A Survey of Approximability Results for Traveling Salesman Problems using the TSP-T3CO Definition Scheme

TL;DR

This paper delivers the first systematic survey of the best known (in)approximability results across a wide spectrum of TSP variants and introduces TSP-T3CO 2025, a formal five-field scheme (Traveler, Targets, Tour, Costs, Objectives) to standardize definitions. By providing both longhand and shorthand grammars, plus 60+ confirmed definitions, the authors create a concise, comparable landscape of results and clearly delineate the assumptions behind each bound. The framework enables easier cross-variant understanding, detection of open gaps, and straightforward extension to VRP through new α-field attributes. Collectively, the work has substantial practical impact for researchers and practitioners seeking to map approximation guarantees to specific TSP formulations and to translate findings across related problems.

Abstract

The traveling salesman (or salesperson) problem, short TSP, is a problem of strong interest to many researchers from mathematics, economics, and computer science. Manifold TSP variants occur in nearly every scientific field and application domain: engineering, physics, biology, life sciences, and manufacturing just to name a few. Several thousand papers are published on theoretical research or application-oriented results each year. This paper provides the first systematic survey on the best currently known approximability and inapproximability results for well-known TSP variants such as the "standard" TSP, Path TSP, Bottleneck TSP, Maximum Scatter TSP, Generalized TSP, Clustered TSP, Traveling Purchaser Problem, Profitable Tour Problem, Quota TSP, Prize-Collecting TSP, Orienteering Problem, Time-dependent TSP, TSP with Time Windows, and the Orienteering Problem with Time Windows. The foundation of our survey is the definition scheme T3CO, which we propose as a uniform, easy-to-use and extensible means for the formal and precise definition of TSP variants. Applying T3CO to formally define the variant studied by a paper reveals subtle differences within the same named variant and also brings out the differences between the variants more clearly. We achieve the first comprehensive, concise, and compact representation of approximability results by using T3CO definitions. This makes it easier to understand the approximability landscape and the assumptions under which certain results hold. Open gaps become more evident and results can be compared more easily.

Figures (6)

• Figure 1: Example of an undirected graph with its edge and node costs.
• Figure 2: EBNF for the TSP-T3CO longhand notation.
• Figure 3: EBNF for the TSP-T3CO shorthand notation. Rules for non-terminal symbols not defined here can be found in Figure \ref{['fig:ebnf-long']}.
• Figure 4: TSP-T3CO definition of the Capacitated Vehicle Routing Problem CVRP described in dantzig1959truck.
• Figure 5: Recap of Figure \ref{['fig:ebnf-long']}: EBNF for TSP-T3CO longhand notation.

Theorems & Definitions (3)

• Definition 1: Walk
• Definition 2: Parts of a Walk
• Definition 3: Shoreline Metric psaraftis1990routing