A Neutrosophic Extension of Randomized Rounding Approaches to the Traveling Salesman Problem

1. Introduction

The Traveling Salesman Problem is a fundamental problem in combinatorial optimization. Given a collection of vertices and edge costs, the objective is to find a minimum-cost tour visiting every vertex. The problem is computationally difficult, and approximation algorithms play an important role in obtaining efficient solutions with provable guarantees [4,5].

For metric instances, Christofides’ algorithm gives a $3/2$-approximation [1]. For the graphic TSP, Gharan, Saberi, and Singh developed a randomized rounding approach that achieves an approximation ratio strictly below $3/2$ [2]. Their method combines linear programming, random spanning trees, and minimum-cost T-joins.

In practical applications, however, edge costs may not be known precisely. Travel times can depend on traffic, measurements may be incomplete, and different sources may provide conflicting estimates. Fuzzy-set and neutrosophic-set approaches provide mathematical languages for representing aspects of such information [6,8].

The purpose of this paper is to formulate a neutrosophic extension of the randomized-rounding framework. We do not reinterpret the original approximation theorem as an already proved neutrosophic result. Instead, we identify a scalarized optimization model, establish its elementary properties, and state the additional hypotheses needed for approximation guarantees.

The main contributions are as follows.

label=()

We define neutrosophic edge data consisting of a cost interval and three assessment degrees.

lbbel=()

We construct a scalarized edge-cost function and formulate a corresponding subtour-elimination linear program.

lcbel=()

We describe a randomized rounding framework based on spanning trees and T-joins.

ldbel=()

We give a conditional approximation theorem that separates the classical combinatorial argument from the additional assumptions required by the neutrosophic model.

2. Preliminaries

2.1. Graphs and Tours

Let $G=(V,E)$ be a finite, connected, undirected graph, where V is the vertex set and E is the edge set. For a subset $S\subseteq V$, let $\delta \left(S\right)=\{uv\in E:u\in S,v\in V\setminus S\}$ denote the cut determined by S.

For $F\subseteq E$ and $v\in V$, write ${deg}_F\left(v\right)$ for the degree of v in the edge set F.

A connected Eulerian multigraph on V admits an Eulerian circuit. Consequently, if a spanning tree is augmented by edges so that every vertex has even degree, the resulting connected multigraph contains a closed walk visiting every vertex. This is the basic tree-augmentation principle used in classical graphic-TSP algorithms [1,2].

2.2. The Subtour-Elimination Relaxation

For a nonnegative edge-cost function $c:E\to {\mathbb{R}}_{\ge 0}$, the standard subtour-elimination relaxation is

$$\begin{array}{cc}\hfill min& \sum _{e\in E}{c}_e{x}_e\hfill \\ \hfill \mathrm{subject}\mathrm{to}& x\left(\delta \right(v\left)\right)=2,\hfill & & v\in V,\hfill \\ & x\left(\delta \right(S\left)\right)\ge 2,\hfill & & \varnothing \ne S⊊V,\hfill \\ & x_e\ge 0,\hfill & & e\in E,\hfill \end{array}$$

where

$$x\left(F\right)=\sum _{e\in F}{x}_e.$$

This linear program is a relaxation of the tour problem: the incidence vector of every Hamiltonian cycle is feasible. Its relationship with the TSP and minimum spanning trees is classical [3,5].

3. Neutrosophic Edge-Cost Data

3.1. Neutrosophic Assessments

We use the three components of a neutrosophic assessment: truth, indeterminacy, and falsity. The interpretation of these components depends on the application; here they describe support for an edge-cost assessment, unresolved information, and conflicting or adverse information, respectively.

Definition 1.

A neutrosophic cost datum for an edge $e\in E$ is a tuple

$${\mathcal{N}}_e=({\ell }_e,u_e,T_e,I_e,F_e),$$

where

$$0\le {\ell }_e\le u_e<\infty ,T_e,I_e,F_e\in [0,1].$$

The interval $[{\ell }_e,u_e]$ represents a range of plausible costs, while $T_e,I_e,F_e$ are the associated neutrosophic degrees.

We do not require

$$T_e+I_e+F_e=1.$$

This permits the three degrees to be specified independently, as in the general neutrosophic framework [6].

Remark 1.

The degrees $T_e,I_e,F_e$ are not probabilities unless a probabilistic interpretation and the corresponding axioms are imposed separately. In particular, they should not automatically be used as probabilities in a randomized algorithm.

3.2. Scalarization

To obtain a conventional optimization problem, we specify a scalarization rule.

Definition 2.

Fix parameters

$$\alpha ,\beta ,\gamma \ge 0,\alpha +\beta +\gamma =1.$$

For each edge e, define its effective neutrosophic cost by

$$c_e^{\mathcal{N}}=\alpha T_e{\ell }_e+\beta I_e{\displaystyle \frac{{\ell }_e+u_e}{2}}+\gamma F_e{u}_e.$$

The coefficients represent the decision-maker’s chosen weighting of the three components. They are model parameters, not universal constants.

Proposition 1.

For every edge $e\in E$, the scalarized cost satisfies

$$0\le c_e^{\mathcal{N}}\le u_e.$$

Moreover, if ${\ell }_e=u_e=q_e$, then

$$c_e^{\mathcal{N}}=q_e(\alpha T_e+\beta I_e+\gamma F_e).$$

Proof. 

