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This paper investigates the placement of a finite set of web cache proxies in a computer network with a planar graph topology, based on a dynamic programming approach. Three criteria are defined for the web proxy placement: 1) the min-sum strategy with consolidated request forwarding (MS-CRF); 2) the min-sum criterion with immediate request forwarding (MS-IRF); and 3) the minmax with immediate request forwarding (MM-IRF). Optimal placement is obtained under MS-CRF. Under the MS-IRF and MM-IRF, a near optimal placement is achieved. This can be improved through a refinement strategy. The paper deals with the cylindrical mesh topology as a general topology on which any planar graph can be embedded.