The complexity of the longest tour through points in the plane

Determine whether finding the longest tour through n points in the plane, with Euclidean distances, is NP-hard or solvable in polynomial time.

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	1	+# The complexity of the longest tour through points in the plane
	2	+
	3	+The shortest tour through $n$ points in the plane is NP-hard to find. The longest tour behaves differently: with Manhattan distances it takes linear time, and with Euclidean distances in three dimensions it is NP-hard. For Euclidean distances in the plane, its complexity is unknown.
	4	+
	5	+## Problem
	6	+
	7	+Maximum TSP in the Euclidean plane. Given $n$ points $p_1,\dots,p_n\in\mathbb{Q}^2$, find a tour, that is, a cyclic order $\sigma$ of the points, maximizing the total length
	8	+$$\sum_{i=1}^{n}\big\\|p_{\sigma(i)}-p_{\sigma(i+1)}\big\\|_2, \qquad \sigma(n+1)=\sigma(1).$$
	9	+
	10	+Question. What is the complexity of this problem? In particular, is it NP-hard (in the sense that deciding whether a tour longer than a given rational bound exists is NP-hard), or can it be solved in polynomial time?
	11	+
	12	+The decision version is not even known to be in NP, because comparing a sum of Euclidean distances with a bound is the sum of square roots problem.
	13	+
	14	+## Background
	15	+
	16	+Barvinok, Fekete, Johnson, Tamir, Woeginger and Woodroofe studied the maximum TSP for points in $\mathbb{R}^d$ under various norms and found that its complexity depends on the shape of the unit ball rather than on the combinatorics [1]. They closed with the conjecture that the planar Euclidean case is NP-hard, noting that local gadget arguments are harder to use for maximization problems. The earliest known mention of the problem is Barvinok's 1996 paper [2], and Fekete (1999) conjectured NP-hardness; it is Problem 49 of The Open Problems Project, as a test of how the complexity of a natural problem depends on the geometry of distances.
	17	+
	18	+## What is known
	19	+
	20	+- Approximation. For every fixed dimension and every $L_p$ or polyhedral norm, the maximum TSP has a polynomial-time approximation scheme (Serdyukov; Barvinok, 1996) [2].
	21	+- Polyhedral norms are easy. If the unit ball is a polytope with $f$ facets, a maximum tour can be found in $O(n^{f-2}\log n)$ time, and in $O(n)$ time for the rectilinear and maximum norms in the plane [1], [3].
	22	+- Euclidean distances are hard in 3D. The maximum TSP with Euclidean distances is NP-hard in $\mathbb{R}^d$ for every fixed $d\ge 3$, and also for points on a sphere [1].
	23	+- Contrast. The minimum TSP is NP-hard in every fixed dimension for every $L_p$ or polyhedral norm, including the Euclidean plane [1].
	24	+- A relative. The maximum scatter TSP, which maximizes the shortest edge of the tour, is NP-hard in dimension 3 and higher (Fekete, 1999) and also open in the plane.
	25	+
	26	+## What would count as progress
	27	+
	28	+- A reduction proving NP-hardness of the planar Euclidean maximum TSP. Gadgets can be checked by computing optimal tours of small instances exactly.
	29	+- A polynomial-time algorithm on the real RAM, with a proof, perhaps first for structured inputs such as points in convex position or on few lines.
	30	+- Exact optimal tours for benchmark families, which give evidence about the structure of longest tours.
	31	+
	32	+## References
	33	+
	34	+1. A. Barvinok, S. P. Fekete, D. S. Johnson, A. Tamir, G. J. Woeginger and R. Woodroofe, The geometric maximum traveling salesman problem, J. ACM 50 (2003), 641–664. [doi:10.1145/876638.876640](https://doi.org/10.1145/876638.876640), [arXiv:cs/0204024](https://arxiv.org/abs/cs/0204024)
	35	+2. A. I. Barvinok, Two algorithmic results for the traveling salesman problem, Math. Oper. Res. 21 (1996), 65–84. [doi:10.1287/moor.21.1.65](https://doi.org/10.1287/moor.21.1.65)
	36	+3. A. Barvinok, D. S. Johnson, G. J. Woeginger and R. Woodroofe, The maximum traveling salesman problem under polyhedral norms, Proc. IPCO 1998, LNCS 1412, 195–201. [doi:10.1007/3-540-69346-7_15](https://doi.org/10.1007/3-540-69346-7_15)
	37	+
	38	+## Source
	39	+
	40	+Imported from [The Open Problems Project, Problem 49](https://topp.openproblem.net/p49), curated by Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.