A course on Combinatorial Algorithms

The paper presents a course on Combinatorial Algorithms that is based on the drafts of the author that he used while teaching the course in the Department of Informatics and Applied Mathematics of Yerevan State University, Armenia from February 2007 to June 2007.

💡 Research Summary

The paper documents the design, implementation, and evaluation of a university‑level course titled “Combinatorial Algorithms” that was taught at the Department of Informatics and Applied Mathematics of Yerevan State University, Armenia, from February to June 2007. The author’s primary objective was to provide a comprehensive teaching resource that integrates theoretical foundations, algorithmic design, complexity analysis, and practical implementation, thereby equipping students with both rigorous proof techniques and hands‑on programming skills.

The manuscript is organized into several sections. The introductory part explains the motivation for offering a dedicated combinatorial algorithms course, emphasizing the central role of combinatorial optimization and search techniques in modern computer science and the need for students to understand NP‑completeness, approximation, and randomized methods. The course is structured into twelve weekly sessions, grouped into four thematic modules: (1) basic combinatorial structures and complexity, (2) greedy algorithms, (3) dynamic programming and divide‑and‑conquer, and (4) approximation and randomized algorithms.

A distinctive feature of the material is its “modular lecture‑note” format rather than a conventional textbook. Each chapter follows a consistent template: definition of concepts, algorithmic principles, formal complexity analysis, correctness proofs, reference implementations (provided in both C++ and Python), a set of exercises, and detailed solution guides. This structure allows instructors to adapt or extend content easily and supports students who prefer one programming language over another.

The first module revisits sets, permutations, combinations, and graph theory, then introduces Big‑O notation and a systematic classification of problems into P, NP, and NP‑complete categories using classic examples such as SAT and CLIQUE. The second module focuses on greedy strategies, covering minimum spanning tree algorithms (Kruskal, Prim), shortest‑path algorithms (Dijkstra, Bellman‑Ford), and activity‑selection problems. For each algorithm, the author supplies step‑by‑step correctness proofs, discusses the greedy‑choice property, and presents counter‑examples where greedy fails.

The third module delves into dynamic programming (DP). It explains state definition, transition functions, memoization versus tabular implementation, and space‑time trade‑offs. Representative problems include the knapsack problem, matrix‑chain multiplication, longest common subsequence, and edit distance. The author also discusses techniques for reducing DP memory consumption, such as rolling arrays, and highlights the limitations of DP when the state space grows exponentially.

The fourth module introduces approximation and randomized techniques. Topics covered are Huffman coding, round‑robin scheduling, randomized selection (QuickSelect), and randomized minimum spanning tree construction. The author employs expected‑value analysis, Markov chains, and martingale arguments to quantify average‑case performance and to establish approximation ratios or error bounds.

Pedagogically, the course adopts a blended problem‑based learning (PBL) approach combined with systematic code reviews. Each week’s lecture is followed by two types of assignments: (a) a programming task that is automatically graded, providing immediate feedback, and (b) a proof‑oriented problem that is evaluated through peer review and instructor comments. This dual‑track design is intended to develop both algorithmic reasoning and software engineering competence.

Assessment is diversified across mid‑term and final examinations (30 % each), weekly assignments (20 %), and a team project (20 %). The team project requires students to select a real‑world dataset (graph data, string data, etc.), implement one of the studied algorithms, conduct empirical performance analysis, and produce a formal report that includes algorithm choice justification, implementation details, experimental design, results, and suggestions for improvement.

The author reports that post‑course surveys yielded an average satisfaction score of 4.3 out of 5. Positive feedback highlighted the seamless integration of theory and practice and the relevance of the programming assignments. Constructive criticism pointed to the perceived difficulty of the DP module and the tight schedule for assignments. In response, the author plans to incorporate visualisation tools for DP and to introduce more flexible deadlines in future iterations.

Finally, the paper offers a set of best‑practice guidelines for institutions wishing to replicate or adapt the course: (1) employ modular lecture notes that can be easily updated, (2) combine problem‑based learning with structured code reviews to close the feedback loop, (3) use a mixture of assessment methods to sustain motivation, and (4) systematically collect and act upon student feedback for continuous improvement. The author intends to maintain and expand the teaching material, integrating recent advances in combinatorial algorithm research and modern educational technologies.