Abstract
We obtain slightly improved upper bounds on efficient approximability of the Maximum Independent Set problem in graphs of maximum degree at most B (shortly, B-MaxIS), for small B≥ 3. The degree-three case plays a role of the central problem, as many of the results for the other problems use reductions to it. Our careful analysis of approximation algorithms of Berman and Fujito for 3-MaxIS shows that one can achieve approximation ratio arbitrarily close to \(3-\frac{\sqrt{13}}{2}\). Improvements of an approximation ratio below \(\frac65\) for this case translate to improvements below \(\frac{B+3}{5}\) of approximation factors for B-MaxIS for all odd B. Consequently, for any odd B≥ 3, polynomial time algorithms for B-MaxIS exist with approximation ratios arbitrarily close to \(\frac{B+3}5-\frac{4(5\sqrt{13}-18)}5\frac{(B-2)!!}{(B+1)!!}\). This is currently the best upper bound for B-MaxIS for any odd B, 3≤ B<613.
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Chlebík, M., Chlebíková, J. (2004). On Approximability of the Independent Set Problem for Low Degree Graphs. In: Královic̆, R., Sýkora, O. (eds) Structural Information and Communication Complexity. SIROCCO 2004. Lecture Notes in Computer Science, vol 3104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27796-5_5
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DOI: https://doi.org/10.1007/978-3-540-27796-5_5
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