2026-07-21T14:18:37Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10964 2024-03-06T10:46:56Z ddc:004 open_access

Approximating Activation Edge-Cover and Facility Location Problems Nutov, Zeev Kortsarz, Guy Shalom, Eli generalized min-covering problem activation edge-cover facility location minimum power approximation algorithm What approximation ratio can we achieve for the Facility Location problem if whenever a client u connects to a facility v, the opening cost of v is at most theta times the service cost of u? We show that this and many other problems are a particular case of the Activation Edge-Cover problem. Here we are given a multigraph G=(V,E), a set R subseteq V of terminals, and thresholds {t^e_u,t^e_v} for each uv-edge e in E. The goal is to find an assignment a={a_v:v in V} to the nodes minimizing sum_{v in V} a_v, such that the edge set E_a={e=uv: a_u >= t^e_u, a_v >= t^e_v} activated by a covers R. We obtain ratio 1+max_{x>=1}(ln x)/(1+x/theta)~= ln theta - ln ln theta for the problem, where theta is a problem parameter. This result is based on a simple generic algorithm for the problem of minimizing a sum of a decreasing and a sub-additive set functions, which is of independent interest. As an application, we get the same ratio for the above variant of {Facility Location}. If for each facility all service costs are identical then we show a better ratio 1+max_{k in N}(H_k-1)/(1+k/theta), where H_k=sum_{i=1}^k 1/i. For the Min-Power Edge-Cover problem we improve the ratio 1.406 of [Calinescu et al, 2019] (achieved by iterative randomized rounding) to 1.2785. For unit thresholds we improve the ratio 73/60~=1.217 of [Calinescu et al, 2019] to 1555/1347~=1.155. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Zeev Nutov and Guy Kortsarz and Eli Shalom 2019 Is Part Of LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2019.20 urn:nbn:de:0030-drops-109642 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.20 eng https://creativecommons.org/licenses/by/3.0/legalcode