Edit this page Backlinks This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ===== Lab 12: NP-hard and NP-complete problems ===== Consider the following problems: ===k-Independent Set=== Let $math[G=(V,E)] be an undirected graph and $math[k] be a natural number. **k-Independent-Set** asks if there exists a subset $math[C\subseteq V] of size $math[k] of nodes from $math[G] such that all nodes from $math[C] are **independent**: $math[\forall u,v \in C, (u,v)\not\in E]. **Remark:** Let $math[G] be a **social network** where each edge $math[(u,v)\in E] models friendship between participants $math[u] and $math[v]. **k-independent-set** asks if there exist a group of size $math[k] such that no two members of it are friends. ===Subset Sum=== Let $math[a_1, a_2, \ldots, a_n, b] be natural numbers. **Subset Sum** asks if there exist a subset of $math[a_1, a_2, \ldots, a_n] whose **sum** equals $math[b]. **Remark:** Imagine $math[a_1, a_2, \ldots, a_n] to be **weights** of various items identified as $math[1,2, \ldots, n] and that $math[b] is the **capacity** of a rucksack. **Subset sum** asks if you can pick a **combination** of items such that the rucksack can be **completely full**. ===Partition=== Let $math[a_1, a_2, \ldots, a_n] be natural numbers. **Partition** asks if there exists a partitioning of $math[{a_1, \ldots, a_n}] into $math[P_1,P_2] such that the sum of elements from one element of the partition equals that of the other. * how is a **partition** formally defined? **Remark:** Imagine you have items each having a certain **value** $math[a_1, a_2, \ldots, a_n]. **Partition** asks you to split those item s between two people such that each one receives **equal** value. ==== 1. Reductions ==== 1.1. Show that SAT $math[\leq_p] Subset Sum. Group discussion on the reduction in class. 1.2. Prove that Partition is NP-hard. What are the steps to take? Build the reduction on your own. 1.3. Prove that Independent set is NP-hard. 1.4. Show that Partition $math[\leq_p] Subset Sum. ==== 2.Properties of NP-hard and NP-complete problems ==== 2.1. Identify a problem which is NP-hard but not NP-complete. Justify your answer. 2.2. Suppose $math[f \leq_p g], $math[f] is NP-hard and $math[g] is in NP. Is it true that $math[g \leq_p f] ? 2.3. What is an **equivalence class**? Show that the set of NP-complete problems is an equivalence class. ==== 3. P vs NP ==== 3.1. Suppose you find an algorithm which solves $math[SAT] in $math[O(n^4)] time. Is it true that $math[P = NP]? Justify your answer. What does this entail? 3.2. Suppose that you find an algorithm $math[A] which can solve the problem of finding if **all subsets of size k** of nodes from a graph are **not** cliques in polynomial time. Does that entail $math[P = NP] or $math[P \neq NP]? Justify your answer. ==== 4. Further work ==== 4.1. How would you define the concept of a $math[P]-complete problem? How difficult (informally) would these problems be? 4.2. Give an example of a $math[P]-complete problem.