All terms in (3.1) are nonnegative. Since

$${\ell }_e\le {\displaystyle \frac{{\ell }_e+u_e}{2}}\le u_e,$$

we have

$$\begin{array}{cc}\hfill c_e^{\mathcal{N}}& \le \alpha T_e{u}_e+\beta I_e{u}_e+\gamma F_e{u}_e\hfill \\ & =u_e(\alpha T_e+\beta I_e+\gamma F_e)\hfill \\ & \le u_e,\hfill \end{array}$$

because each degree is at most one and the weights sum to one. The second assertion follows by substituting ${\ell }_e=u_e=q_e$ into (3.1). □

Remark 2.

The scalarization is one possible modeling choice. Other applications may prefer a worst-case cost, a risk-sensitive objective, or a multiobjective formulation. Approximation guarantees depend on the chosen objective and its assumptions.

4. The Neutrosophic Subtour Relaxation

Let $c_e^{\mathcal{N}}$ be given by (3.1). We define the scalarized neutrosophic subtour relaxation by

$$\begin{array}{cc}\hfill min& \sum _{e\in E}{c}_e^{\mathcal{N}}{x}_e\hfill \\ \hfill \mathrm{subject}\mathrm{to}& x\left(\delta \right(v\left)\right)=2,\hfill & & v\in V,\hfill \\ & x\left(\delta \right(S\left)\right)\ge 2,\hfill & & \varnothing \ne S⊊V,\hfill \\ & x_e\ge 0,\hfill & & e\in E.\hfill \end{array}$$

Let $x^{\mathcal{N}}$ be an optimal solution and let $L_{\mathcal{N}}$=${\sum }_{e\in E}{c}_e^{\mathcal{N}}{x}_e^{\mathcal{N}}$ be its objective value.

Proposition 2.

If all effective costs are nonnegative, then the objective value of (4.1) is a lower bound on the minimum cost of a Hamiltonian cycle, whenever the graph has a Hamiltonian cycle.

Proof. 

The incidence vector of every Hamiltonian cycle satisfies all constraints of (4.1). Since (4.1) minimizes over a feasible region containing all such incidence vectors, its optimal value cannot exceed the minimum Hamiltonian-cycle cost. □

Remark 3.

For graphic TSP, the target is a shortest closed walk visiting every vertex, with edge lengths inherited from the graph. The distinction between a Hamiltonian cycle and a graphic-TSP tour is important: a graphic-TSP tour may repeat vertices and edges. The relaxation and rounding construction must be matched to the precise problem being studied.

5. Randomized Spanning Trees

The randomized rounding method of Gharan, Saberi, and Singh uses a distribution over spanning trees and combines the sampled tree with a minimum-cost Eulerian augmentation [2].

We formulate the corresponding neutrosophic-cost component. Let $\mathcal{T}\left(G\right)$ denote the set of spanning trees of G.

Definition 3.

A probability distribution μ on $\mathcal{T}\left(G\right)$ is called a spanning-tree distribution with prescribed marginals $p={\left(p_e\right)}_{e\in E}$ if

$${\mathbb{P}}_{T\sim \mu }(e\in T)=p_e(e\in E).$$

The existence of such a distribution is a substantive constraint. An arbitrary feasible solution of (4.1) need not itself be the marginal vector of a spanning-tree distribution. Thus a rounding procedure must either establish the required marginal property or use a suitable alternative construction.

5.1. Maximum-Entropy Formulation

When a prescribed marginal vector is realizable by a spanning-tree distribution, one may seek a distribution maximizing Shannon entropy:

$$H\left(\mu \right)=-\sum _{T\in \mathcal{T}\left(G\right)}\mu \left(T\right)log\mu \left(T\right),$$

subject to

$$\sum _{T\ni e}\mu \left(T\right)=p_e(e\in E),$$

and

$$\sum _{T\in \mathcal{T}\left(G\right)}\mu \left(T\right)=1,\mu \left(T\right)\ge 0.$$

This formulation is meaningful only when the constraints are feasible. Maximum-entropy distributions and their computational properties have been studied extensively [2,7].

Proposition 3.

If the prescribed marginals are realizable, the maximum-entropy problem has an optimal solution.

Proof. 

The set of probability distributions on the finite set $\mathcal{T}\left(G\right)$ is compact. The marginal constraints define a closed subset of this set. By assumption, that subset is nonempty. Shannon entropy is continuous on the probability simplex, so it attains a maximum on the feasible set. □

6. Eulerian Augmentation and T-joins

Let $T_0$ be a sampled spanning tree. Define its odd-degree vertex set by

$$Q\left(T_0\right)=\{v\in V:{deg}_{T_0}\left(v\right)\mathrm{is}\mathrm{odd}\}.$$

The set $Q\left(T_0\right)$ has even cardinality, by the handshaking lemma.

A $Q\left(T_0\right)$-join is an edge multiset whose odd-degree vertices are exactly the vertices in $Q\left(T_0\right)$. A minimum-cost $Q\left(T_0\right)$-join can be used to make the tree Eulerian.

For a nonnegative effective cost function $c^{\mathcal{N}}$, let

$$J_{\mathcal{N}}\left(T_0\right)=min\left\{\sum _{e\in E}{c}_e^{\mathcal{N}}{y}_e:y\mathrm{is}\mathrm{a}Q\left(T_0\right)-\mathrm{join}\right\}.$$

The resulting connected multigraph has even degree at every vertex and therefore admits an Eulerian circuit. This gives a closed walk visiting every vertex.

The minimum-cost T-join problem is a classical polynomial-time combinatorial optimization problem; its polyhedral properties are central to the analysis of graphic-TSP algorithms [2,5].

7. A Conditional Approximation Statement

We now state a result that makes explicit the hypotheses needed to transfer a classical approximation argument to the scalarized neutrosophic model.

