[Paper Review] Improved integrality gap upper bounds for TSP with distances one and two

Matthias Mnich, Tobias Mömke|arXiv (Cornell University)|Dec 9, 2013

Advanced Graph Theory Research17 references3 citations

TL;DR

This paper improves integrality gap upper bounds for the traveling salesperson problem (TSP) with edge costs one and two by analyzing the support structure of solutions to the subtour elimination relaxation (SER). It introduces a modified linear program, SER⁺, with a single cutting plane inequality, enabling polynomial-time algorithms to find 2-matchings with few components, leading to tight integrality gap bounds: 5/4 for undirected TSP, 7/6 for half-integral solutions, and 10/9 for subcubic support graphs; for directed TSP, bounds of 3/2 and 4/3 are achieved.

ABSTRACT

We study the structure of solutions to linear programming formulations for the traveling salesperson problem (TSP). We perform a detailed analysis of the support of the subtour elimination linear programming relaxation, which leads to algorithms that find 2-matchings with few components in polynomial time. The number of components directly leads to integrality gap upper bounds for the TSP with distances one and two, for both undirected and directed graphs. Our main results concern the subtour elimination relaxation with one additional cutting plane inequality: - For undirected instances we obtain an integrality gap upper bound of 5/4 without any further restrictions, of 7/6 if the optimal LP solution is half-integral. - For instances of order n where the fractional LP value has a cost of n, we obtain a tight integrality gap upper bound of 10/9 if there is an optimal solution with subcubic support graph. The latter property that the graph is subcubic is implied if the solution is a basic solution in the fractional 2-matching polytope. - For directed instances we obtain an integrality gap upper bound of 3/2, and of 4/3 if given an optimal 1/2-integral solution. In the case of undirected graphs, we can avoid to add the cutting plane inequality if we accept slightly increased values. For the tight result, the cutting plane is not required. Additionally, we show that relying on the structure of the support is not an artefact of our algorithm, but is necessary under standard complexity-theoretic assumptions: we show that finding improved solutions via local search is W[1]-hard for k-edge change neighborhoods even for the TSP with distances one and two, which strengthens a result of Dániel Marx.

Motivation & Objective

• To improve integrality gap upper bounds for the symmetric and asymmetric TSP with edge costs one and two.
• To analyze the support structure of solutions to the subtour elimination relaxation (SER) to enable better approximation guarantees.
• To develop polynomial-time algorithms that find 2-matchings with few components, directly translating to improved integrality gaps.
• To establish tight bounds under structural assumptions such as half-integrality and subcubic support graphs.
• To prove that local search for k-edge changes is W[1]-hard, showing structural analysis is necessary under complexity-theoretic assumptions.

Proposed method

• Augment the standard subtour elimination relaxation (SER) with a single cutting plane inequality to form SER⁺, ensuring integer solutions.
• Analyze the support graph of optimal SER⁺ solutions to identify structural properties enabling improved 2-matching construction.
• Design polynomial-time algorithms to transform 2-matchings into equivalent forms with fewer components, avoiding isolated vertices and using alternating paths.
• Apply improvements based on alternating paths and path transformations to reduce the number of components in 2-matchings to at most n/4.
• Leverage known structural properties of basic solutions in the fractional 2-matching polytope—specifically half-integrality and subcubic degrees—to derive tighter bounds.
• Use a reduction from the t-Clique problem to prove W[1]-hardness of k-edge change local search, establishing the necessity of structural analysis.

Experimental results

Research questions

• RQ1 Can the integrality gap of the subtour elimination relaxation be improved for TSP with distances one and two using structural insights into the support graph?
• RQ2 What integrality gap bounds can be achieved when the optimal SER solution is half-integral or has subcubic support?
• RQ3 Is the use of a single cutting plane inequality in SER⁺ sufficient to achieve tighter integrality gap bounds without sacrificing polynomial-time solvability?
• RQ4 Can improved approximation guarantees be derived for both undirected and directed TSP variants under specific structural constraints?
• RQ5 Is local search for k-edge changes in (1,2)-TSP fixed-parameter tractable, or is it W[1]-hard under standard complexity assumptions?

Key findings

• For undirected (1,2)-STSP, the integrality gap of SER⁺ is at most 5/4 without additional constraints, and 7/6 when the optimal solution is half-integral.
• For undirected (1,2)-STSP with optimal LP value equal to n and subcubic support graph, the integrality gap is tightly bounded by 10/9.
• For directed (1,2)-ATSP, the integrality gap of SER⁺ is at most 3/2, and 4/3 when the optimal solution is half-integral.
• The cutting plane inequality is not required for the 5/4 bound in undirected TSP, but is essential for achieving the tight 10/9 bound under subcubic support.
• Finding improved solutions via k-edge change local search is W[1]-hard for (1,2)-TSP, indicating that structural analysis is not an artifact but a necessity.
• Under the Exponential Time Hypothesis, no f(k)·n^{o(√[3]{k})}-time algorithm exists for k-edge change (1,2)-TSP, implying strong lower bounds on the exponent of n.