Optimization Problems in Correlated Networks

TL;DR

This paper introduces models for correlated link weights in networks and studies the complexity of shortest path and min-cut problems under these models, revealing NP-hardness in some cases and polynomial solutions in others.

Contribution

It proposes two correlated link-weight models and analyzes the complexity of fundamental network problems under these models, providing new insights and solution methods.

Findings

NP-hardness of shortest path and min-cut under deterministic correlated model

Polynomial-time solutions for these problems under constrained models

Convex optimization approaches for stochastic correlated model

Abstract

Solving the shortest path and the min-cut problems are key in achieving high performance and robust communication networks. Those problems have often beeny studied in deterministic and independent networks both in their original formulations as well as in several constrained variants. However, in real-world networks, link weights (e.g., delay, bandwidth, failure probability) are often correlated due to spatial or temporal reasons, and these correlated link weights together behave in a different manner and are not always additive. In this paper, we first propose two correlated link-weight models, namely (i) the deterministic correlated model and (ii) the (log-concave) stochastic correlated model. Subsequently, we study the shortest path problem and the min-cut problem under these two correlated models. We prove that these two problems are NP-hard under the deterministic correlated model, and even cannot be approximated to arbitrary degree in polynomial time. However, these two problems are polynomial-time solvable under the (constrained) nodal deterministic correlated model, and can be solved by convex optimization under the (log-concave) stochastic correlated model.