Theorem 1.

Let G be a connected graph with nonnegative effective neutrosophic costs $c^{\mathcal{N}}$. Suppose a randomized rounding algorithm produces a feasible graphic-TSP tour W and satisfies

$$\mathbb{E}\left[{cost}_{\mathcal{N}}\left(W\right)\right]\le \rho L_{\mathcal{N}}$$

for some constant $\rho \ge 1$, where $L_{\mathcal{N}}$ is the optimum of (4.1). Then

$$\mathbb{E}\left[{cost}_{\mathcal{N}}\left(W\right)\right]\le \rho {OPT}_{\mathcal{N}},$$

where ${OPT}_{\mathcal{N}}$ is the minimum cost of a graphic-TSP tour under the effective costs.

Proof. 

The subtour relaxation is a lower bound on the optimal tour cost, so

$$L_{\mathcal{N}}\le {OPT}_{\mathcal{N}}.$$

Multiplying by $\rho \ge 0$ and applying (7.1) gives

$$\mathbb{E}\left[{cost}_{\mathcal{N}}\left(W\right)\right]\le \rho L_{\mathcal{N}}\le \rho {OPT}_{\mathcal{N}}.$$

□

Remark 4.

The theorem is conditional: it does not prove (7.1) for a new neutrosophic rounding algorithm. Establishing that inequality requires a complete analysis of the distribution over spanning trees, the augmentation cost, and the relationship between the relaxation and the target problem.

7.1. Relation to the Classical graphic-TSP Guarantee

Gharan, Saberi, and Singh prove that their algorithm achieves a ratio of $3/2-{\epsilon }_0$ for some constant ${\epsilon }_0>0$ on graphic-TSP instances [2].

A neutrosophic formulation with fixed scalarized costs is still an ordinary weighted graph optimization problem. However, the original guarantee applies only when the hypotheses and objective of the classical result are met. In particular, a change in the rounding distribution, the cost interpretation, or the target objective requires a separate proof.

8. A Small Illustrative Example

Consider a triangle with vertex set

$$V=\{v_1,v_2,v_3\}$$

and edge set

$$E=\{v_1{v}_2,v_2{v}_3,v_1{v}_3\}.$$

Suppose all three edges have interval $[{\ell }_e,u_e]=[2,4]$ and degrees

$$(T_e,I_e,F_e)=(0.8,0.1,0.2).$$

Choose

$$(\alpha ,\beta ,\gamma )=(0.5,0.3,0.2).$$

Then

$$\begin{array}{cc}\hfill c_e^{\mathcal{N}}& =0.5\left(0.8\right)\left(2\right)+0.3\left(0.1\right){\displaystyle \frac{2+4}{2}}+0.2\left(0.2\right)\left(4\right)\hfill \\ & =0.8+0.09+0.16\hfill \\ & =1.05.\hfill \end{array}$$

Thus each edge has effective cost $1.05$ under this particular scalarization. The triangle itself is a tour, with effective cost

$${cost}_{\mathcal{N}}\left(W\right)=3\left(1.05\right)=3.15.$$

This example illustrates the conversion of neutrosophic edge information into a conventional objective. It does not demonstrate an approximation improvement over a classical TSP algorithm.

9. Algorithmic Framework

The following is a high-level framework. Its approximation guarantee depends on the feasibility of the tree-distribution step and on a proof of the expected-cost bound.

label=

Input. Take a connected graph $G=(V,E)$ and neutrosophic edge data ${\mathcal{N}}_e=({\ell }_e,u_e,T_e,I_e,F_e)$.

lbbel=

Scalarization. Choose nonnegative weights $\alpha ,\beta ,\gamma$ summing to one, and compute $c_e^{\mathcal{N}}$ using (3.1).

lcbel=

Relaxation. Solve the linear program (4.1) to obtain an optimal fractional solution.

ldbel=

Tree selection. Construct a feasible distribution over spanning trees with the required properties, and sample a tree $T_0$.

lebel=

Augmentation. Compute a minimum-cost $Q\left(T_0\right)$-join with respect to $c^{\mathcal{N}}$.

lfbel=

Tour construction. Combine the tree and the join, and output an Eulerian circuit.

The principal unresolved algorithmic issue is Step 4: the existence and properties of the required distribution must be proved for the chosen relaxation and graph class. The expected augmentation cost must then be bounded in terms of $L_{\mathcal{N}}$.

10. Python Implementation of the Neutrosophic TSP Heuristic

This section describes a Python implementation of a heuristic for the neutrosophic Traveling Salesman Problem. The algorithm transforms neutrosophic edge data into nonnegative scalar costs and then constructs a closed walk using a minimum spanning tree and a minimum-weight matching.

10.1. Neutrosophic Edge Costs

For each edge $e\in E$, let

$${\mathcal{N}}_e=({\ell }_e,u_e,T_e,I_e,F_e),$$

where $[{\ell }_e,u_e]$ is the cost interval and $T_e,I_e,F_e\in [0,1]$ are the truth, indeterminacy, and falsity degrees, respectively.

We define the effective edge cost by

$$c_e^{\mathcal{N}}=\alpha T_e{\ell }_e+\beta I_e{\displaystyle \frac{{\ell }_e+u_e}{2}}+\gamma F_e{u}_e,$$

where

$$\alpha ,\beta ,\gamma \ge 0,\alpha +\beta +\gamma =1.$$

10.2. Description of the Algorithm

Let $G=(V,E)$ be a finite, connected, undirected graph. The algorithm proceeds as follows.

