Simultaneous Network Design with Restricted Link Usage

2025-07-12

AI Summary

This paper studies the minimum-cost network design problem with color constraints on directed graphs: given a source node (s), a sink node (t), a conservative cost function, and a collection of (not necessarily disjoint) color classes, the goal is to select a minimum-cost arc subset such that the intersection with each color class contains an (s o t) directed path. This model captures practical constraints in multicommodity flow settings where each commodity is restricted to links of designated colors. The authors propose the first unified combinatorial optimization framework for this problem; characterize polynomially solvable special cases—e.g., when color classes form a laminar family; design a fixed-parameter tractable (FPT) algorithm parameterized by the number of multicolored arcs; and systematically establish computational complexity boundaries, including NP-hardness proofs and conditions for fixed-parameter tractability.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

Given a digraph with two terminal vertices $s$ and $t$ as well as a conservative cost function and several not necessarily disjoint color classes on its arc set, our goal is to find a minimum-cost subset of the arcs such that its intersection with each color class contains an $s$-$t$ dipath. Problems of this type arise naturally in multi-commodity network design settings where each commodity is restricted to use links of its own color only. We study several variants of the problem, deriving strong hardness results even for restricted cases, but we also identify cases that can be solved in polynomial time. The latter ones include the cases where the color classes form a laminar family, or where the underlying digraph is acyclic and the number of color classes is constant. We also present an FPT algorithm for the general case parameterized by the number of multi-colored arcs.

Problem

• Find minimum-cost arc subset with color-restricted paths
• Study hardness and polynomial-time solvable variants
• Develop FPT algorithm for multi-colored arcs parameter

Innovation

• Minimum-cost arc subset with color constraints
• Polynomial solutions for laminar color families
• FPT algorithm for multi-colored arcs