consider the problem of computing a maximum density pattern H with weight at least W and and length at most L in a host G. We call this problem the bi-constrained density maximization problem. In doing so, we compute a single Pareto-optimal solution with the best weight per cost ratio subject to additional constraints further narrowing down feasible solutions for the underlying bi-objective network construction problem. The problems expressible within this framework include maximization of return on investment for network construction problems in the presence of a limited budget and a target profit. We consider this problem for different classes of hosts and patterns. We show that it is NPhard even if the host has treewidth 2 and the pattern is a path. However, it can be solved in pseudo-polynomial linear time if the host has bounded treewidth and the pattern is a graph from a given minor-closed family of graphs. Finally, we present an FPTAS for a relaxation of the density maximization problem, in which we are allowed to violate the upper bound on the length at the cost of some penalty.

Exact and Approximation Algorithms for the Expanding Search Problem

“…Observe that this objective function is the inverse of the search density. Unfortunately, the approach used to obtain the PTAS does not generalize to the case where T has to contain the root r. Kao et al (2013) describe both exact and approximation algorithms for different variants of MDSP, but their methods also do not provide an algorithm for MDSP as defined above.…”