[Paper Review] Minmax-Regret $k$-Sink Location on a Dynamic Tree Network with Uniform Capacities

Mordecai J. Golin, Sai Sandeep|arXiv (Cornell University)|Jun 11, 2018

Facility Location and Emergency ManagementBusiness, Management and Accounting21 references4 citations

TL;DR

This paper presents the first polynomial-time algorithm for the minmax-regret $k$-sink location problem on dynamic tree networks with uniform edge capacities, achieving an $Oigl( ext{max}(k^2, ext{log}^2 n)igr)k^2n^2 ext{log}^5n$ time complexity for both discrete and continuous sink placements. The approach decomposes the global regret function into local regret components and leverages tree partitioning to efficiently compute the optimal sink configuration under scenario uncertainty.

ABSTRACT

A dynamic flow network $G$ with uniform capacity $c$ is a graph in which at most $c$ units of flow can enter an edge in one time unit. If flow enters a vertex faster than it can leave, congestion occurs. The evacuation problem is to evacuate all flow to sinks assuming that all flow is confluent, i.e., all flow passing through a particular vertex must follow the same exit edge. The $k$-sink location problem is to place $k$-sinks so as to minimize this evacuation time. Although the $k$-sink location problem is NP-Hard on a general graph it can be solved in $ ilde O(k^2 n)$ time on trees. The concept of minmax-regret arises from robust optimization. For each source, a range of possible flow values is provided and any scenario with flow values in those ranges might occur. The goal is to find a sink placement that minimizes, over all possible scenarios, the difference between the evacuation time to those sinks and the minimal evacuation time of that scenario. The Minmax-Regret $k$-Sink Location on a Dynamic Path Networks with uniform capacities is polynomial solvable in $n$ and $k$. Similarly, the Minmax-Regret $k$-center problem on trees is polynomial solvable in $n$ and $k$. Prior to this work, polynomial time solutions to the Minmax-Regret $k$-Sink Location on Dynamic Tree Networks with uniform capacities were only known for $k=1$. This paper solves this problem, for general $k,$ in time $$O\Bigl( \max(k^2 \log^2 k,\log ^2n)\, k^4 n^2 \log^5 n\Bigr)$$

Motivation & Objective

• To address the robust $k$-sink location problem in dynamic tree networks where source flow values are uncertain and given as intervals.
• To extend prior work limited to $k=1$ and path networks by providing a solution for general $k \geq 1$ on trees with uniform capacities.
• To design an efficient algorithm that computes the minmax-regret sink placement minimizing the worst-case deviation from the optimal evacuation time across all possible flow scenarios.

Proposed method

• The algorithm expresses the global minmax-regret cost as a function of local regret functions defined on subtrees.
• It uses a tree-partitioning technique from prior work to recursively decompose the tree into $k$ subtrees, each containing one sink.
• Local minmax-regret functions are computed efficiently using dynamic programming and range queries, leveraging the uniform capacity property.
• The method handles both discrete sinks (at vertices) and continuous sinks (on edges) by extending the same structural decomposition.
• It applies a binary search-like strategy over candidate sink locations, guided by the local regret functions, to identify the optimal configuration.
• The algorithm integrates results from robust optimization and dynamic network flows, particularly focusing on confluent flows and evacuation time minimization.

Experimental results

Research questions

• RQ1 Can the minmax-regret $k$-sink location problem on dynamic tree networks with uniform capacities be solved in polynomial time for $k > 1$?
• RQ2 How can the global regret function, which depends on all possible flow scenarios, be efficiently computed without enumerating all scenarios?
• RQ3 What structural properties of uniform-capacity dynamic trees allow for a decomposition of the regret function into local components?
• RQ4 Is it possible to extend the minmax-regret framework to continuous sink placements on edges while maintaining polynomial-time complexity?
• RQ5 What is the computational complexity of the minmax-regret $k$-sink problem on general dynamic tree networks with non-uniform capacities?

Key findings

• The paper presents the first polynomial-time algorithm for the minmax-regret $k$-sink location problem on dynamic tree networks with uniform capacities, solving a long-open problem.
• The algorithm runs in $O\bigl(\text{max}(k^2, \log^2 n)\bigr)k^2n^2\log^5n$ time, which is polynomial in $n$ and $k$, and applies to both discrete and continuous sink placements.
• The key insight is that the global minmax-regret function can be expressed as a combination of local regret functions, enabling efficient computation via tree decomposition.
• The method successfully generalizes prior results from $k=1$ and path networks to general $k \geq 1$ on trees with uniform capacities.
• The approach is extendable to other minmax-regret problems, such as the weighted $k$-center problem, though it does not improve on existing bounds for standard cases.
• The algorithm's efficiency relies on restricting the set of worst-case scenarios to a manageable size, a property that fails in the general-capacity case, highlighting a key limitation of the approach.