Parameterized Algorithms

🎯 Learning Objectives

• Explain fixed-parameter tractability (FPT)
• Design bounded search tree algorithms
• Apply kernelization techniques
• Understand the W-hierarchy
• Identify FPT vs. W-hard problems

1. Parameterized Complexity

1.1 Intuition

1.2 Definitions

2. Vertex Cover (Bounded Search Tree)

2.1 Algorithm

python
Run Inline
def vertex_cover(G, k):
    if k < 0: return None
    if not G.edges(): return set()
    (u, v) = any edge in G
    # Branch: either u or v must be in the cover
    result1 = vertex_cover(G - {u}, k-1)
    if result1 is not None:
        return result1 ∪ {u}
    result2 = vertex_cover(G - {v}, k-1)
    if result2 is not None:
        return result2 ∪ {v}
    return None

2.2 Kernelization

1. Remove isolated vertices (degree 0)
2. If any vertex has degree > k, include it in cover, decrease k
3. If remaining graph has > $k^2$ edges, reject (more than $k^2$ edges needs > k vertices)
4. Kernel size: $O(k^2)$

3. Feedback Vertex Set (FVS)

3.1 Problem

3.2 FPT Algorithm

• Branch on vertices in cycles
• Use iterative compression technique
• Running time: $O(5^k n^{O(1)})$

4. The W-Hierarchy

5. Common Pitfalls

Pitfall: Confusing XP with FPT

6. Key Concepts Reference

7. 📝 Practice Questions

Q1: Run vertex cover search tree on a triangle graph with k=2.

Answer: Pick edge (a,b):

• Branch 1: Include a, recurse on {b,c} with k=1. Pick edge (b,c). Include b (k=0, no edges = valid). Cover: {a,b}.
• Branch 2: Include b, recurse on {a,c} with k=1. Pick edge (a,c). Include a (k=0). Cover: {b,a} = same. Q2: Explain the Buss kernel for vertex cover.

Answer: (1) Remove isolated vertices (degree 0 — never needed). (2) If any vertex has degree > k, it MUST be in any k-cover (if not, its k+1 neighbors need k+1 vertices to cover). Add it to cover, reduce k by 1. (3) After removing high-degree vertices, remaining max degree ≤ k, so max edges = n*k/2. But we can bound n ≤ k²+k (since otherwise > k² edges). Kernel size: O(k²) vertices and edges. Q3: Why is Clique not believed to be FPT?

Answer: Clique is W[1]-complete. If Clique ∈ FPT, then W[1] = FPT, which would mean all W[1]-hard problems (Independent Set, Partial Vertex Cover) are FPT — widely considered unlikely. The hardness comes from the parameter k being in the exponent of n in the naive $O(n^k)$ brute force, and no $f(k)n^{O(1)}$ algorithm is known. Q4: What is the bounded search tree technique?

Answer: At each step, identify a structure (edge, cycle, formula clause) that requires branching into 2+ cases. Each branch reduces the parameter k. The tree depth is at most k, branching factor b, so total nodes ≤ b^k. If each node processes in polynomial time, total = O(b^k × n^{O(1)}).

8. 🔗 Cross-References

• Week 6 - NP-Completeness: Why parameterization helps
• Week 11 - Kernelization: Kernelization details
• BSCS4020 (DSA): Graph algorithms Join Discord PreviousOnline & ParallelNextKernelization