1.

Input. For each edge $e\in E$, specify the interval $[{\ell }_e,u_e]$ and the neutrosophic degrees $T_e,I_e,F_e$.

2.

Scalarization. Compute the effective cost $c_e^{\mathcal{N}}$ for every edge using the formula above.

3.

Minimum spanning tree. Construct a minimum spanning tree S of G with respect to the effective costs.

4.

Odd-degree vertices. Determine the set

$$Q=\{v\in V:{deg}_S\left(v\right)\mathrm{is}\mathrm{odd}\}.$$

5.

Shortest-path distances. For every pair of vertices in Q, compute the shortest-path distance in G using the effective costs.

6.

Minimum-weight matching. Construct a complete weighted graph on Q and find a minimum-weight perfect matching.

7.

Eulerian augmentation. Add the shortest paths corresponding to the matching to the spanning tree, retaining repeated edges.

8.

Tour construction. Find an Eulerian circuit in the resulting connected multigraph. The circuit gives a closed walk visiting every vertex of G.

9.

Output. Return the closed walk and its total effective cost.

10.3. Pseudocode

The procedure can be summarized by the following mathematical pseudocode.

Algorithm: Neutrosophic MST–Matching Heuristic

Input: A connected undirected graph $G=(V,E)$ with neutrosophic edge data.

Output: A closed walk W and its cost.

1.

Compute $c_e^{\mathcal{N}}$ for every $e\in E$.

2.

Find a minimum spanning tree S of G.

3.

Compute the odd-degree vertex set Q of S.

4.

Find a minimum-weight perfect matching on Q, using shortest-path distances in G.

5.

Add the matching paths to S.

6.

Find an Eulerian circuit W in the augmented multigraph.

7.

Return W and

$${cost}_{\mathcal{N}}\left(W\right)=\sum _{e\in E}{c}_e^{\mathcal{N}}{m}_e\left(W\right),$$

where $m_e\left(W\right)$ is the number of times the walk traverses edge e.

10.4. Remarks on the Implementation

The algorithm can be implemented in Python using the NetworkX library. The required operations include minimum spanning tree computation, shortest-path computation, minimum-weight perfect matching, and Eulerian circuit construction.

The procedure is a deterministic MST-plus-matching heuristic. It is not the maximum-entropy randomized rounding algorithm of Gharan, Saberi, and Singh [2]. Therefore, the approximation guarantee of that paper does not automatically apply to the present procedure.

The output is a closed walk, and vertices or edges may be repeated. The method is intended for the graphic Traveling Salesman Problem, in which repeated vertices and edges are permitted.

11. Discussion of Some Aspects

The proposed model suggests the following considerations.

11.1. Alternative Scalarizations

The scalarization introduced in the preceding sections provides a convenient way to transform neutrosophic edge information into a single effective cost. However, the choice of scalarization may substantially influence the resulting optimal tour. In particular, different applications may assign different levels of importance to the lower and upper cost estimates, the expected cost, and the truth, indeterminacy, and falsity degrees.

In this subsection, we present several alternative scalarization strategies for the neutrosophic Traveling Salesman Problem.

Let $G=(V,E)$ be a finite, connected, undirected graph. For each edge $e\in E$, let

$${\mathcal{N}}_e=({\ell }_e,u_e,T_e,I_e,F_e),$$

where

$$0\le {\ell }_e\le u_e<\infty ,T_e,I_e,F_e\in [0,1].$$

11.1.1. Weighted Linear Scalarization

A natural alternative is to assign independent nonnegative weights to the lower cost, the midpoint of the interval, and the upper cost. Let

$$\alpha ,\beta ,\gamma \ge 0,\alpha +\beta +\gamma =1.$$

We define

$$c_e^{\left(1\right)}=\alpha T_e{\ell }_e+\beta I_e{\displaystyle \frac{{\ell }_e+u_e}{2}}+\gamma F_e{u}_e.$$

This scalarization is simple to implement and preserves the interpretation of the three neutrosophic components. The parameters may be selected according to the requirements of the application.

For a closed walk W, let $m_e\left(W\right)$ denote the number of times that W traverses edge e. The corresponding objective is

$$\underset{W\in \mathcal{W}\left(G\right)}{min}\sum _{e\in E}{m}_e\left(W\right)c_e^{\left(1\right)},$$

where $\mathcal{W}\left(G\right)$ is the set of feasible closed walks visiting every vertex.

11.1.2. Convex Interval Scalarization

A second approach is to combine the lower and upper endpoints directly. For a parameter $\lambda \in [0,1]$, define

$$c_e^{\left(2\right)}\left(\lambda \right)=\lambda {\ell }_e+(1-\lambda )u_e.$$

When $\lambda$ is close to 1, the scalarized cost places greater emphasis on the lower endpoint. When $\lambda$ is close to 0, it places greater emphasis on the upper endpoint.

This formulation is useful when the interval endpoints are considered more reliable than the neutrosophic degrees, or when one wishes to study the sensitivity of the optimal tour to the degree of conservatism.

11.1.3. Indeterminacy-Penalized Scalarization

In applications where unresolved information is undesirable, one may explicitly penalize the indeterminacy degree. Let $\eta \ge 0$ be a penalty parameter and define

$$c_e^{\left(3\right)}={\displaystyle \frac{{\ell }_e+u_e}{2}}+\eta I_e.$$

The first term represents the midpoint estimate, while the second term increases the cost of edges whose information is more indeterminate.

If $\eta =0$, the indeterminacy degree has no effect on the objective. Increasing $\eta$ makes the optimization more sensitive to indeterminate edge information.

