Worked Examples: Advanced Reductions

These examples are designed to step through the logical process of applying the algorithmic concepts.

Example 1: Clique Gadget

❓ Problem: Nodes for $(x \lor y \lor z)$?

💡 Solution: 3 nodes connected in triangle (pick 1).

Example 2: Consistency Edge

❓ Problem: Between $x$ and $\neg x$?

💡 Solution: No edge (can't pick both).

Example 3: Indep. Set -> VC

❓ Problem: Relation.

💡 Solution: $IS = V - VC$.

Example 4: Subset Sum

❓ Problem: Problem statment.

💡 Solution: Subset adds to Target T?

Example 5: Hamiltonian Cycle

❓ Problem: vs Euler Tour.

💡 Solution: Euler (Edges) is P. Ham (Nodes) is NPC.

Example 6: Knapsack NPC

❓ Problem: Why?

💡 Solution: Subset sum is special case of Knapsack.