Ultimate greedy approximation of independent sets in subcubic graphs We study the approximability of the maximum size independent set (MIS) problem in bounded degree graphs. This is one of the most classic and widely studied NPhard optimization problems. We focus on the well known minimum degree greedy algorithm for this problem. This algorithm iteratively chooses a minimum degree vertex in the graph, adds it to the solution and removes its neighbors, until the remaining graph is empty. The approximation ratios of this algorithm have been very widely studied, where it is augmented with an advice that tells the greedy which minimum degree vertex to choose if it is not unique. Our main contribution is a new mathematical theory for the design of such greedy algorithms with efficiently computable advice and for the analysis of their approximation ratios. With this new theory we obtain the ultimate approximation ratio of 54 for greedy on graphs with maximum degree 3, which completely solves the open problem from the paper by Halldorsson and Yoshihara (1995). Our algorithm is the fastest currently known algorithm with this approximation ratio on such graphs. We apply our new algorithm to the minimum vertex cover problem on graphs with maximum degree 3 to obtain a substantially faster 65approximation algorithm than the one currently known. We complement our positive, upper bound results with negative, lower bound results which prove that the problem of designing good advice for greedy is computationally hard and even hard to approximate on various classes of graphs. These results significantly improve on such previously known hardness results. Moreover, these results suggest that obtaining the upper bound results on the design and analysis of greedy advice is nontrivial.

Subset feedback vertex set on graphs of bounded independent set size Abstract   The ( Weighted )  Subset Feedback Vertex Set  problem is a generalization of the classical  Feedback Vertex Set  problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although  Subset Feedback Vertex Set  and  Feedback Vertex Set  exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that  Weighted Feedback Vertex Set  can be solved in polynomial time. We provide a dichotomy result with respect to the size xcexb1 of a maximum independent set. In particular we show that  Weighted Subset Feedback Vertex Set  can be solved in polynomial time for graphs with    xcexb1  xe2x89xa4  3   , whereas we prove that the problem remains NPhard for graphs with    xcexb1  xe2x89xa5  4   . Moreover, we show that the (unweighted)  Subset Feedback Vertex Set  problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time      n    O  (  xcexb1  )     . To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size.  Node Multiway Cut  is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for  Subset Feedback Vertex Set , we settle the complexity of  Node Multiway Cut  as well as its variants where nodes are weighted andor the terminals are deletable, for every value of the given independent set number.