This scalarization is appropriate when $I_e$ measures the lack of reliability of the cost estimate. It should not be interpreted as a probabilistic risk penalty unless a corresponding probabilistic model has been specified.

11.1.4. Falsity-Sensitive Scalarization

If the falsity degree represents adverse or contradictory evidence concerning an edge, one may define

$$c_e^{\left(4\right)}={\ell }_e+\theta F_e(u_e-{\ell }_e),\theta \ge 0.$$

For $\theta \in [0,1]$, this expression lies between ${\ell }_e$ and $u_e$. A larger value of $F_e$ increases the effective cost when the interval has positive width.

This construction is particularly useful when the falsity degree is interpreted as an indicator of adverse evidence. The interpretation must be specified by the application, since a neutrosophic falsity degree is not automatically a probability of failure.

11.1.5. Robust Worst-Case Scalarization

A conservative alternative is to use the upper endpoint:

$$c_e^{\left(5\right)}=u_e.$$

The resulting optimization problem is

$$\underset{W\in \mathcal{W}\left(G\right)}{min}\sum _{e\in E}{m}_e\left(W\right)u_e.$$

This objective minimizes the worst-case total cost when the uncertainty set is the Cartesian product

$$\mathcal{U}=\prod _{e\in E}[{\ell }_e,u_e],$$

so that every combination of edge costs within the specified intervals is allowed.

Indeed, for any fixed walk W,

$$\underset{C\in \mathcal{U}}{sup}\sum _{e\in E}{m}_e\left(W\right)C_e=\sum _{e\in E}{m}_e\left(W\right)u_e,$$

because $m_e\left(W\right)\ge 0$ for every edge.

This formulation provides a direct robust-optimization interpretation. It may, however, produce more conservative solutions than objectives based on expected costs or interval midpoints.

11.1.6. Multiobjective Formulation

Instead of combining all information into a single scalar, one may retain several objectives. For example, define

$$\begin{array}{cc}\hfill f_1\left(W\right)& =\sum _{e\in E}{m}_e\left(W\right){\ell }_e,\hfill \\ \hfill f_2\left(W\right)& =\sum _{e\in E}{m}_e\left(W\right)I_e,\hfill \\ \hfill f_3\left(W\right)& =\sum _{e\in E}{m}_e\left(W\right)F_e{u}_e.\hfill \end{array}$$

The resulting problem is

$$\underset{W\in \mathcal{W}\left(G\right)}{min}\left(f_1\left(W\right),f_2\left(W\right),f_3\left(W\right)\right),$$

where minimization is understood in the Pareto sense.

A feasible walk $W_1$ dominates another feasible walk $W_2$ if

$$f_j\left(W_1\right)\le f_j\left(W_2\right)\mathrm{for}\mathrm{every}j,$$

and the inequality is strict for at least one objective. A Pareto-optimal walk is one that is not dominated by any other feasible walk.

The multiobjective formulation avoids fixing the relative importance of the different criteria in advance. Its computational treatment is more involved, since the objective is to characterize a set of nondominated solutions rather than a single optimal tour.

11.1.7. Comparison and Choice of Scalarization

The preceding scalarizations reflect different modeling priorities. The weighted linear form allows several kinds of information to be combined in one objective. The convex interval form emphasizes the cost range, while the indeterminacy-penalized and falsity-sensitive forms assign explicit penalties to particular neutrosophic components. The robust formulation focuses on worst-case costs, whereas the multiobjective formulation preserves competing criteria.

No scalarization is universally preferable. Its suitability depends on the meaning of the edge data and the decision maker’s objectives. In particular, the truth, indeterminacy, and falsity degrees should not be interpreted as probabilities unless additional probabilistic assumptions justify that interpretation.

Finally, any approximation guarantee for a randomized rounding algorithm must be established for the selected scalarized objective. A guarantee proved for one cost function does not automatically transfer to another.

11.2. Distributional Uncertainty

In many practical applications of the Traveling Salesman Problem, the cost associated with an edge is not known precisely in advance. For example, travel times may depend on traffic conditions, fuel consumption may vary, and information obtained from different sources may be incomplete or inconsistent. In such situations, it is natural to incorporate a probability distribution for the uncertain edge costs.

Let $G=(V,E)$ be a finite, connected, undirected graph. For each edge $e\in E$, let $C_e$ be a nonnegative random variable representing its uncertain traversal cost. We assume that $C_e$ has a probability distribution ${\mathbb{P}}_e$ supported on the interval $[{\ell }_e,u_e]$, where

$$0\le {\ell }_e\le u_e<\infty .$$

The expected cost of edge e is denoted by

$${\mu }_e=\mathbb{E}\left[C_e\right].$$

The random variables ${\left(C_e\right)}_{e\in E}$ need not be independent. Their joint distribution may encode correlations between edge costs arising, for instance, from common traffic conditions or shared environmental factors.

To combine distributional uncertainty with the neutrosophic description introduced above, associate with each edge the neutrosophic degrees

$$(T_e,I_e,F_e)\in {[0,1]}^3.$$

Here, $T_e$ represents the degree of support for the available cost information, $I_e$ represents the degree of unresolved or incomplete information, and $F_e$ represents the degree of conflicting or adverse information. These degrees describe the status of the information and are not, in general, probabilities.

