---
title: Polychromatic Traveling Salesman Problem (PCTSP)
url: https://www.emergentmind.com/topics/polychromatic-traveling-salesman-problem-pctsp
type: topic
---

# Polychromatic Traveling Salesman Problem (PCTSP)

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 \(P=NP\) [2507.04974].

## 1. Formal model

A polychromatic graph is a tuple
\[
(G,\mathcal{P},w),
\]
where \(G\) is a graph, \(\mathcal{P}=\{V_1,\dots,V_k\}\) is a partition of \(V(G)\) into \(k\) equal-sized color classes,
\[
|V_1|=\cdots=|V_k|,
\]
and \(w:E(G)\to \mathbb{R}_{\ge 0}\) is a nonnegative edge-weight function. If \(n=|V(G)|\), then each class has size \(n/k\). For a permutation
\[
\sigma=(\sigma(0),\sigma(1),\dots,\sigma(k-1))
\]
of \([k]=\{1,\dots,k\}\), a cycle
\[
C=(v_0,v_1,\dots,v_r), \quad v_r=v_0
\]
is a \(\sigma\)-cycle if
\[
v_i \in V_{\sigma(i \bmod k)} \quad \text{for all } i.
\]
The weight of a path or cycle \(\pi\) is
\[
w(\pi)=\sum_{e\in E(\pi)} w(e),
\]
and the objective is to find a minimum-weight Hamiltonian cycle that is a \(\sigma\)-cycle for some permutation \(\sigma\). Equivalently,
\[
\arg\min_{\pi\in \mathcal{H}_G} w(\pi),
\]
where \(\mathcal{H}_G\) denotes the set of all Hamiltonian polychromatic cycles. A Hamiltonian polychromatic cycle is also called a \(\sigma\)-tour [2507.04974].

Two standard specializations are explicit. In **metric PCTSP**, \(G\) is complete and satisfies the triangle inequality
\[
w(u,v)\le w(u,x)+w(x,v)
\]
for all vertices \(u,v,x\). In **Euclidean PCTSP**, the vertices are points \(P\subset \mathbb{R}^d\), each color class is a subset \(P_i\), and weights are Euclidean distances,
\[
w((v_p,v_{p'}))=\|p-p'\|_2.
\]
The model interpolates between alternating-color routing and unconstrained Hamiltonian routing: if \(k=2\), the tour alternates between the two classes; if \(k=n\), each class has size \(1\), so the problem becomes ordinary TSP [2507.04974].

## 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 \(G=(V,E)\) with metric edge lengths \(c_e\), a distinguished root \(r\in V\), and nonnegative penalties \(\pi_v\) for vertices in \(V\setminus\{r\}\). The task is to find a cycle \(C=(V_C,E_C)\) containing \(r\) that minimizes
\[
\sum_{e\in E_C} c_e \;+\; \sum_{v\in V\setminus V_C} \pi_v.
\]
Here vertices may be omitted at a penalty, and the natural LP relaxation uses edge variables \(x_e\) and coverage variables \(y_v\) [2212.03776].

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 \(S=\{s_1,\dots,s_n\}\) of planar points partitioned into \(k\) color classes, and the goal is to find a polygon
\[
P=(p_1,p_2,\dots,p_k,p_1)
\]
of minimum perimeter such that \(p_i\in S\) for all \(i\) and \(C(p_i)\neq C(p_j)\) for all \(i\neq j\). In other words, one selects exactly one representative point from each color class and connects those selected points into the shortest possible cycle [2401.03297].

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 \(\sigma\), the metric approximation algorithm first writes, without loss of generality,
\[
\sigma=(1,2,\dots,k),
\]
and sets \(V_{k+1}=V_1\). For each adjacent pair of classes \((V_i,V_{i+1})\), it computes a minimum-weight perfect matching
\[
M_i = \textsc{MinMatching}(G,w,V_i,V_{i+1}).
\]
Since all classes have the same size and \(G\) is complete, such a matching exists. Let
\[
H = \left(V(G), \bigcup_{i=1}^k M_i\right).
\]
Each connected component of \(H\) is itself a \(\sigma\)-cycle. From each component \(C\), one chooses a representative vertex \(v_C\in V(C)\cap V_1\), lets \(V_1'\) denote the set of representatives, solves TSP on the induced subgraph \(G[V_1']\) using the \((1.5-10^{-36})\)-approximation for metric TSP, and then glues the components together according to that representative tour order. The fixed-order analysis uses two bounds:
\[
\sum_{i=1}^k \sum_{e\in M_i} w(e)\le \sum_{e\in E(R)} w(e)
\]
for any Hamiltonian \(\sigma\)-cycle \(R\), and
\[
\sum_{e\in E(S)} w(e)\le \sum_{i=1}^k \sum_{e\in M_i} w(e) + (1.5-10^{-36})\cdot \mathsf{tsp}(G)
\]
for the glued solution \(S\). Together with \(\mathsf{tsp}(G)\le \text{OPT}_{\text{PCTSP}}\), this yields a \((2.5-10^{-36})\)-approximation for a fixed \(\sigma\) [2507.04974].

