# A Ternary Gamma Semiring Framework for Solving Multi-Objective Network Optimization Problems

arXiv: https://arxiv.org/abs/2511.18099 · HTML: https://mlsanity.com/paper/2511.18099/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2025.

## Built on by (most cited first, 1 of 1)

- [Ternary Gamma Semirings: From Neural Implementation to Categorical Foundations](https://mlsanity.com/paper/2603.19317/) (2026)

## Abstract (arXiv)

Classical shortest-path methods rely on binary tropical semirings (min,+), whose dyadic structure limits them to pairwise cost interactions. However, many real-world systems, including logistics, supply chains, communication networks, and reliability-aware infrastructures, exhibit inherently ternary dependencies among cost, time, and risk that cannot be decomposed into pairwise components. This paper introduces the Ternary Tropical Gamma Semiring (TTGS), a Gamma-indexed algebraic structure that generalizes tropical semirings by replacing binary additive composition with a non-separable ternary operator. We establish the axioms of TTGS, prove associativity, distributivity, and monotonicity, and show that TTGS forms a well-structured foundation for multi-parameter optimization. Building on this framework, we develop TTGS-Pathfinder, a ternary analogue of the Bellman--Ford algorithm. We derive its dynamic-programming recurrence, prove correctness through an invariant-based argument, analyze convergence under the TTGS order, and obtain an O(n2 m) complexity bound. Applications demonstrate that TTGS naturally models systems whose behaviour depends on triadic cost interactions, offering a principled alternative to binary tropical, vector, or scalarized multi-objective methods.