Abstract
Important variants of theVERTEX COVER problem (among others, CONNECTED VERTEX COVER, CAPACITATED VERTEX COVER, and MAXIMUM PARTIAL VERTEX COVER) have been intensively studied in terms of polynomial-time approximability. By way of contrast, their parameterized complexity has so far been completely open. We close this gap here by showing that, with the size of the desired vertex cover as the parameter, CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER are both fixed-parameter tractable while MAXIMUM PARTIAL VERTEX COVER is W[1]-complete. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered variants of VERTEX COVER behave very similar in terms of constant factor approximability, they display a wide range of different characteristics when investigating their parameterized complexities.
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Guo, J., Niedermeier, R. & Wernicke, S. Parameterized Complexity of Vertex Cover Variants. Theory Comput Syst 41, 501–520 (2007). https://doi.org/10.1007/s00224-007-1309-3
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DOI: https://doi.org/10.1007/s00224-007-1309-3