Polychromatic Traveling Salesman Problem (PCTSP)

• Polychromatic TSP is defined as finding a minimum-weight Hamiltonian cycle through vertices partitioned into k equal-sized color classes, following a fixed cyclic order.
• It generalizes classical TSP and Bipartite TSP by enforcing specific color constraints, with metric cases achieving a (3-2·10⁻³⁶)-approximation.
• The Euclidean variant is APX-hard, highlighting the complexity differences and challenges in approximation and geometric formulations.

Searching arXiv for the cited PCTSP-related papers to ground the article in current research. Polychromatic Traveling Salesman Problem (PCTSP) is a colored generalization of the Traveling Salesman Problem in which the vertex set is partitioned into $k$ equal-sized color classes and the objective is to find a minimum-weight Hamiltonian cycle that visits the classes in a fixed cyclic order, for some permutation $\sigma$ of the classes. In a $\sigma$-cycle, vertex $v_i$ must belong to $V_{\sigma(i \bmod k)}$, so the tour repeatedly traverses the classes in the order $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$ until all vertices are visited exactly once. This formulation generalizes Bipartite TSP when $k=2$ and classical TSP when $k=n$; for the metric case, a polynomial-time $(3-2\cdot 10^{-36})$-approximation is known, while Euclidean PCTSP in $\mathbb{R}^2$ is APX-hard and does not admit a PTAS unless $\sigma$0 (Schibler et al., 7 Jul 2025).

1. Formal model

A polychromatic graph is a tuple

$\sigma$1

where $\sigma$2 is a graph, $\sigma$3 is a partition of $\sigma$4 into $\sigma$5 equal-sized color classes,

$\sigma$6

and $\sigma$7 is a nonnegative edge-weight function. If $\sigma$8, then each class has size $\sigma$9. For a permutation

$\sigma$0

of $\sigma$1, a cycle

$\sigma$2

is a $\sigma$3-cycle if

$\sigma$4

The weight of a path or cycle $\sigma$5 is

$\sigma$6

and the objective is to find a minimum-weight Hamiltonian cycle that is a $\sigma$7-cycle for some permutation $\sigma$8. Equivalently,

$\sigma$9

where $v_i$0 denotes the set of all Hamiltonian polychromatic cycles. A Hamiltonian polychromatic cycle is also called a $v_i$1-tour (Schibler et al., 7 Jul 2025).

Two standard specializations are explicit. In metric PCTSP, $v_i$2 is complete and satisfies the triangle inequality

$v_i$3

for all vertices $v_i$4. In Euclidean PCTSP, the vertices are points $v_i$5, each color class is a subset $v_i$6, and weights are Euclidean distances,

$v_i$7

The model interpolates between alternating-color routing and unconstrained Hamiltonian routing: if $v_i$8, the tour alternates between the two classes; if $v_i$9, each class has size $V_{\sigma(i \bmod k)}$0, so the problem becomes ordinary TSP (Schibler et al., 7 Jul 2025).

2. Terminological scope and neighboring uses of “PCTSP”

The acronym “PCTSP” is not unique across the TSP literature. In Prize-Collecting Traveling Salesperson Problem, the input is a complete undirected graph $V_{\sigma(i \bmod k)}$1 with metric edge lengths $V_{\sigma(i \bmod k)}$2, a distinguished root $V_{\sigma(i \bmod k)}$3, and nonnegative penalties $V_{\sigma(i \bmod k)}$4 for vertices in $V_{\sigma(i \bmod k)}$5. The task is to find a cycle $V_{\sigma(i \bmod k)}$6 containing $V_{\sigma(i \bmod k)}$7 that minimizes

$V_{\sigma(i \bmod k)}$8

Here vertices may be omitted at a penalty, and the natural LP relaxation uses edge variables $V_{\sigma(i \bmod k)}$9 and coverage variables $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$0 (Blauth et al., 2022).

A different nearby formulation appears in planar colored point sets. In Colored Points Traveling Salesman Problem, which is described as essentially the Polychromatic Traveling Salesman Problem in the plane, the input is a set $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$1 of planar points partitioned into $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$2 color classes, and the goal is to find a polygon

$V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$3

of minimum perimeter such that $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$4 for all $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$5 and $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$6 for all $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$7. In other words, one selects exactly one representative point from each color class and connects those selected points into the shortest possible cycle (Asaeedi, 2024).

Accordingly, polychromatic TSP in the sense of Hamiltonian class-ordered tours should be distinguished from both prize-collecting formulations, where omission is penalized, and representative-selection formulations, where only one point per color is visited.

3. Approximation algorithm for the metric case

For a fixed order $V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$8, the metric approximation algorithm first writes, without loss of generality,

$V_{\sigma(0)},V_{\sigma(1)},\dots,V_{\sigma(k-1)}$9

and sets $k=2$0. For each adjacent pair of classes $k=2$1, it computes a minimum-weight perfect matching

$k=2$2

Since all classes have the same size and $k=2$3 is complete, such a matching exists. Let

$k=2$4

Each connected component of $k=2$5 is itself a $k=2$6-cycle. From each component $k=2$7, one chooses a representative vertex $k=2$8, lets $k=2$9 denote the set of representatives, solves TSP on the induced subgraph $k=n$0 using the $k=n$1-approximation for metric TSP, and then glues the components together according to that representative tour order. The fixed-order analysis uses two bounds: $k=n$2 for any Hamiltonian $k=n$3-cycle $k=n$4, and

$k=n$5

for the glued solution $k=n$6. Together with $k=n$7, this yields a $k=n$8-approximation for a fixed $k=n$9 (Schibler et al., 7 Jul 2025).

