### Obtaining optimal $\textit{k}-cardinality$ trees fastObtaining optimal $\textit{k}-cardinality$ trees fast

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 Author Chimani, Markus ♦ Kandyba, Maria ♦ Ljubi, Ivana ♦ Mutzel, Petra Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2009 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Computer programming, programs & data Subject Keyword (prize collecting) Steiner tree ♦ k-cardinality tree ♦ Exact algorithm ♦ Branch and cut ♦ Comparison with metaheuristics Abstract Given an undirected graph $\textit{G}$ = $(\textit{V},\textit{E})$ with edge weights and a positive integer number $\textit{k},$ the $\textit{k}-cardinality$ tree problem consists of finding a subtree $\textit{T}$ of $\textit{G}$ with exactly $\textit{k}$ edges and the minimum possible weight. Many algorithms have been proposed to solve this NP-hard problem, resulting in mainly heuristic and metaheuristic approaches. In this article, we present an exact ILP-based algorithm using directed cuts. We mathematically compare the strength of our formulation to the previously known ILP formulations of this problem, and show the advantages of our approach. Afterwards, we give an extensive study on the algorithm's practical performance compared to the state-of-the-art metaheuristics. In contrast to the widespread assumption that such a problem cannot be efficiently tackled by exact algorithms for medium and large graphs (between 200 and 5,000 nodes), our results show that our algorithm not only has the advantage of proving the optimality of the computed solution, but also often outperforms the metaheuristic approaches in terms of running time. ISSN 10846654 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2010-01-05 Publisher Place New York e-ISSN 10846654 Journal Journal of Experimental Algorithmics (JEA) Volume Number 14 Page Count 19 Starting Page 2.5 Ending Page 2.23

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Source: ACM Digital Library