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7 Complexity Theory

# 7 Complexity Theory - IEE 598 Lecture 11(I.5 Complexity...

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IEE 598 - Lecture 11 (I.5) Complexity Theory Muhong Zhang DEPARTMENT OF INDUSTRIAL ENGINEERING Feb. 24, 2009

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Complexity Theory How to measure an ”efﬁcient algorithm”? What kind of problems are ”hard” and what are ”easy”? Zhang IEE 376 Introduction to OR 2 / 1
Decision Problem In complexity theory, we address the ”decision problem”. A decision problem or feasibility problem is a problem for which the answer is either yes or no . For an optimization problem: max { c ( x ) : x S } , the corresponding decision problem is Is there an x S with value c ( x ) k ? Example: Traveling salesman problem (TSP) Given a graph G = ( V , A ) and traveling time c ij for all ( i , j ) A . Optimization problem: Determine the tour from one city, visiting every city exactly once and back to the city with the smallest total time. Decision problem: For a given constant K , determine whether there is a tour from one city, visiting every city exactly once and back to the city with the total time at most K . Zhang IEE 376 Introduction to OR 3 / 1

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Roughly speaking, a problem is a speciﬁed input and output relationship. An instance of a problem consists of the input needed to compute a solution to the problem. An algorithm is a tool/method to solve a well-speciﬁed computational problem. Example: Consider the model and data in OPL, their relationship is similar to the relationship between a problem and an instance. The method used by OPL is an algorithm for the problem. The size of an instance is the length of the input of the binary representation. For example: In the TSP problem, the size of an instance is
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7 Complexity Theory - IEE 598 Lecture 11(I.5 Complexity...

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