An improved lower bound for approximating the Minimum Integral Solution Problem with Preprocessing over $$\ell _\infty $$ ℓ ∞ norm

Abstract

In this paper, we study the approximation complexity of the Minimum Integral Solution Problem with Preprocessing introduced by Alekhnovich et al. (FOCS, pp. 216---225, 2005). We show that the Minimum Integral Solution Problem with Preprocessing over $$\ell _\infty $$l? norm ($$\hbox {MISPP}_\infty $$MISPP?) is NP-hard to approximate to within a factor of $$(\log n)^{1/2-\epsilon },$$(logn)1/2-∈, unless $$\mathbf{NP}\subseteq \mathbf{DTIME}(2^{poly log(n)}).$$NP⊆DTIME(2polylog(n)). This improves on the best previous result. The best result so far gave $$\sqrt{2}-\epsilon $$2-∈ factor hardness for any $$\epsilon >0.$$∈>0.