# Knapsack Problems

## Hans Kellerer

Language: English

Pages: 548

ISBN: 3540402861

Format: PDF / Kindle (mobi) / ePub

Thirteen years have passed since the seminal book on knapsack problems by Martello and Toth appeared. On this occasion a former colleague exclaimed back in 1990: "How can you write 250 pages on the knapsack problem?" Indeed, the definition of the knapsack problem is easily understood even by a non-expert who will not suspect the presence of challenging research topics in this area at the first glance. However, in the last decade a large number of research publications contributed new results for the knapsack problem in all areas of interest such as exact algorithms, heuristics and approximation schemes. Moreover, the extension of the knapsack problem to higher dimensions both in the number of constraints and in the num ber of knapsacks, as well as the modification of the problem structure concerning the available item set and the objective function, leads to a number of interesting variations of practical relevance which were the subject of intensive research during the last few years. Hence, two years ago the idea arose to produce a new monograph covering not only the most recent developments of the standard knapsack problem, but also giving a comprehensive treatment of the whole knapsack family including the siblings such as the subset sum problem and the bounded and unbounded knapsack problem, and also more distant relatives such as multidimensional, multiple, multiple-choice and quadratic knapsack problems in dedicated chapters.

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x j = 0 and in a similar way U} an upper bound on the branch Xj = 1. Moreover, assume that an incumbent solution value z has been found in some way, e.g. by using the algorithm Greedy from Section 2.1. If U~ ~ z, then we know that the branch Xj = 0 does not lead to an improved solution and thus we may fix the variable to Xj = 1. In a similar way U} ~ z implies that Xj = 0 in every improved solution. All variables fixed at their optimal value may be removed from the problem thus decreasing the

................. 266 9.5.5 Metaheuristics ..................................... 268 9.6 The Two-Dimensional Knapsack Problem ..................... 269 9.7 The Cardinality Constrained Knapsack Problem ................ 271 9.8 10. XIII 9.7.1 Related Problems .................................. 272 9.7.2 Branch-and-Bound ................................. 273 9.7.3 Dynamic Programming ............................. 273 9.7.4 Approximation Algorithms .......................... 276 The

implementations. For medium sized problems of pthree and even odd we even observe an improvement of 200-400 times. This may seem surprising as the word size is 94 4. The Subset Sum Problem algorithm n 10 30 100 Belltab 300 1000 3000 10000 10 30 100 Be/lstate 300 1000 3000 10000 10 30 100 Wordsubsum 300 1000 3000 10000 pthree psix 0.0000 0.3836 0.0005 3.9784 0.0067 44.0179 0.1335 395.2377 3.2165 29.1376 0 323.8119 0.0000 0.0000 0.0010 1.7560 0.0166 70.4943 0.5756 6.8077 61.7440 - - 0

induction moving "upwards" in the tree, i.e. beginning with its leaves and applying induction to the inner nodes. We start with the leaves of the tree, i.e. executions of Divide-and-Conquer with no further recursive calls. Therefore, we have = 0 after resolving Rt and by considering the condition for not calling the recursion, YF Resolving R2 we either have U2 previous inequality or we get = 0 and hence yx = yf and we are done with the and hence with yfC = 0 For all other nodes we show that

g(2,·) and h(2,·) are obtained using algorithm Wordmerge. Table g'(I,·) is obtained by shifting g(I,·) right by W2 = 1 while table h'(I,·) is obtained by shifting h(1,·) right by P2 = 3. The bottom table shows each step of algorithm Wordmerge and the corresponding emitted values. Proof To start with the time complexity, we notice that each iteration of the main loop emits at least one bit and thus increments either 19 or lh. As 19 :::; c and lh :::; Z we immediately get the stated bound. To