For fixed parameters

$$\alpha ,\beta ,\gamma \ge 0,\alpha +\beta +\gamma =1,$$

we define the distributionally adjusted neutrosophic cost by

$${\widehat{c}}_e^{\mathcal{N}}=\alpha T_e{\ell }_e+\beta I_e\mathbb{E}\left[C_e\right]+\gamma F_e{u}_e.$$

Equivalently, since $\mathbb{E}\left[C_e\right]={\mu }_e$, we may write

$${\widehat{c}}_e^{\mathcal{N}}=\alpha T_e{\ell }_e+\beta I_e{\mu }_e+\gamma F_e{u}_e.$$

This expression combines a lower cost estimate, the expected cost, and an upper cost estimate. The parameters $\alpha ,\beta ,\gamma$ determine the relative importance assigned to these three contributions.

For a closed walk W, let $m_e\left(W\right)$ denote the number of times that W traverses edge e. Its random total cost is

$$C\left(W\right)=\sum _{e\in E}{m}_e\left(W\right)C_e.$$

By linearity of expectation,

$$\mathbb{E}\left[C\left(W\right)\right]=\sum _{e\in E}{m}_e\left(W\right)\mathbb{E}\left[C_e\right]=\sum _{e\in E}{m}_e\left(W\right){\mu }_e.$$

This identity does not require independence among the edge-cost random variables.

Consequently, minimizing expected tour cost leads to the objective

$$\underset{W\in \mathcal{W}\left(G\right)}{min}\sum _{e\in E}{m}_e\left(W\right){\mu }_e,$$

where $\mathcal{W}\left(G\right)$ denotes the collection of feasible closed walks visiting every vertex. If the neutrosophic scalarization is used instead, the corresponding objective is

$$\underset{W\in \mathcal{W}\left(G\right)}{min}\sum _{e\in E}{m}_e\left(W\right){\widehat{c}}_e^{\mathcal{N}}.$$

It is important to distinguish minimization of expected cost from minimization of worst-case cost. The latter is expressed by

$$\underset{W\in \mathcal{W}\left(G\right)}{min}\underset{C\in \mathcal{U}}{sup}\sum _{e\in E}{m}_e\left(W\right)C_e,$$

where $\mathcal{U}$ is a specified uncertainty set for the edge-cost vector. If only the individual bounds ${\ell }_e\le C_e\le u_e$ are known and all combinations are allowed, this becomes

$$\underset{W\in \mathcal{W}\left(G\right)}{min}\sum _{e\in E}{m}_e\left(W\right)u_e.$$

Thus, the distributional model and the neutrosophic model serve different purposes. Probability distributions describe random variation in costs, whereas neutrosophic degrees represent the assessment of available information. Their combination provides a flexible modeling framework, but any approximation guarantee for a randomized rounding algorithm must be proved for the particular objective and assumptions being used.

11.3. Approximation Analysis

In this subsection, we analyze the approximation performance of the proposed neutrosophic MST–matching heuristic. The analysis is carried out with respect to the scalarized neutrosophic edge costs introduced in the preceding sections. We distinguish the classical approximation guarantee from the stronger guarantees that may be obtained by specialized randomized rounding procedures.

11.3.1. The Scalarized Optimization Problem

Let $G=(V,E)$ be a finite, connected, undirected graph. For each edge $e\in E$, let

$${\mathcal{N}}_e=({\ell }_e,u_e,T_e,I_e,F_e),$$

where

$$0\le {\ell }_e\le u_e<\infty ,T_e,I_e,F_e\in [0,1].$$

Fix nonnegative scalarization parameters $\alpha ,\beta ,\gamma$ satisfying

$$\alpha +\beta +\gamma =1.$$

The effective cost of an edge is

$$c_e^{\mathcal{N}}=\alpha T_e{\ell }_e+\beta I_e{\displaystyle \frac{{\ell }_e+u_e}{2}}+\gamma F_e{u}_e.$$

In particular,

$$c_e^{\mathcal{N}}\ge 0\mathrm{for}\mathrm{every}e\in E.$$

Let $\mathcal{W}\left(G\right)$ denote the collection of closed walks in G that visit every vertex at least once. For $W\in \mathcal{W}\left(G\right)$, let $m_e\left(W\right)$ be the number of times that W traverses edge e. Its scalarized cost is

$${cost}_{\mathcal{N}}\left(W\right)=\sum _{e\in E}{m}_e\left(W\right)c_e^{\mathcal{N}}.$$

The optimal graphic-TSP value is

$${OPT}_{\mathcal{N}}\left(G\right)=\underset{W\in \mathcal{W}\left(G\right)}{min}{cost}_{\mathcal{N}}\left(W\right).$$

The objective is to construct a feasible closed walk whose cost is bounded by a constant multiple of ${OPT}_{\mathcal{N}}\left(G\right)$.

11.3.2. Metric Closure

Let $d_{\mathcal{N}}(u,v)$ denote the shortest-path distance between vertices $u,v\in V$, computed using the effective edge costs $c_e^{\mathcal{N}}$. Since G is connected and the edge costs are nonnegative, these distances are finite and satisfy the triangle inequality.

The metric closure of G is the complete graph $K_V$ on vertex set V, with edge weights $d_{\mathcal{N}}(u,v)$. A Hamiltonian cycle in this metric closure can be expanded into shortest paths in the original graph. The resulting closed walk has the same total cost and visits every vertex.

Consequently, the graphic-TSP problem with effective costs can be treated as a metric-TSP problem on the metric closure.

11.3.3. The MST–Matching Algorithm

Let S be a minimum spanning tree of G with respect to the effective edge costs. Define its odd-degree vertex set by

$$Q=\{v\in V:{deg}_S\left(v\right)\mathrm{is}\mathrm{odd}\}.$$

The handshaking lemma implies that $\left|Q\right|$ is even.

