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| - | ===== Lab 12: NP-hard and NP-complete problems ===== | + | ====== TDA-uri și inducție structurală ====== |
| + | | ||
| - | Consider the following problems: | + | 1. Definiți axiome pentru următorii operatori pe tipul ''List'': |
| - | ===k-Independent Set=== | + | * ''reverse'' (inversează elementele dintr-o listă) |
| + | * ''filterEven'' (elimină elementele impare dintr-o listă) | ||
| - | 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[G] are **independent**: $math[\forall u,v \in V . (u,v)\not\in E]. | + | 2. Definiți axiome pentru următorii operatori pe tipul ''BTree'': |
| - | **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. | + | * ''mirror : BTree → BTree'' (arborele oglindit pe verticală, i.e. pentru orice nod, copilul stâng devine copilul drept și vice-versa) |
| + | * ''flatten : BTree → List'' (lista cu toate elementele din arbore; observați că există mai multe ordini posibile) | ||
| - | ===Subset Sum=== | + | 3. Definiți axiome pentru următorii operatori pe tipul ''Map'': |
| - | 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]. | + | * ''update : Map × K × V → Map'' (un Map cu o nouă asociere cheie:element) |
| + | * ''delete : Map × K → Map'' (șterge cheia și valoarea asociată) | ||
| - | **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**. | + | 4. Demonstrați următoarele propoziții, folosind inducție structurală: |
| - | ===Partition=== | + | * $math[\forall t \in \texttt{BTree}. size(t) = size(mirror(t))] |
| + | * $math[\forall t \in \texttt{BTree}. size(t) = length(flatten(t))] | ||
| + | * $math[\forall l \in \texttt{List}. append(l, Empty) = l] | ||
| + | * $math[\forall l_1, l_2, l_3 \in \texttt{List}. append(l_1, append(l_2, l_3)) = append(append(l_1, l_2), l_3))] | ||
| + | * $math[\forall l_1, l_2 \in \texttt{List}. length(append(l_1, l_2)) = length(append(l_2, l_1))] | ||
| + | * $math[\forall l_1, l_2 \in \texttt{List}. reverse(append(l_1, l_2)) = append(reverse(l_2), reverse(l_1))]. | ||
| - | 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 items 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. | ||
| + | <note> | ||
| + | Soluțiile acestui laborator se găsesc [[https://ocw.cs.pub.ro/ppcarte/doku.php?id=aa:lab:sol:12|aici]] | ||
| + | </note> | ||