IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Exemplar: An exploration of the Travelling Salesman Problem

Caption for Fig 1 is placed far from its reference in the text and lacks integration; reposition and explicitly discuss the figure’s relevance.

The greedy-method description ends mid-sentence and leaves the approach incomplete; ensure text is fully included and coherent.

Figure 9’s image is misnumbered and lacks a caption; ensure consistent numbering and descriptive caption matching text references.

The conclusion section is nearly duplicated on consecutive pages, indicating unnecessary repetition; consolidate to improve conciseness.

The bibliography contains duplicate entries and inconsistent formatting of titles and dates; apply a uniform citation style and remove repeats.

The final references lack access dates and author details for online sources, especially Wikipedia; include full citation details consistently.

The Big-O chart is generic and not explicitly tied to TSP growth; clarify how the axes correspond to number of cities and operations.

Notation “1·10° 13” is unclear and mis-formatted; use proper exponent syntax such as $$1\times 10^{13}$$ consistently.

The NN heuristic complexity is stated as $O(n^2)$ but the source citation is ambiguous; provide a clearer reference and brief derivation.

The Prim’s algorithm figure lacks guided commentary; describe each highlighted step to improve clarity.

One-tree concept is introduced without a formal definition or labeled diagram; add precise definitions and reference elements in the figure.

Fig 7 is not interpreted in the text; discuss what the example demonstrates and how it supports the greedy explanation.

The Eulerian tour sequence is shown but not explained; interpret why vertices repeat and how the shortcutting produces a valid TSP tour.

Reward‐update formula is shown without explaining edge normalization or limits on pheromone saturation; add clarity on practical implementation.

The student relies heavily on a Wikipedia definition without paraphrasing or adding their own interpretation, limiting personal engagement and originality.

A quantitative comparison of greedy performance (1.17 vs 1.25) is cited without data or calculations; include actual tour weights.

The summary lacks any evaluation of Christofides’ 1.1× ratio impact on specific datasets; reflect on actual performance implications.

The discussion of k-opt issues is conceptual; include a small example showing how a local minimum traps 2-opt to illustrate the point.

Reflective comments are generic and describe learning broadly; link insights to specific methods’ strengths, weaknesses, or results obtained.

The Big O explanation is generic; include a concrete TSP example (e.g. 17 cities) to show how each complexity class applies.

The derivation of the halving factor for symmetric tours is asserted but not justified; explain why dividing by 2 eliminates mirror tours.

The heuristic definition is too general; illustrate with a simple TSP example to show how trial-and-error steps proceed.

The approximation-ratio formula is given without a worked example; calculate the ratio for a small TSP instance to demonstrate its use.

The claim that removing one edge from a TSP tour yields an MST is oversimplified; discuss conditions under which the resulting tree is actually minimal.

Prim’s algorithm description is clear but a pseudocode listing or stepwise outline would strengthen mathematical rigour.

The description of perfect matching cites polynomial methods but omits naming or summarizing them; include at least one matching algorithm.

The statement that an Eulerian graph “doesn't allow revisiting vertices” is incorrect; clarify that Eulerian tours traverse each edge once but revisit vertices.

The algorithm steps are listed clearly but lack commentary on computational complexity or implementation considerations for each step.

A pseudocode outline and complexity analysis for 2-opt would deepen mathematical rigour, rather than a text-only description.

The ant‐colony probability formula lacks mention of pheromone evaporation or decay; discuss how pheromone levels are regulated over iterations.

The acceptance probability formula $$e^{-\Delta /T}$$ is given without discussing how to choose or schedule T; elaborate on parameter tuning.