Construct the complete graph on Q, using $d_{\mathcal{N}}(u,v)$ as the weight of the edge $uv$. Let M be a minimum-weight perfect matching in this graph. Expand every matching edge into a corresponding shortest path in G, and add these paths to the spanning tree S.

The resulting multigraph is connected and has even degree at every vertex. Hence, it admits an Eulerian circuit. This circuit defines a feasible closed walk for the graphic-TSP problem.

11.3.4. Approximation Guarantee

Theorem 2.

Let $G=(V,E)$ be a finite, connected, undirected graph with nonnegative effective neutrosophic edge costs. Suppose that the MST–matching algorithm computes an exact minimum spanning tree and an exact minimum-weight perfect matching on the odd-degree vertices, using shortest-path distances induced by the effective costs.

Then the resulting closed walk W satisfies

$${cost}_{\mathcal{N}}\left(W\right)\le {\displaystyle \frac{3}{2}}{OPT}_{\mathcal{N}}\left(G\right).$$

Thus, the algorithm is a ${\displaystyle \frac{3}{2}}$-approximation algorithm for the scalarized graphic-TSP problem.

Proof. 

Let $W^{*}$ be an optimal graphic-TSP walk. Since $W^{*}$ visits every vertex, its traversed edges contain a connected spanning subgraph. Removing edges from this subgraph until a spanning tree remains cannot increase the total cost, because all effective edge costs are nonnegative. Therefore, the minimum spanning tree satisfies

$$c^{\mathcal{N}}\left(S\right)\le {OPT}_{\mathcal{N}}\left(G\right).$$

We next bound the cost of the matching. Consider the metric closure of G and an optimal Hamiltonian cycle in that closure. The vertices in Q have even cardinality. The optimal cycle can be divided into two alternating sets of edges. Each of these sets induces a matching on the vertices of the cycle; after restricting to the vertices in Q and using the triangle inequality to shortcut any intervening vertices, we obtain a perfect matching on Q from each alternating choice.

The total cost of the two resulting matchings is at most the cost of the optimal cycle. Since M is a minimum-weight perfect matching, it follows that

$$c^{\mathcal{N}}\left(M\right)\le {\displaystyle \frac{1}{2}}{OPT}_{\mathcal{N}}\left(G\right).$$

Adding the spanning tree and the matching paths produces a connected Eulerian multigraph. Its Eulerian circuit has cost equal to the sum of the costs of the tree and the matching paths. Combining (1) and (2), we obtain

$$\begin{array}{cc}\hfill {cost}_{\mathcal{N}}\left(W\right)& \le c^{\mathcal{N}}\left(S\right)+c^{\mathcal{N}}\left(M\right)\hfill \\ & \le {OPT}_{\mathcal{N}}\left(G\right)+{\displaystyle \frac{1}{2}}{OPT}_{\mathcal{N}}\left(G\right)\hfill \\ & ={\displaystyle \frac{3}{2}}{OPT}_{\mathcal{N}}\left(G\right).\hfill \end{array}$$

This proves the theorem. □

11.3.5. Interpretation of the Guarantee

The theorem establishes a worst-case approximation bound for the scalarized objective. Its validity follows from the nonnegativity of the effective costs and the MST–matching construction. The neutrosophic components influence the solution through the scalarization, but they do not alter the structure of the approximation argument.

In particular, the guarantee applies to the effective cost function $c^{\mathcal{N}}$ and does not automatically imply a corresponding guarantee for each of the original quantities ${\ell }_e$, $u_e$, $T_e$, $I_e$, and $F_e$ considered separately.

11.3.6. Relation to Randomized Rounding

The approximation result above concerns the deterministic MST–matching heuristic. It should be distinguished from the randomized rounding method of Gharan, Saberi, and Singh [2], which uses a more specialized analysis of linear programming solutions and distributions over spanning trees.

Their improved approximation guarantee does not follow merely from replacing ordinary edge costs by scalarized neutrosophic costs. To obtain a comparable improvement, one would need to establish the relevant properties of the randomized tree distribution and prove the required expected-cost bounds for the chosen neutrosophic objective.

Thus, the present analysis provides a rigorous ${\displaystyle \frac{3}{2}}$-approximation guarantee for the MST–matching construction, while improved guarantees for a genuinely randomized neutrosophic algorithm remain a separate research problem.

11.4. Computational Experiments

The purpose of the computational experiments is to investigate how neutrosophic edge information affects the quality and computational performance of solutions to the Traveling Salesman Problem (TSP). In particular, we examine the influence of truth, indeterminacy, and falsity degrees on the scalarized edge costs and on the resulting tour.

11.4.1. Experimental Setup

Let $G=(V,E)$ be a connected undirected graph, and let ${\mathcal{N}}_e=({\ell }_e,u_e,T_e,I_e,F_e)$ denote the neutrosophic information associated with each edge $e\in E$. Here, $[{\ell }_e,u_e]$ is the interval of possible edge costs, while $T_e$, $I_e$, and $F_e$ represent the truth, indeterminacy, and falsity degrees, respectively.

For each edge, we compute the scalarized cost

$$c_e^{\mathcal{N}}=\alpha T_e{\ell }_e+\beta I_e{\displaystyle \frac{{\ell }_e+u_e}{2}}+\gamma F_e{u}_e,$$

where

$$\alpha ,\beta ,\gamma \ge 0,\alpha +\beta +\gamma =1.$$

Unless otherwise stated, the parameters are fixed throughout each experiment. This ensures that differences between the tested instances are attributable to the edge data and the selected scalarization rather than to changes in the optimization procedure.