The second step approximates the best cyclic order of the classes. For each pair \(i\neq j\), one computes a minimum matching \(M_{i,j}\) between \(V_i\) and \(V_j\), and defines a complete graph \(G'\) on the classes \(V_1,\dots,V_k\) with edge weight
\[
w'((V_i,V_j))=\sum_{e\in M_{i,j}} w(e).
\]
This graph is metric. For a permutation \(\sigma=(r_1,\dots,r_k)\), the quantity
\[
w_\sigma = \sum_{i=1}^k w'((V_{r_i},V_{r_{i+1}}))
\]
is exactly the weight of the corresponding Hamiltonian cycle in \(G'\). Applying the \((1.5-10^{-36})\)-approximation for metric TSP to \(G'\) gives an approximately optimal order \(\hat{\sigma}\), and feeding \(\hat{\sigma}\) into the fixed-order routine yields the final
\[
\boxed{(3-2\cdot 10^{-36})\text{-approximation}.}
\]
More generally, if matching can be computed in time \(M(n)\) and metric TSP can be \(\alpha\)-approximated in time \(T(n)\), then PCTSP admits a \(2\alpha\)-approximation with running time
\[
O\big(k^2 M(2n/k)+T(k)+T(n/k)\big)
\]
[2507.04974].

## 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 \(\sigma\) itself to encode a truth assignment. The construction introduces
\[
k=3n+1
\]
color classes
\[
R_1,\dots,R_n,\quad T_1,\dots,T_n,\quad F_1,\dots,F_n,\quad R_{n+1},
\]
together with a notion of **valid** permutation \(\sigma\) satisfying
\[
R_\alpha \prec_\sigma T_\alpha,\quad R_\alpha \prec_\sigma F_\alpha \qquad (1\le \alpha\le n)
\]
and
\[
T_\alpha \prec_\sigma R_\beta,\quad F_\alpha \prec_\sigma R_\beta \qquad (1\le \alpha<\beta\le n+1).
\]
These constraints leave exactly one binary choice per variable,
\[
T_\alpha \prec_\sigma F_\alpha \quad \text{or} \quad F_\alpha \prec_\sigma T_\alpha,
\]
so there are exactly \(2^n\) valid permutations, corresponding naturally to truth assignments [2507.04974].

For each clause \(c_i=x_\alpha\vee x_\beta\), a geometric gadget is created so that, for any valid \(\sigma\), the shortest \(\sigma\)-path through that gadget has cost
\[
c + 2a
\]
if \(\sigma\) satisfies the clause, and
\[
c + 2\sqrt{a^2+4}
\]
otherwise, where
\[
c = 5 + \sqrt{a^2+4} + \sqrt{a^2+1} + \sqrt{49a^2+1} + \sqrt{(18a)^2+1}.
\]
The full instance is \(P_\Pi=S\cup C\), where \(C\) is the union of clause gadgets and \(S\) is a line of auxiliary points, one in each color class, placed on the line \(y=-2W-1\). If a valid \(\sigma\) satisfies exactly \(k\) clauses, the optimal tour cost is
\[
\|T^*\| = 5W + (m-1)b + k(c+2a) + (m-k)(c+2\sqrt{a^2+4}).
\]
Two structural lemmas complete the reduction: if a valid \(\sigma\)-tour is short enough, then \(\sigma\) must satisfy many clauses; and if \(\sigma\) is invalid, it can be repaired without increasing cost. Hence Euclidean PCTSP in \(\mathbb{R}^2\) is APX-hard and does not admit a PTAS unless \(P=NP\) [2507.04974].

## 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
\[
x_1\cdot x_2\cdot \dots \cdot x_k
\]
representative selections, with worst-case complexity
\[
O\!\left(\left(\frac{n}{k}\right)^k\right),
\]
and an approximation algorithm based on the **minimum color-spanning circle** and **onion peeling**. If \(r\) is the radius of the smallest color-spanning circle, the circle can be computed in
\[
O(k^3 n \log n)
\]
time, each onion layer contributes at most \(2\pi r\), the worst case has \(k/3\) layers, and with the lower bound \(\text{perimeter}(P)\ge n\) for grid points, the paper states the approximation factor
\[
\boxed{\frac{2\pi r}{3}}.
\]
The overall running time is polynomial and dominated by the color-spanning-circle computation [2401.03297].

Another geometric relative is the **two-colored noncrossing Euclidean TSP**. In that problem, the input consists of red and blue terminals in \(\mathbb{R}^2\), and the solution is a pair of pairwise disjoint closed curves, one visiting all red terminals and one visiting all blue terminals, minimizing
\[
wt(\pi_1)+wt(\pi_2).
\]
The central constraint is noncrossing rather than cyclic color order. The Euclidean problem admits a randomized \((1+\varepsilon)\)-approximation scheme in
\[
2^{O(1/\varepsilon)}\, n\, \mathrm{polylog}(n)
\]
time, and this dependence on \(\varepsilon\) is Gap-ETH-tight. The key technical ingredient is a patching lemma for two noncrossing curves: given a segment \(S\) and noncrossing closed curves \(R,B\), there exist modified noncrossing curves \(R',B'\) such that
\[
|I(R',S)\cup I(B',S)| \le 10
\]
and
\[
wt(R')+wt(B') \le wt(R)+wt(B)+20\cdot wt(S),
\]
with modification confined to an infinitesimal neighborhood of \(S\). The same work also gives a PTAS for the problem in plane unweighted graphs [2209.08904].

## 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 [2507.04974] [2401.03297] [2212.03776] [2209.08904].

| Variant | Defining constraint | Objective |
|---|---|---|
| Polychromatic TSP | Visit all vertices in a cyclic class order | Minimum-weight Hamiltonian \(\sigma\)-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.

Source: https://www.emergentmind.com/topics/polychromatic-traveling-salesman-problem-pctsp