{"ID":2838301,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.18099","arxiv_id":"2511.18099","title":"A Ternary Gamma Semiring Framework for Solving Multi-Objective Network Optimization 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.","short_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 amo...","url_abs":"https://arxiv.org/abs/2511.18099","url_pdf":"https://arxiv.org/pdf/2511.18099v1","authors":"[\"Chandrasekhar Gokavarapu\",\"D. Madhusudhana Rao\"]","published":"2025-11-22T15:52:00Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[\"Large Language Model\"]","has_code":false}