- The bin packing decision problem is that given an unlimited number of bins, each of capacity 1,

and n objects with sizes s1, s2, . . . , sn, where 0 < si ≤ 1, do the objects fit in k bins? where k is

a given integer.

The bin packing optimization problem is to find the smallest number of bins into which the objects

can be packed.

Show that if the decision problem can be solved in polynomial time, then the optimization problem

can also be solved in polynomial time. - Show that if the hamiltonian cycle decision problem can be solved, then the problem of listing the

vertices of a hamiltonian cycle in order is also solvable. - Suppose that we had a polynomial-time subprogram TSP to solve the traveling saleperson decision

problem (i.e., given a complete weighted graph and an integer k, it determines whether there is a

tour of total weight at most k.)

(a) Show how to use the TSP subprogram to determine the weight of an optimal tour in polynomial

time.

(b) Show how to use the TSP subprogram to find an optimal tour in polynomial time.

Sample Solution

n (1990) word-level test or checklist (Richards and Renandya, 2002: 259). Some researchers argue that for some group students, the 2000 word is not enough. It is considered that university students learning language for a specific purpose require more than 1000 words of learning with a high frequency of use (Celce- Murcia, 2001: 287). 4.3.2 Strategies for Learning Vocabulary with Teaching Path Techniques used to learn vocabulary through instruction should be focused directly on the word. It is important that the words are frequently exposed with clues to facilitate understanding and given in a specific context and the word to be learned. Words associated with each other should be taught with words and phrases along the lines and in parts. Many computer programs that have been used recently, including spoken words, spoken words, and spoken in a certain context, reflect the basic principles of such learning (celce-Murcia, 2001: 286-287). Deep processing is very important for learning vocabulary through teaching. It is known that short-term and long-term memory is important for learning. The short-term memory has a small storage capacity and holds words for a short time during processing (usually about 23 seconds). To start the deep processing process, it is necessary to transfer the information from the memory to the memory for a short period of time. The more often learners face word, the more often they think about t>

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