Let's break down the solution step by step as requested. --- ### **Step 1: Introduction to the Concept** **NP-Complete Problems:** A decision problem is **NP-complete** if: - It is in NP (nondeterministic polynomial time). - Every problem in NP can be reduced to it in polynomial time. We are given four known NP-complete problems: 1. **SubsetSum** 2. **Clique** 3. **HamiltonianCycle** 4. **2-Partition** We need to prove that each of the following problems is NP-complete: a) 5-Partition b) Bin Packing c) Hamiltonian Path For each, we will show that it is in NP and then reduce a known NP-complete problem to it. --- ### **Step 2: Proving 5-Partition is NP-Complete** #### **a) 5-Partition** **Restatement:** Given a set of numbers \( S \), can it be partitioned into 5 subsets with equal sums? **Proof:** - *In NP*: A proposed partition can be checked for disjointness and equal sums in polynomial time. - *Reduction*: Reduce from **2-Partition** (which is NP-complete). - Given an instance of 2-Partition (\( S \)), create an instance for 5-Partition by adding three zeros to \( S \), making it possible to partition into 5 subsets with equal sums if and only if the original set can be partitioned into 2 subsets with equal sums. **Conclusion:** 5-Partition is NP-complete. --- ### **Step 3: Proving Bin Packing is NP-Complete** #### **b) Bin Packing** **Restatement:** Given numbers \( S \), bins \( k \), and bin size \( W \), can \( S \) be partitioned into \( k \) subsets with sum at most \( W \) per subset? **Proof:** - *In NP*: A proposed packing can be checked in polynomial time. - *Reduction*: Reduce from **3-Partition** (which is known to be NP-complete and can be reduced from 2-Partition). - Given a 3-Partition instance, use the same numbers, set \( k \) to the required number of subsets, and set \( W \) to the required subset sum. A solution exists if and only if the items can be packed exactly as in 3-Partition. **Conclusion:** Bin Packing is NP-complete. --- ### **Step 4: Proving Hamiltonian Path is NP-Complete** #### **c) Hamiltonian Path** **Restatement:** Given a directed graph \( G = (V, E) \) and vertices \( s, t \), is there a path from \( s \) to \( t \) visiting every vertex exactly once? **Proof:** - *In NP*: A proposed path can be checked in polynomial time. - *Reduction*: Reduce from **HamiltonianCycle** (which is NP-complete). - Given a graph \( G \) for HamiltonianCycle, construct \( G' \) by splitting a node \( v \) into \( s \) and \( t \), redirecting edges accordingly. \( G \) has a Hamiltonian cycle if and only if \( G' \) has a Hamiltonian path from \( s \) to \( t \). **Conclusion:** Hamiltonian Path is NP-complete. --- ### **Step 5: Final Answer in a Few Sentences** All three problems (5-Partition, Bin Packing, and Hamiltonian Path) are **NP-complete**. This is shown by demonstrating each is in NP and by providing a polynomial-time reduction from a known NP-complete problem (2-Partition or HamiltonianCycle) to each. --- ### **Step 6: What We Did in Each Step** 1. Explained the concept of NP-completeness and listed the given NP-complete problems. 2. For 5-Partition, showed it is in NP and reduced 2-Partition to it. 3. For Bin Packing, showed it is in NP and reduced 3-Partition to it. 4. For Hamiltonian Path, showed it is in NP and reduced Hamiltonian Cycle to it. 5. Summarized the final answer. 6. Briefly explained the approach taken in each step. --- **This step-by-step method ensures clarity and completeness in demonstrating the NP-completeness of each problem.**