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Given a k-dimensional subspace M\subseteq \R^n and a full rank integerlattice L\subseteq \R^n, the \emph{subspace avoiding problem} SAP is to find ashortest vector in L\setminus M. Treating k as a parameter, we obtain newparameterized approximation and exact algorithms for SAP based on the AKSsieving technique. More precisely, we give a randomized$(1+\epsilon)$-approximation algorithm for parameterized SAP that runs in time2^{O(n)}.(1/\epsilon)^k, where the parameter k is the dimension of the subspaceM. Thus, we obtain a 2^{O(n)} time algorithm for \epsilon=2^{-O(n/k)}. We alsogive a 2^{O(n+k\log k)} exact algorithm for the parameterized SAP for any\ell_p norm.  Several of our algorithms work for all gauge functions as metric with somenatural restrictions, in particular for all \ell_p norms. We also prove an\Omega(2^n) lower bound on the query complexity of AKS sieving based exactalgorithms for SVP that accesses the gauge function as oracle.

Lattice Problems, Gauge Functions and Parameterized Algorithms