Minsum Problem for Discrete and Weighted Set Flow on Dynamic Path Network

TL;DR

The paper studies the minsum flow problem on dynamic path networks with flows represented as discrete weighted sets, denoted $MS-DWSF$, to capture group constraints and priorities in evacuations. It develops a two-phase approach by reducing to a related minsum bin packing with ready times and weights, $MS-BPWRT$, and presents a $2$-approximation pseudo-polynomial time algorithm for that problem, which is then used to derive a $2$-approximation for $MS-DWSF$ on path networks with uniform capacity and a single facility. It also proves NP-hardness of $MS-DWSF$ via a partition reduction and notes a two-node equivalence to the weighted minsum bin packing problem, highlighting core hardness. The work provides provable approximation guarantees for a dynamic, non-separable evacuation model on path networks, informing practical planning where group cohesion and prioritization matter.

Abstract

In this research, we examine the minsum flow problem in dynamic path networks where flows are represented as discrete and weighted sets. The minsum flow problem has been widely studied for its relevance in finding evacuation routes during emergencies such as earthquakes. However, previous approaches often assume that individuals are separable and identical, which does not adequately account for the fact that some groups of people, such as families, need to move together and that some groups may be more important than others. To address these limitations, we modify the minsum flow problem to support flows represented as discrete and weighted sets. We also propose a 2-approximation pseudo-polynomial time algorithm to solve this modified problem for path networks with uniform capacity.

Figures (1)

• Figure 1: (a) An instance of the dynamic flow problem (b) An instance of the problem for discrete and weighted set flow on dynamic network