[Paper Review] Edge Coloring with Minimum Reload/Changeover Costs

Didem Gözüpek, Mordechai Shalom|arXiv (Cornell University)|Jul 22, 2016

Advanced Graph Theory Research11 references3 citations

TL;DR

This paper introduces and studies two novel problems in network optimization: minimizing reload and changeover costs through proper edge coloring of a graph. It presents dynamic programming algorithms for sparse block graphs with bounded-degree cut vertices, proving polynomial-time solvability under these structural constraints, while establishing NP-hardness and inapproximability results in general cases.

ABSTRACT

In an edge-colored graph, a traversal cost occurs at a vertex along a path when consecutive edges with different colors are traversed. The value of the traversal cost depends only on the colors of the traversed edges. This concept leads to two global cost measures, namely the \emph{reload cost} and the \emph{changeover cost}, that have been studied in the literature and have various applications in telecommunications, transportation networks, and energy distribution networks. Previous work focused on problems with an edge-colored graph being part of the input. In this paper, we formulate and focus on two pairs of problems that aim to find an edge coloring of a graph so as to minimize the reload and changeover costs. The first pair of problems aims to find a proper edge coloring so that the reload/changeover cost of a set of paths is minimized. The second pair of problems aim to find a proper edge coloring and a spanning tree so that the reload/changeover cost is minimized. We present several hardness results as well as polynomial-time solvable special cases.

Motivation & Objective

• To address the gap in the literature by formulating edge coloring problems that minimize reload and changeover costs, rather than assuming a fixed coloring.
• To study the complexity of finding a proper edge coloring that minimizes global traversal costs over paths or spanning trees.
• To identify structural conditions under which these problems become tractable, particularly focusing on bounded-degree cut vertices and sparse block graphs.
• To provide exact algorithms and complexity bounds for two new families of network design problems involving traversal cost minimization.

Proposed method

• Formulates two pairs of problems: one minimizing reload/changeover cost over a given set of paths, and another minimizing cost from a root to all vertices via a spanning tree.
• Applies dynamic programming on the block tree of the graph, recursively computing optimal costs at blocks and cut vertices.
• For each block, computes minimum-cost colorings using min-weight perfect matching in complete bipartite graphs derived from edge colorings and subtree costs.
• For cut vertices, exhaustively guesses edge colorings and optimal subtrees, combining results via a sum of traversal costs and subtree costs.
• Uses a recursive formulation where the cost at a vertex depends on the color of incoming edges and the costs of its child subtrees.
• Employs a postorder traversal of the block tree to ensure that subproblems are solved before their superproblems, enabling bottom-up computation.

Experimental results

Research questions

• RQ1 Is it possible to find a proper edge coloring that minimizes the total reload cost over a given set of paths, and what is the complexity of this problem?
• RQ2 Can the changeover cost be minimized via edge coloring when a spanning tree is also to be selected, and under what conditions is this solvable in polynomial time?
• RQ3 What structural properties of the graph (e.g., bounded-degree cut vertices, sparse blocks) enable polynomial-time algorithms for these problems?
• RQ4 How hard is it to approximate the minimum reload or changeover cost when edge coloring is part of the decision?
• RQ5 What is the impact of bounded vs. unbounded traversal costs on the approximability and complexity of the edge coloring problems?

Key findings

• The reload and changeover cost minimization problems via edge coloring are NP-hard even when the underlying graph is a star and traversal costs are bounded.
• The problems are hard to approximate within any polynomial-time computable function of the input length when traversal costs are unbounded.
• Polynomial-time solvability is achieved for the changeover cost problem in graphs with sparse blocks and bounded-degree cut vertices using dynamic programming.
• The algorithm computes optimal solutions by combining min-weight perfect matching and recursive cost aggregation over the block tree structure.
• For the MinRCPTEC problem, the innermost term in the cost function includes a multiplicative factor of path count, reflecting the reload cost's dependence on commodity flow.
• The time complexity of computing the optimal cost at a cut vertex is polynomial in the input size, bounded by |X|^{c₁} · |E(G)|^{c₂+1} for constants c₁, c₂ related to vertex degrees and block sizes.