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Exact Algorithms for Resource Reallocation Under Budgetary Constraints

Arun Kumar Das, Sandip Das, Sweta Das, Foivos Fioravantes, Nikolaos Melissinos

TL;DR

This work introduces the Red-Blue Reinforcement (R-BR) problem on red-blue graphs, formalizing the task of minimizing client reallocations under a server budget by reducing the number of required servers via targeted blue-node removals. It advances the algorithmic frontier by presenting three exact fixed-parameter algorithms parameterized by distance to cluster ($3^{dc}$), modular-width ($2^{mw}$), and clique-width ($4^{cw}$), each accompanied by a concrete DP/covering- reduction framework. The key techniques include reducing R-BR to Maximum Price Coverage (MPC) for the distance-to-cluster case, and performing modular-decomposition and clique-width–based dynamic programming to exploit structural graph properties. The results yield practical, scalable solutions for networks with rural, hierarchical, or dense topologies, and the clique-width algorithm is argued to be asymptotically optimal under the Strong Exponential Time Hypothesis. Overall, the paper connects resource-reallocation problems to established dominance-related parameters, offering both theoretical insight and pathways toward practical implementations and heuristics in network design.

Abstract

Efficient resource (re-)allocation is a critical challenge in optimizing productivity and sustainability within multi-party supply networks. In this work, we introduce the \textsc{Red-Blue Reinforcement} (R-BR) problem, where a service provider under budgetary constraints must minimize client reallocations to reduce the required number of servers they should maintain by a specified amount. We conduct a systematic algorithmic study, providing three exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our algorithms are efficient for topologies that model rural road networks (bounded distance to cluster), modern transportation systems (bounded modular-width), or have bounded clique-width, a parameter that is of great theoretical importance.

Exact Algorithms for Resource Reallocation Under Budgetary Constraints

TL;DR

This work introduces the Red-Blue Reinforcement (R-BR) problem on red-blue graphs, formalizing the task of minimizing client reallocations under a server budget by reducing the number of required servers via targeted blue-node removals. It advances the algorithmic frontier by presenting three exact fixed-parameter algorithms parameterized by distance to cluster (), modular-width (), and clique-width (), each accompanied by a concrete DP/covering- reduction framework. The key techniques include reducing R-BR to Maximum Price Coverage (MPC) for the distance-to-cluster case, and performing modular-decomposition and clique-width–based dynamic programming to exploit structural graph properties. The results yield practical, scalable solutions for networks with rural, hierarchical, or dense topologies, and the clique-width algorithm is argued to be asymptotically optimal under the Strong Exponential Time Hypothesis. Overall, the paper connects resource-reallocation problems to established dominance-related parameters, offering both theoretical insight and pathways toward practical implementations and heuristics in network design.

Abstract

Efficient resource (re-)allocation is a critical challenge in optimizing productivity and sustainability within multi-party supply networks. In this work, we introduce the \textsc{Red-Blue Reinforcement} (R-BR) problem, where a service provider under budgetary constraints must minimize client reallocations to reduce the required number of servers they should maintain by a specified amount. We conduct a systematic algorithmic study, providing three exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our algorithms are efficient for topologies that model rural road networks (bounded distance to cluster), modern transportation systems (bounded modular-width), or have bounded clique-width, a parameter that is of great theoretical importance.
Paper Structure (15 sections, 4 theorems, 6 equations)

This paper contains 15 sections, 4 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Preliminaries
  4. Parameterized Complexity
  5. Structural Parameters
  6. Efficient Algorithm I: Distance to Cluster
  7. An efficient algorithm for MPC.
  8. Reducing R-BR to MPC.
  9. Efficient Algorithm II: Modular-Width
  10. Efficient Algorithm III: Clique-Width
  11. Vertex-Introduce Nodes.
  12. Rename Nodes.
  13. Edge-Introduce Nodes.
  14. Union Nodes.
  15. Conclusion

Key Result

Theorem 1

Let $(\mathcal{U}, \mathcal{S}, \mathcal{X}, P, K)$ be an instance of MPC. There is an algorithm that computes a set $C\subseteq \mathcal{S}$ of maximum price in time $2^{|\mathcal{U}|}(|\mathcal{U}|+|\mathcal{S}|)^{O(1)}$.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • proof