The Lexicographic Method for the Threshold Cover Problem The lexicographic method is a technique that was introduced by Hell and Huang Journal of Graph Theory, 20(3): 361xe2x80x93374, 1995 as a way to simplify the problems of recognizing and obtaining representations of comparability graphs, proper circulararc graphs and proper interval graphs. This method gives rise to conceptually simple recognition algorithms and leads to much simpler proofs for some characterization theorems for these classes. Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size k if its edges can be covered using k threshold graphs. Chvatal and Hammer conjectured in 1977 that given a graph G, a suitably constructed auxiliary graph (Gu0027) has chromatic number equal to the minimum size of a threshold cover of G. Although this conjecture was shown to be false in the general case by Cozzens and Leibowitz, it was shown to be true for graphs having a threshold cover of size 2 by Raschle and Simon Proceedings of the Twentyseventh Annual ACM Symposium on Theory of Computing, STOC xe2x80x9995, pages 650xe2x80x93661, 1995. That is, a graph G has a threshold cover of size 2 if and only if (Gu0027) is bipartitexe2x80x94this is the only known forbidden structure characterization of graphs having a threshold cover of size 2. We show how the lexicographic method can be used to obtain a completely new and much simpler proof for this result. This method also gives rise to a simple new LexBFSbased algorithm for recognizing graphs having a threshold cover of size 2. Although this algorithm is not the fastest known, it is a certifying algorithm that matches the time complexity of the fastest known certifying algorithm for this problem. The algorithm can also be easily adapted to give a certifying recognition algorithm for bipartite graphs that can be covered by two chain subgraphs.

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.