Difference between revisions of "Chapter 12"
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===Special Cases of Hard Problems=== | ===Special Cases of Hard Problems=== | ||
| − | :[[12.1]] | + | :[[12.1]]. Dominos are tiles represented by integer pairs <math>(x_i, y_i)</math>, where each of the values <math>x_i</math> and <math>y_i</math> are integers between 1 and <math>n</math>. Let <math>S</math> be a sequence of m integer pairs <math>[(x_1, y_1),(x_2, y_2), ...,(x_m, y_m)]</math>. The goal of the game is to create long chains <math>[(x_{i1}, y_{i1}),(x_{i2}, y_{i2}), ...,(x_{it}, y_{it})]</math> such that <math>y_{ij} = x_{i(j+1)}</math>. Dominos can be flipped, so <math>(x_i, y_i)</math> equivalent to <math>(y_i, x_i)</math>. For <math>S = [(1, 3),(4, 2),(3, 5),(2, 3),(3, 8)]</math>, the longest domino sequences include <math>[(4, 2),(2, 3),(3, 8)]</math> and <math>[(1, 3),(3, 2),(2, 4)]</math>. |
| + | ::(a) Prove that finding the longest domino chain is NP-complete. | ||
| + | ::(b) Give an efficient algorithm to find the longest domino chain where the numbers increase along the chain. For S above, the longest such chains are <math>[(1, 3),(3, 5)]</math> and <math>[(2, 3),(3, 5)]</math>. | ||
| + | [[12.1|Solution]] | ||
| − | :12.2 | + | :12.2. Let <math>G = (V, E)</math> be a graph and <math>x</math> and <math>y</math> be two distinct vertices of <math>G</math>. Each vertex <math>v</math> contains a given number of tokens <math>t(v)</math> that you can collect if you visit <math>v</math>. |
| + | ::(a) Prove that it is NP-complete to find the path from <math>x</math> to <math>y</math> where you can collect the greatest possible number of tokens. | ||
| + | ::(b) Give an efficient algorithm if <math>G</math> is a directed acyclic graph (DAG). | ||
| − | :[[12.3]] | + | :[[12.3]]. The ''Hamiltonian completion problem'' takes a given graph <math>G</math> and seeks an algorithm to add the smallest number of edges to <math>G</math> so that it contains a Hamiltonian cycle. This problem is NP-complete for general graphs; however, it has an efficient algorithm if <math>G</math> is a tree. Give an efficient and provably correct algorithm to add the minimum number of possible edges to tree <math>T</math> so that <math>T</math> plus these edges is Hamiltonian. |
| − | + | [[12.3|Solution]] | |
===Approximation Algorithms=== | ===Approximation Algorithms=== | ||
Revision as of 20:30, 10 September 2020
=Dealing with Hard Problems=\
Contents
Special Cases of Hard Problems
- 12.1. Dominos are tiles represented by integer pairs [math]\displaystyle{ (x_i, y_i) }[/math], where each of the values [math]\displaystyle{ x_i }[/math] and [math]\displaystyle{ y_i }[/math] are integers between 1 and [math]\displaystyle{ n }[/math]. Let [math]\displaystyle{ S }[/math] be a sequence of m integer pairs [math]\displaystyle{ [(x_1, y_1),(x_2, y_2), ...,(x_m, y_m)] }[/math]. The goal of the game is to create long chains [math]\displaystyle{ [(x_{i1}, y_{i1}),(x_{i2}, y_{i2}), ...,(x_{it}, y_{it})] }[/math] such that [math]\displaystyle{ y_{ij} = x_{i(j+1)} }[/math]. Dominos can be flipped, so [math]\displaystyle{ (x_i, y_i) }[/math] equivalent to [math]\displaystyle{ (y_i, x_i) }[/math]. For [math]\displaystyle{ S = [(1, 3),(4, 2),(3, 5),(2, 3),(3, 8)] }[/math], the longest domino sequences include [math]\displaystyle{ [(4, 2),(2, 3),(3, 8)] }[/math] and [math]\displaystyle{ [(1, 3),(3, 2),(2, 4)] }[/math].
- (a) Prove that finding the longest domino chain is NP-complete.
- (b) Give an efficient algorithm to find the longest domino chain where the numbers increase along the chain. For S above, the longest such chains are [math]\displaystyle{ [(1, 3),(3, 5)] }[/math] and [math]\displaystyle{ [(2, 3),(3, 5)] }[/math].
- 12.2. Let [math]\displaystyle{ G = (V, E) }[/math] be a graph and [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] be two distinct vertices of [math]\displaystyle{ G }[/math]. Each vertex [math]\displaystyle{ v }[/math] contains a given number of tokens [math]\displaystyle{ t(v) }[/math] that you can collect if you visit [math]\displaystyle{ v }[/math].
- (a) Prove that it is NP-complete to find the path from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math] where you can collect the greatest possible number of tokens.
- (b) Give an efficient algorithm if [math]\displaystyle{ G }[/math] is a directed acyclic graph (DAG).
- 12.3. The Hamiltonian completion problem takes a given graph [math]\displaystyle{ G }[/math] and seeks an algorithm to add the smallest number of edges to [math]\displaystyle{ G }[/math] so that it contains a Hamiltonian cycle. This problem is NP-complete for general graphs; however, it has an efficient algorithm if [math]\displaystyle{ G }[/math] is a tree. Give an efficient and provably correct algorithm to add the minimum number of possible edges to tree [math]\displaystyle{ T }[/math] so that [math]\displaystyle{ T }[/math] plus these edges is Hamiltonian.
Approximation Algorithms
- 12.4
- 12.6
- 12.8
- 12.10
Combinatorial Optimization
- 12.12
- 12.14
- 12.16
- 12.18
"Quantum" Computing
- 12.20
- 12.22
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