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Jean B. Lasserre (lasserrelaas.fr) Abstract: Given vectors $a,c\in Z^n$ and $b\in Z$, we consider the (unbounded) knapsack optimization problem $P:\,\min\{c'x\,\vert\, a'x=b;\,x\in N^n\}$. We compute the minimum value $p^*$ using techniques from complex analysis, namely Cauchy residue technique to integrate a function in $C^2$, the $Z$transform of an appropriate function related to $P$. The computational complexity depends on $s:=\sum_{a_j} a_j$, not on the magnitude of $b$ as in dynamic programming based approaches. We also completely characterize the number of solutions with value less than $p$, as a function of $p$. Keywords: knapsack problem; Ztransform; counting problems Category 1: Integer Programming Category 2: Combinatorial Optimization (Polyhedra ) Citation: Operations Research Letters 30 (2002), 394400 http://www.elsevier.com/homepage/sae/orms/orl/menu.htm Download: Entry Submitted: 07/13/2002 Modify/Update this entry  
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