The double exponential runtime is tight for 2-stage stochastic ILPs

Abstract We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A stochastic ILP is an integer program the form $$\min \{c^T x \mid {\mathcal {A}} = b, \ell \le u, \in {\mathbb {Z}}^{r + ns} \}$$ min { c T x ? A = b , ? ? u ? Z r + n s } where constraint matrix $${\mathcal {Z}}^{nt \times r +ns}$$ t × consists n matrices $$A_i {Z}}^{t r}$$ i on vertical line $$B_i s}$$ B diagonal aside. show stronger hardness result for problem Quadratic Congruences objective compute $$z \gamma $$ z ? satisfying $$z^2 \equiv \alpha \bmod \beta 2 ? ? mod ? given $$\alpha , {Z}}$$ . This was proven be NP-hard already in 1978 by Manders Adleman. However, this only applies instances prime factorization $$\beta admits large multiplicities each number. circumvent necessity proving that remains NP-hard, even if occurs constantly often. Using new $$\textsc {Quadratic Congruences}$$ Q U D R I C O N G E S problem, we prove lower bound $$2^{2^{\delta (s+t)}} |I|^{O(1)}$$ ? ( ) | 1 some $$\delta > 0$$ > 0 running time any algorithm solving ILPs assuming Exponential Time Hypothesis (ETH). Here, | I encoding length instance. holds $$||b||_{\infty }$$ ? $$||c||_{\infty }, ||\ell ||_{\infty largest absolute value $$\varDelta ? {A}}$$ are constant. shows state-of-the-art algorithms nearly tight. Further, it proves suspicion these indeed harder solve than closely related $$n$$ -fold transpose