Precoloring extension with demands on paths

2025-09-23

AI Summary

This paper studies the Demand-based Precolored Extension with Distance Constraints (DPED) problem on paths: given a set of precolored vertices, a minimum monochromatic distance constraint (d), and exact color demands (d_i) for each of (c) colors, find a (c)-coloring satisfying all constraints. DPED models broadcast scheduling where content repetition must be avoided while respecting positional constraints. We formally define DPED and prove it is NP-complete and W[1]-hard parameterized by either the number of colors or the number of precolored vertices—even on paths. Our contributions include: (i) a polynomial-time exact algorithm for instances where precolored vertices lie only at path endpoints; (ii) an additive-error bounded approximation algorithm; (iii) multiple fixed-parameter tractable algorithms parameterized by distance (d), number of colors (c), and size of the precolored set; and (iv) a generalization to the distance-list coloring variant, where each vertex imposes individual distance constraints per color.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Let $G$ be a graph with a set of precolored vertices, and let us be given an integer distance parameter $d$ and a set of integer demands $d_1,dots,d_c$. The Distance Precoloring Extension with Demands (DPED) problem is to compute a vertex $c$-coloring of $G$ such that the following three conditions hold: (i) the resulting coloring respects the colors of the precolored vertices, (ii) the distance of two vertices of the same color is at least $d$, and (iii) the number of vertices colored by color $i$ is exactly $d_i$. This problem is motivated by a program scheduling in commercial broadcast channels with constraints on content repetition and placement, which leads precisely to the DPED problem for paths. In this paper, we study DPED on paths and present a polynomial time exact algorithm when precolored vertices are restricted to the two ends of the path and devise an approximation algorithm for DPED with an additive approximation factor polynomially bounded by $d$ and the number of precolored vertices. Then, we prove that the Distance Precoloring Extension problem on paths, a less restrictive version of DPED without the demand constraints, and then DPED itself, is NP-complete. Motivated by this result, we further study the parameterized complexity of DPED on paths. We establish that the DPED problem on paths is $W[1]$-hard when parameterized by the number of colors and the distance. On the positive side, we devise a fixed parameter tractable (FPT) algorithm for DPED on paths when the number of colors, the distance, and the number of precolored vertices are considered as the parameters. Moreover, we prove that Distance Precoloring Extension is FPT parameterized by the distance. As a byproduct, we also obtain several results for the Distance List Coloring problem on paths.

Problem

• Extends precoloring to entire paths while satisfying distance constraints
• Ensures exact color demands are met in vertex coloring
• Addresses program scheduling with repetition and placement constraints

Innovation

• Exact polynomial algorithm for paths with end precoloring
• FPT algorithm parameterized by colors, distance, precolored vertices
• Proved NP-completeness and W[1]-hardness for path coloring problems