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MCS 312: NP Completeness and Approximation algorithms Instructor Neelima Gupta

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Table of Contents Generalizations of Subset Sum 0-1 Knapsack Bin Packing

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The Knapsack Problem The classic Knapsack problem is: A thief breaks into a store and wants to fill his knapsack of capacity K with goods of as much value as possible. Decision version: Does there exist a collection of items that fits into his knapsack and whose total value is >= W?

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The Knapsack Problem Input –Capacity K –n items with weights w i and values v i Output: a set of items S such that the sum of weights of items in S is at most K and the sum of values of items in S is maximized

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Some Simplest Versions… Fractional Knapsack Problem: Can items be picked up partially? The thief’s knapsack can hold 100 gms and has to choose from: 30 gms of gold dust at Rs 1000 /gm 60 gms of silver dust at Rs 500/gm 30 gms of platinum dust at Rs 1500/gm Note: Optimal fills the Knapsack upto full capacity. Proof: Else the remaining capacity can be filled with some item, picking it partially if the need be.

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Greedy Algorithm for fractional Knapsack 1.Sort the items in the increasing order of value/weight ratio (cost effectiveness). 2.If the next item cannot fit into the knapsack, break it and pick it partially just to fill the knapsack.

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Fractional Knapsack has greedy choice property That is, if v1/w1 is maximum, then there exists an optimal solution that contains item x1 upto the extent of min{w1, W}. Proof: Suppose not. Let O be an optimal solution that does not contain x1. Let xt be the item with maximum weight wt in O. If wt > w1, replacing w1 amount of xt by w1 amount of x1, value of the solution will improve (since v1/w1 > vt/wt). Let S’ be a subset of items in O whose is > w1. Replacing w1 of this total weight by w1 of x1 will improve the value of the solution. If no such set S’ exists then (sum of all the sets in O =) W < = w1. Replace all the sets in O by W amount of x1 and the value of the solution will improve.

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Other Simple versions Are all of the weights or total values identical? The thief breaks into a ring shop where all of the rings weight 10 gms. He can hold 25 gms; which should he take?

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0-1 Knapsack An item can either be picked or left. It cannot be picked partially. For example gold coins, diamond rings, TV etc.

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Greedy doesn’t work for 0-1 Knapsack Capacity 200 Items : X1 : v1/w1 = 12, weight : 50 X2 : v2/w2 = 10, weight : 55 X3 : v3/w3 = 8, weight : 10 X4 : v4/w4 = 6, weight : 100 Value of Greedy : 50 * *10 + 8* 10= 1230 Optimal : 8 * *10 + 8*10= 1430

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Dynamic Programming Solution Let V(i,w) is the value of the set of items from the first i items that maximizes the value subject to the constraint that the sum of the values of the items in the set is <= w Value of the original problem corresponds to V(n, K)

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Recurrence Relation V(i,w) = max (V(i-1,w-w i ) + v i, V(i-1, w)) –Fisrt term corresponds to the case when xi is included in the solution and –The second term corresponds to the case when xi is not included V(0,w) = 0 (no items to choose from) V(i,0) = 0 (no weight allowed)

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Time Analysis O(nK) It is exponential in the input size.

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Generalizations of Subset Sum The Subset Sum problem we have studied and shown that it is NPC is the following: Given a finite set S of natural numbers and a target t є N, does there exist a subset S’ of S whose elements sum up to t. Its generalization is: Gen1_SS: Given a finite set S ={x1 … xn} of n numbers, does there exist a subset S’ of S that whose sum is = K By Generalization it is NPC

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Further generalization of Subset Sum Gen1_SS: Given a finite set S ={x1 … xn} of n numbers, does there exist a subset S’ of S that whose sum is = K. Gen2_SS: Given a finite set S ={x1 … xn} of n numbers, find a subset S’ of S that maximizes the sum with the constraint that the sum is <= K. Its decision version is: Given a finite set S ={x1 … xn} of n numbers, does there exist a subset S’ of S whose sum >= W with the constraint that the sum is <= K.

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0-1 Knapsack is NP-Complete Gen2_SS: Given a finite set S ={x1 … xn} of n numbers, find a subset S’ of S that maximizes the sum with the constraint that the sum is <= K. The above subset sum problem is a particular case of 0-1 Knapsack by putting wi = vi = xi. i.e. 0-1 Knapsack is a generalization of the above subset sum problem. Hence, by generalization 0-1 Knapsack is NPC

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Bin Packing Problem Given a set of items S = {x1…xn} each with some weight wi, pack maximum number of items into a collection of finite number of bins each with some capacity Bi using minimum number of bins. Knapsack problem is a particular case of Bin-packing when the number of bins is 1 and its capacity is K

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Bin Packing is NPC By Generalization of Knapsack.

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