A Ternary Gamma Semiring Framework for Solving Multi-Objective Network Optimization Problems
Chandrasekhar Gokavarapu, D. Madhusudhana Rao
TL;DR
The paper addresses the inadequacy of binary tropical semirings to model genuinely ternary cost interactions among attributes like cost, time, and risk in network optimization. It introduces the Ternary Tropical Gamma Semiring (TTGS), a Gamma-indexed algebraic structure with a monotone non-separable ternary operation, and develops TTGS-Pathfinder, a Bellman-Ford–style dynamic-programming algorithm that relaxes two-step fragments and converges to a TTGS-optimal fixed point with complexity $O(n^2 m)$. Theoretical results establish ternary associativity, distributivity over $\oplus=\min$, and monotonicity, ensuring the DP recurrence is well-defined within TTGS and leading to finite convergence in the absence of improving ternary cycles. The framework enables principled modeling of triadic dependencies in logistics, reliability-aware network design, and computational engineering, extending algebraic optimization beyond the binary tropical paradigm and offering a new tool for multi-parameter decision problems.
Abstract
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 \emph{Ternary Tropical Gamma Semiring} (TTGS), a $Γ$-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 $O(n^2 m)$ 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.