[Paper Review] Shortest path and maximum flow problems in planar flow networks with additive gains and losses

Saber Mirzaei, A. J. Kfoury|arXiv (Cornell University)|Mar 29, 2016

Advanced Graph Theory Research3 citations

TL;DR

This paper proves that both the shortest path and maximum flow problems in planar additive flow networks—where edges have additive gains or losses—are NP-hard in the strong sense. Despite the planar structure, which often enables polynomial-time solutions, the presence of additive gains/losses disrupts standard flow properties, rendering these fundamental network problems computationally intractable even with bounded input values.

ABSTRACT

In contrast to traditional flow networks, in additive flow networks, to every edge e is assigned a gain factor g(e) which represents the loss or gain of the flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is less than the designated capacity of e, then f(e) + g(e) = 0 units of flow reach the end point of e, provided e is used, i.e., provided f(e) != 0. In this report we study the maximum flow problem in additive flow networks, which we prove to be NP-hard even when the underlying graphs of additive flow networks are planar. We also investigate the shortest path problem, when to every edge e is assigned a cost value for every unit flow entering edge e, which we show to be NP-hard in the strong sense even when the additive flow networks are planar.

Motivation & Objective

• To investigate the computational complexity of shortest path and maximum flow problems in additive flow networks with planar underlying graphs.
• To determine whether the planarity of the network structure enables polynomial-time solutions for these problems.
• To extend prior NP-hardness results from general graphs to the more constrained case of planar graphs.
• To explore the implications of additive gains and losses on classical network flow properties such as flow conservation and duality.
• To lay groundwork for future research on approximation algorithms and fixed-parameter tractable solutions.

Proposed method

• Reduction from 3-SAT to construct a planar additive flow network where feasible flow corresponds to a satisfying assignment.
• Design of variable and clause gadgets to encode logical variables and clauses in the flow network structure.
• Use of edge capacities, gain factors, and cost functions to model logical constraints and flow behavior.
• Construction of a reversed flow network to relate in-flow and out-flow maximization problems.
• Formal proof of NP-hardness in the strong sense using polynomially bounded input values.
• Leveraging the structure of planar graphs and the properties of additive gains/losses to show that standard algorithms fail.

Experimental results

Research questions

• RQ1 Is the shortest path problem in planar additive flow networks NP-hard in the strong sense?
• RQ2 Is the maximum in-flow or out-flow problem in planar additive flow networks NP-hard in the strong sense?
• RQ3 Can the standard max-flow/min-cut duality or augmenting path methods be applied to additive flow networks?
• RQ4 Do additive gains and losses invalidate the optimality of subpaths in shortest path problems?
• RQ5 Can approximation algorithms or fixed-parameter tractable algorithms be developed for these problems?

Key findings

• The shortest path problem in planar additive flow networks is NP-hard in the strong sense, even when all edge costs, gains, and capacities are bounded by polynomials in the input size.
• The maximum in-flow problem in planar additive flow networks is NP-hard in the strong sense, as shown via a 3-SAT reduction with planar construction.
• The maximum out-flow problem is also NP-hard in the strong sense, established through a duality argument using reversed flow networks.
• The presence of additive gains and losses breaks classical properties such as flow conservation, subpath optimality, and max-flow/min-cut duality.
• The results hold even when all input parameters (cost, gain, capacity) are polynomially bounded in the size of the network.
• The findings suggest that no known polynomial-time algorithms exist for these problems in planar additive networks, motivating future work on approximation and fixed-parameter algorithms.