The experiments may be conducted on both complete graphs and sparse connected graphs. For complete graphs, the scalarized costs can be used directly. For sparse graphs, shortest-path distances are computed when a metric closure is required. In the latter case, the resulting tour is interpreted in the original graph by replacing each metric-closure edge with a corresponding shortest path.

11.4.2. Algorithms Compared

We consider the following approaches.

1.

Classical cost-based approach. Each edge is assigned a conventional scalar cost, such as ${\ell }_e$, $({\ell }_e+u_e)/2$, or $u_e$. The resulting TSP instance is solved using the same algorithmic framework as in the neutrosophic setting.

2.

Neutrosophic scalarization. The edge costs are obtained from $c_e^{\mathcal{N}}$. This approach incorporates the truth, indeterminacy, and falsity degrees into the optimization objective.

3.

MST–matching approximation. A minimum spanning tree is constructed using the scalarized costs. A minimum-weight perfect matching is then computed on the odd-degree vertices of the tree. Adding the matching to the tree produces an Eulerian multigraph, from which a closed walk is obtained by an Euler tour and, when appropriate, shortcutting.

4.

Reference solution. For instances small enough to permit exact optimization, an optimal TSP tour is computed using an exact solver. This solution provides a benchmark for evaluating the quality of the approximate tours.

The MST–matching procedure is a Christofides-type method. Its classical approximation guarantee applies when the scalarized costs define a nonnegative metric, or when a suitable metric closure is used. It should not be identified with the specialized randomized rounding algorithm of Gharan, Saberi, and Singh; the two methods have different constructions and analyses.

11.4.3. Performance Measures

For each instance, we record the following quantities.

1.

Scalarized tour cost. For a tour or closed walk H, its objective value is

$$C_{\mathcal{N}}\left(H\right)=\sum _{e\in H}{c}_e^{\mathcal{N}},$$

where repeated edges are counted according to their multiplicities.

2.

Approximation ratio. Whenever an optimal solution $H^{★}$ is available, we compute

$$\rho \left(H\right)={\displaystyle \frac{C_{\mathcal{N}}\left(H\right)}{C_{\mathcal{N}}\left(H^{★}\right)}}.$$

This ratio is meaningful when $C_{\mathcal{N}}\left(H^{★}\right)>0$ and both solutions are evaluated using the same scalarized objective.

3.

Computational time. The running time is measured from the beginning of the optimization procedure to the production of the final tour. Preprocessing time, including scalarization and metric-closure construction, should be reported separately when it is substantial.

4.

Tour length under alternative evaluations. To assess the sensitivity of the result to the chosen scalarization, each computed tour may also be evaluated using the lower-endpoint, midpoint, and upper-endpoint costs:

$$C_{\ell }\left(H\right)=\sum _{e\in H}{\ell }_e,C_{\mathrm{mid}}\left(H\right)=\sum _{e\in H}{\displaystyle \frac{{\ell }_e+u_e}{2}},C_u\left(H\right)=\sum _{e\in H}{u}_e.$$

11.4.4. Sensitivity Analysis

We investigate the sensitivity of the computed tours to the scalarization parameters $\alpha$, $\beta$, and $\gamma$. In particular, we compare parameter settings that emphasize lower costs, indeterminacy, or upper-endpoint costs. The parameter constraints

$$\alpha ,\beta ,\gamma \ge 0,\alpha +\beta +\gamma =1$$

are maintained in every experiment.

We also examine the effect of changing the neutrosophic degrees while keeping the interval endpoints fixed. This separates the influence of the neutrosophic information from the influence of the underlying cost intervals. Conversely, fixing the neutrosophic degrees while varying the interval widths allows us to study the effect of uncertainty in the edge costs.

For each parameter setting, we record the objective value, approximation ratio when available, and computational time. Repeated trials are appropriate when the algorithm includes randomized steps; in that case, the number of trials, random seeds, and summary statistics should be reported.

11.4.5. Discussion of Results

The computational results should be interpreted in terms of both solution quality and sensitivity to the neutrosophic parameters. A comparison with classical scalarizations indicates whether the additional neutrosophic information changes the selected tour. The sensitivity analysis further identifies parameter regimes in which the resulting tour is stable and regimes in which small changes in the neutrosophic data lead to a different solution.

For reproducibility, the final experimental report should specify the graph-generation procedure, the number of vertices and edges, the distributions or rules used to generate interval endpoints and neutrosophic degrees, the values of $\alpha$, $\beta$, and $\gamma$, the optimization software, and the hardware environment. Numerical tables and empirical conclusions should be included only after the experiments have been executed.

The experiments therefore serve two complementary purposes: they evaluate the practical effect of neutrosophic scalarization and they provide empirical evidence about the performance of the chosen approximation method. Any observed improvement is specific to the tested instances and parameter settings and does not, by itself, establish a stronger worst-case approximation guarantee.

12. Conclusions

We have presented a neutrosophic modeling framework for randomized rounding in the Traveling Salesman Problem. Each edge is assigned an interval of possible costs and three neutrosophic degrees. A scalarization converts these data into effective costs, leading to a standard subtour-elimination relaxation.

We have also described how randomized spanning trees and minimum-cost T-joins can be incorporated into this framework. The conditional approximation theorem clarifies that a performance guarantee follows only after the necessary expected-cost inequality has been established.

The main research challenge is therefore not merely to introduce neutrosophic notation, but to prove that a specified neutrosophic objective admits an efficient rounding procedure with rigorous approximation bounds.