In this paper we show that ~(n) variables are needed for. first-order logic with counting to identify graphs on n vertices. The k-variable language with counting is equivalent to the (k-1)-dimensional Weisfeiler Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices.

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For more than 35 years, the fastest known method for integer multiplication has been the Schönhage-Strassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Θ(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n 2 O(log * n). The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.

We define a powerful new approximation technique called semi-local optimization. It provides very natural heuristics that are distinctly more powerful than those based on local optimization. With an appropriate metric, semi-local optimization can still be viewed as a local optimization, but it has the advantage of making global changes to an approximate solution. Semi-local optimization generalizes recent heuristics of Halld6rsson for 3-Set Cover, Color Saving, and k-Set Cover. Greatly improved performance ratios of 4/3 for 3-Set Cover and 6/5 for Color Saving in graphs without independent sets of size 4 are obtained and shown to be the best possible with semi-local optimization. Also, based on the result for 3-Set Cover and a restricted greedy phase for big sets, we can improve the performance ratio for k-Set Cover to ~k -1/2. In Color Saving, when larger independent sets exist, we can improve the performance ratio to ~.

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