A Practical Introduction to Data Structures and Algorithm Analysis

590 Chap. 17 Limits to ComputationINDEPENDENT SETInput: A graph G and an integer k.Output: YES if there is a subset S of the vertices in G of size k orgreater such that no edge connects any two vertices in S, and NO otherwise.Assuming that K-CLIQUE is N P-complete, prove that INDEPENDENTSET is N P-complete.17.11 Define the problem PARTITION as follows:PARTITIONInput: A collection of integers.Output: YES if the collection can be split into two such that the sumof the integers in each partition sums to the same amount. NO otherwise.(a) Assuming that PARTITION is N P-complete, prove that the decisionform of BIN PACKING is N P-complete.(b) Assuming that PARTITION is N P-complete, prove that KNAPSACKis N P-complete.17.12 Imagine that you have a problem P that you know is N P-complete. Forthis problem you have two algorithms to solve it. For each algorithm, someproblem instances of P run in polynomial time and others run in exponentialtime (there are lots of heuristic-based algorithms for real N P-completeproblems with this behavior). You can’t tell beforehand for any given probleminstance whether it will run in polynomial or exponential time on eitheralgorithm. However, you do know that for every problem instance, at leastone of the two algorithms will solve it in polynomial time.(a) What should you do?(b) What is the running time of your solution?(c) What does it say about the question of P = N P if the conditionsdescribed in this problem existed?17.13 Here is another version of the knapsack problem, which we will call EXACTKNAPSACK. Given a set of items each with given integer size, and a knapsackof size integer k, is there a subset of the items which fits exactly withinthe knapsack?Assuming that EXACT KNAPSACK is N P-complete, use a reduction argumentto prove that KNAPSACK is N P-complete.