Lattice Problems, Gauge Functions and Parameterized Algorithms

Given a k-dimensional subspace M ⊆ R and a full rank integer lattice L ⊆ R, the subspace avoiding problem SAP, defined by Blömer and Naewe [BN07], is to find a shortest vector in L\M . Treating k as a parameter (in the sense of parameterized complexity), we obtain new parameterized approximation and exact algorithms for SAP based on the AKS sieving technique [AKS01]. – Our first result is a randomized (1 + ǫ)-approximation algorithm for parameterized SAP that runs in time 2.(1/ǫ) , where the parameter k is the dimension of the subspace M . Thus, we obtain a 2 time algorithm for ǫ = 2. – Several of our algorithms work for all gauge functions as metric with some natural restrictions, in particular for all lp norms. We also prove an Ω(2) lower bound on the query complexity of AKS sieving based exact algorithms for SVP that accesses the gauge function as oracle. – Next, we give a 2 log k) exact algorithm for the parameterized SAP for any lp norm. This implies a 2 time randomized algorithm for computing the i successive minima of rank n lattice for any lp norm if i is O(n/ log n). It is known that computing all n successive minima’s is equivalent to the problems CVP, SIVP [M08]. So our result can be thought of as a step forward in getting 2 time randomized algorithm for CVP. We also give a randomized 2 time algorithm for CVP if the input instance satisfies certain promise. We also give a new algorithm for the Theta-series problem for which parameterized hardness results are shown in [DFVW99]. Furthermore, we study a new parameterized version of SVP, CVP, and SAP and show that these parameterized CVP and SAP have randomized s time algorithms, where k is the parameter and s is the input size, and are hard for the class W[1].