Question about Traveling Salesperson Problem

Consider the symmetric Traveling Salesperson Problem (TSP) on a complete and weighted graph that satisfies the triangle inequality. A student has applied different approaches to find near-optimal tours and has given the following statements:

I. The Closest Neighbor heuristic always produces a tour whose cost is at most twice the optimum, provided that the graph satisfies the triangle inequality.

II. The Cheapest Insertion heuristic typically improves over the Closest Neighbor method but does not guarantee any fixed approximation ratio in the general case.

III. The Rosenkrantz–Stearns–Lewis algorithm, based on doubling the minimum spanning tree and shortcutting, guarantees a solution within a factor of 2 of the optimum.

IV. The Christofides algorithm improves on the MST-doubling approach by adding a minimum-weight perfect matching among the odd-degree vertices of the MST, achieving a 1.5 approximation bound.

Which of the statements above are incorrect?

A) Only I is incorrect.
B) Only IV is incorrect.
C) Only I and II are incorrect.
D) Only I, II, and IV are incorrect.
E) None of the above.


Original idea by: Mateus de Padua Vicente

Comentários

  1. Joao Meidanis2 de novembro de 2025 às 03:17

    Boa questão, mas como avaliar se Cheapest Insertion tipicamente melhora Closest Neighbor?

    ResponderExcluir

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