- Solving the knapsack problem via Z-transform Jean B. Lasserre (lasserrelaas.fr) Eduardo S. Zeron (eszeronmath.cinvestav.mx) 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; Z-transform; counting problems Category 1: Integer Programming Category 2: Combinatorial Optimization (Polyhedra ) Citation: Operations Research Letters 30 (2002), 394-400 http://www.elsevier.com/homepage/sae/orms/orl/menu.htm Download: Entry Submitted: 07/13/2002Entry Accepted: 07/13/2002Entry Last Modified: 03/02/2003Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.