Chapter 12

=Dealing with Hard Problems=\

Special Cases of Hard Problems

12.1. Dominos are tiles represented by integer pairs $(x_{i},y_{i})$, where each of the values $x_{i}$ and $y_{i}$ are integers between 1 and $n$. Let $S$ be a sequence of m integer pairs $[(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{m},y_{m})]$. The goal of the game is to create long chains $[(x_{i1},y_{i1}),(x_{i2},y_{i2}),...,(x_{it},y_{it})]$ such that $y_{ij}=x_{i(j+1)}$. Dominos can be flipped, so $(x_{i},y_{i})$ equivalent to $(y_{i},x_{i})$. For $S=[(1,3),(4,2),(3,5),(2,3),(3,8)]$, the longest domino sequences include $[(4,2),(2,3),(3,8)]$ and $[(1,3),(3,2),(2,4)]$.

(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 $[(1,3),(3,5)]$ and $[(2,3),(3,5)]$.

Solution

12.2. Let $G=(V,E)$ be a graph and $x$ and $y$ be two distinct vertices of $G$. Each vertex $v$ contains a given number of tokens $t(v)$ that you can collect if you visit $v$.

(a) Prove that it is NP-complete to find the path from $x$ to $y$ where you can collect the greatest possible number of tokens.

(b) Give an efficient algorithm if $G$ is a directed acyclic graph (DAG).

12.3. The Hamiltonian completion problem takes a given graph $G$ and seeks an algorithm to add the smallest number of edges to $G$ so that it contains a Hamiltonian cycle. This problem is NP-complete for general graphs; however, it has an efficient algorithm if $G$ is a tree. Give an efficient and provably correct algorithm to add the minimum number of possible edges to tree $T$ so that $T$ plus these edges is Hamiltonian.

Solution

Approximation Algorithms

12.4

12.5

12.6

12.7

12.8

12.9

12.10

12.11

Combinatorial Optimization

12.12

12.13

12.14

12.15

12.16

12.17

12.18

"Quantum" Computing

12.19

12.20

12.21

12.22

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