Approximation Algorithms

🎯 Learning Objectives

• Define approximation ratio and compare to optimal
• Design 2-approximation for vertex cover using maximal matching
• Explain the traveling salesman problem approximations
• Analyze set cover using greedy log(n) approximation

1. Introduction

1.1 Intuition

1.2 Approximation Ratio

2. Vertex Cover

2.1 Problem

2.2 2-Approximation via Maximal Matching

python
Run Inline
def vertex_cover_approx(G):
    C = set()
    E_uncovered = set(G.edges)
    while E_uncovered:
        (u, v) = E_uncovered.pop()
        C.add(u)
        C.add(v)
        # Remove all edges incident to u or v
        for e in list(E_uncovered):
            if u in e or v in e:
                E_uncovered.remove(e)
    return C

3. Traveling Salesman Problem (TSP)

3.1 Metric TSP (Triangle Inequality)

1. Compute MST of the graph
2. Double each edge (Eulerian graph)
3. Find Eulerian tour
4. Take shortcuts to get Hamiltonian cycle Christofides Algorithm (1.5-approximation):
5. Compute MST
6. Add minimum-weight perfect matching on odd-degree vertices
7. Find Eulerian tour
8. Take shortcuts

4. Set Cover

4.1 Greedy Algorithm

python
Run Inline
def set_cover(U, sets):
    covered = set()
    cover = []
    while covered != U:
        # Pick set covering most uncovered elements
        best_set = max(sets, key=lambda s: len(s - covered))
        cover.append(best_set)
        covered |= best_set
    return cover

5. 📝 Practice Questions

Q1: What is the approximation ratio of the maximal matching algorithm for vertex cover?

Answer: 2-approximation. The algorithm picks both endpoints of each matched edge. Since the edges form a matching, any vertex cover must include at least one endpoint per edge. Thus |C_alg| = 2|M| and |OPT| ≥ |M|, giving |C_alg| ≤ 2|OPT|. Q2: Why does the TSP 2-approximation require the triangle inequality?

Answer: Shortcutting in the Eulerian tour uses the triangle inequality (cost of direct edge ≤ cost of path) to ensure the shortcut doesn't increase cost. Without it, the shortcut could be arbitrarily expensive, breaking the approximation guarantee. Q3: What is the greedy set cover approximation ratio?

Answer: H(n) ≈ ln(n) + O(1), where n is the number of elements. This is essentially optimal — no polynomial-time algorithm achieves a better ratio unless P = NP. Q4: Give an example where the greedy vertex cover (always pick highest-degree vertex) is worse than the maximal matching approach.

Answer: A star graph: greedy picks the center (optimal, good). But consider a graph where high-degree vertices are mostly covered by many smaller-degree vertices' edges. The maximal matching approach guarantees 2-approximation regardless. Q5: What is an approximation-preserving reduction?

Answer: A reduction from problem A to problem B such that an α-approximation for B implies a β-approximation for A, where β depends on α. Used to prove inapproximability results.

6. 🔗 Cross-References

• Week 6 - NP-Completeness: Approximation helps cope with NP-hardness
• Week 8 - Randomized Algorithms: Randomized approximation schemes
• BSCS4021 - Week 10 - Parameterized: Alternate approach to hard problems Join Discord PreviousNP-CompletenessNextRandomized Algorithms