The second step approximates the best cyclic order of the classes. For each pair $(3-2\cdot 10^{-36})$0, one computes a minimum matching $(3-2\cdot 10^{-36})$1 between $(3-2\cdot 10^{-36})$2 and $(3-2\cdot 10^{-36})$3, and defines a complete graph $(3-2\cdot 10^{-36})$4 on the classes $(3-2\cdot 10^{-36})$5 with edge weight

$(3-2\cdot 10^{-36})$6

This graph is metric. For a permutation $(3-2\cdot 10^{-36})$7, the quantity

$(3-2\cdot 10^{-36})$8

is exactly the weight of the corresponding Hamiltonian cycle in $(3-2\cdot 10^{-36})$9. Applying the $\mathbb{R}^2$0-approximation for metric TSP to $\mathbb{R}^2$1 gives an approximately optimal order $\mathbb{R}^2$2, and feeding $\mathbb{R}^2$3 into the fixed-order routine yields the final

$\mathbb{R}^2$4

More generally, if matching can be computed in time $\mathbb{R}^2$5 and metric TSP can be $\mathbb{R}^2$6-approximated in time $\mathbb{R}^2$7, then PCTSP admits a $\mathbb{R}^2$8-approximation with running time

$\mathbb{R}^2$9

(Schibler et al., 7 Jul 2025).

4. Euclidean hardness and APX-inapproximability

Euclidean PCTSP remains hard even in low dimension. The hardness reduction is from Max 2-SAT and uses the class order $\sigma$00 itself to encode a truth assignment. The construction introduces

$\sigma$01

color classes

$\sigma$02

together with a notion of valid permutation $\sigma$03 satisfying

$\sigma$04

and

$\sigma$05

These constraints leave exactly one binary choice per variable,

$\sigma$06

so there are exactly $\sigma$07 valid permutations, corresponding naturally to truth assignments (Schibler et al., 7 Jul 2025).

For each clause $\sigma$08, a geometric gadget is created so that, for any valid $\sigma$09, the shortest $\sigma$10-path through that gadget has cost

$\sigma$11

if $\sigma$12 satisfies the clause, and

$\sigma$13

otherwise, where

$\sigma$14

The full instance is $\sigma$15, where $\sigma$16 is the union of clause gadgets and $\sigma$17 is a line of auxiliary points, one in each color class, placed on the line $\sigma$18. If a valid $\sigma$19 satisfies exactly $\sigma$20 clauses, the optimal tour cost is

$\sigma$21

Two structural lemmas complete the reduction: if a valid $\sigma$22-tour is short enough, then $\sigma$23 must satisfy many clauses; and if $\sigma$24 is invalid, it can be repaired without increasing cost. Hence Euclidean PCTSP in $\sigma$25 is APX-hard and does not admit a PTAS unless $\sigma$26 (Schibler et al., 7 Jul 2025).

5. Geometric relatives with different color semantics

A planar representative-selection variant is studied under the name Colored Points Traveling Salesman Problem. There the input is a colored point set in the plane, the objective is minimum perimeter rather than graph-theoretic Hamiltonian weight, and exactly one point from each color class is selected. NP-hardness follows by assigning each point a unique color, so the minimal-perimeter color-visiting polygon coincides with the minimal-perimeter polygon through all points. The paper gives a brute-force exact algorithm enumerating

$\sigma$27

representative selections, with worst-case complexity

$\sigma$28

and an approximation algorithm based on the minimum color-spanning circle and onion peeling. If $\sigma$29 is the radius of the smallest color-spanning circle, the circle can be computed in

$\sigma$30

time, each onion layer contributes at most $\sigma$31, the worst case has $\sigma$32 layers, and with the lower bound $\sigma$33 for grid points, the paper states the approximation factor

$\sigma$34

The overall running time is polynomial and dominated by the color-spanning-circle computation (Asaeedi, 2024).

Another geometric relative is the two-colored noncrossing Euclidean TSP. In that problem, the input consists of red and blue terminals in $\sigma$35, and the solution is a pair of pairwise disjoint closed curves, one visiting all red terminals and one visiting all blue terminals, minimizing

$\sigma$36

The central constraint is noncrossing rather than cyclic color order. The Euclidean problem admits a randomized $\sigma$37-approximation scheme in

$\sigma$38

time, and this dependence on $\sigma$39 is Gap-ETH-tight. The key technical ingredient is a patching lemma for two noncrossing curves: given a segment $\sigma$40 and noncrossing closed curves $\sigma$41, there exist modified noncrossing curves $\sigma$42 such that

$\sigma$43

and

$\sigma$44

with modification confined to an infinitesimal neighborhood of $\sigma$45. The same work also gives a PTAS for the problem in plane unweighted graphs (Dross et al., 2022).

6. Structural distinctions and recurrent confusions

The literature therefore uses closely related names for structurally different problems. The distinction is not terminological only; it changes the feasible set, the role of color classes, and the transferability of approximation and hardness results (Schibler et al., 7 Jul 2025, Asaeedi, 2024, Blauth et al., 2022, Dross et al., 2022).

Variant	Defining constraint	Objective
Polychromatic TSP	Visit all vertices in a cyclic class order	Minimum-weight Hamiltonian $\sigma$46-tour
Colored Points TSP	Choose exactly one point from each color	Minimum-perimeter polygon
Prize-Collecting TSP	Omit vertices by paying penalties	Tour length plus omission penalties
Noncrossing bicolored TSP	One tour per color, tours do not cross	Minimum total Euclidean length

A common misconception is to treat all colored TSP variants as interchangeable because each uses labels or color classes. In fact, they differ along at least three axes: whether every vertex must be visited, whether one selects representatives instead of visiting all input points, and whether geometry enters through Euclidean distances together with noncrossing or through a metric graph together with Hamiltonicity. This suggests that approximation factors, LP relaxations, and hardness transfers must be interpreted within the specific variant under discussion.