Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion

We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph and the problem of finding the chromatic number of a graph. We show that for any constant γ> 0, there is no polynomial time algorithm that approximates these problems within factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n/2^{(\log n)^{3/4+\gamma}}$\end{document} in an n vertex graph, assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rm NP} \nsubseteq {\rm BPTIME}(2^{(\log n)^{O(1)}})$\end{document}. This improves the hardness factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n/2^{(\log n)^{1-\gamma'}}$\end{document} for some small (unspecified) constant γ′ > 0 shown by Khot [20]. Our main idea is to show an improved hardness result for the Min-3Lin-Deletion problem. An instance of Min-3Lin-Deletion is a system of linear equations modulo 2, where each equation is over three variables. The objective is to find the minimum number of equations that need to be deleted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{\Omega(\sqrt{\log n})}$\end{document} for this problem, improving upon the hardness factor of (logn)β shown by Håstad [18], for some small (unspecified) constant β> 0. The hardness results for clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given in [20].