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Approximation algorithms for minimum weight connected 3-path vertex cover. (English) Zbl 1428.05297
Summary: A \(k\)-path vertex cover \(( \text{VCP}_k)\) is a vertex set \(C\) of graph \(G\) such that every path of \(G\) on \(k\) vertices has at least one vertex in \(C\). Because of its background in keeping data integrality of a network, minimum \(\text{VCP}_k\) problem \(( \text{MinVCP}_k)\) has attracted a lot of researches in recent years. This paper studies the minimum weight connected \(\text{VCP}_k\) problem \(( \text{MinWCVCP}_k)\), in which every vertex has a weight and the \(\text{VCP}_k\) found by the algorithm induces a connected subgraph and has the minimum weight. It is known that \(\text{MinWCVCP}_k\) is set-cover-hard. We present two polynomial-time approximation algorithms for \(\text{MinWCVCP}_3\). The first one is a greedy algorithm achieving approximation ratio \(3\ln n\). The difficulty lies in its analysis dealing with a non-submodular potential function. The second algorithm is a 2-stage one, finding a \(\text{VCP}_3\) in the first stage and then adding more vertices for connection. We show that its approximation ratio is at most \(\ln \delta_{\max} + 4 + \ln 2\), where \(\delta_{\max}\) is the maximum degree of the graph. Considering the inapproximability of this problem, this ratio is asymptotically tight.

MSC:
05C85 Graph algorithms (graph-theoretic aspects)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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