Polynomial algorithms for partitioning a tree into single‐center subtrees to minimize flat service costs

Abstract

This paper deals with the following graph partitioning problem: given a graph with n nodes, p of which are prescribed to be centers (the remaining nodes are called units), the goal is to find a partition of the set of nodes into connected components containing only one center each, so as to minimize the total assignment cost of units to centers. This problem is known to be NP-hard in general graphs, and it is shown here to remain such even if the assignment cost is monotone and the graph is bipartite. Therefore, we specialize on tree graphs to derive polynomial time algorithms. For this class of graphs we provide several reformulations of the problem as integer linear programs. Moreover, we develop a dynamic programming algorithm, whose recursion is based on solving sequences of minimum weight closure problems, that solves the problem on trees in O(n 2 p).

Political districting: from classical models to recent approaches

“…If such constraints are included into a lagrangean objective function, the resulting PD problem with updated values for the costs c is corresponds to an MCP. Unfortunately, MCP is NP-hard on general graphs (Apollonio et al 2008;Cordone 2001), thus it cannot be exploited for solving PD in real cases.…”

“…To conclude this section, we report an integer linear programming formulation of the PD problem provided in Apollonio et al (2008) in the special case when the contiguity graph G is a tree. Even if this paper focuses on a more general problem called Minimum Cost Centered Partition Problem (MCP), the motivation of the study originates from the application to PD.…”

“…When G is a tree, MCP can be formulated as a linear program based on order constraints which guarantee that the partition of the tree is connected. For this problem the authors provide a polynomial time solution algorithm (Apollonio et al 2008). Consider the contiguity graph G of a given territory; suppose that a set S ⊂ N of k centers are already selected, and that for every pair i ∈ N and s ∈ S, c is corresponds to the distance between unit i and the district center s. Then, the PD problem can be formulated as an MCP with side population balance constraints.…”

Partitioning a graph into connected components with fixed centers and optimizing cost‐based objective functions or equipartition criteria

“…Some of the most studied graph partitioning problems are those with a cardinality constraint on the number of components. These problems are NP-hard on arbitrary graphs, but, in many cases, they can be solved in polynomial time on trees (see, e.g., [3,6,8,13,17,25]). Polynomial time algorithms are also known for the max-split clustering problem on ladder graphs [26], and, more generally, on outerplanar graphs [28], as well as for equipartition problems on ladder graphs [7].…”

“…In these cases, one has necessarily to identify a set of p centers for the p districts in order to compute the objective function. Application to PD was already considered in [3] where the problem was formulated as a Minimum Cost Centered Partition Problem, i.e., a p-centered partition problem where the objective function is the sum of the unit-center assignment costs. This formulation derives from imposing bounds on the district population to satisfy population equality and adopting inertia to measure compactness.…”