On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete k-Center for Small k

We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in O?(n^{3/2}) time. - We prove a lower bound of ?(n^{4/3-?}) for rectilinear discrete 3-center in 4D, for any constant ? > 0, under a standard hypothesis about triangle detection in sparse graphs. - Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in O?(n^{8/5}) time. We also prove a lower bound of ?(n^{3/2-?}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is O?(n^{7/4}). - We prove a lower bound of ?(n^{2-?}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of O?(n^?), if the matrix multiplication exponent ? is equal to 2. - We similarly prove an ?(n^{k-?}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ? 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ? = 2. - We also prove an ?(n^{2-?}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound

The pp-Center Problem in Tree Networks Revisited

We present two improved algorithms for weighted discrete $p$-center problem for tree networks with $n$ vertices. One of our proposed algorithms runs in $O(n \log n + p \log^2 n \log(n/p))$ time. For all values of $p$, our algorithm thus runs as fast as or faster than the most efficient $O(n\log^2 n)$ time algorithm obtained by applying Cole's speed-up technique [cole1987] to the algorithm due to Megiddo and Tamir [megiddo1983], which has remained unchallenged for nearly 30 years. Our other algorithm, which is more practical, runs in $O(n \log n + p^2 \log^2(n/p))$ time, and when $p=O(\sqrt{n})$ it is faster than Megiddo and Tamir's $O(n \log^2n \log\log n)$ time algorithm [megiddo1983]

Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2 or 3D meshes

We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, which we choose of the Brezzi-Pitk\"aranta type. The scheme features two essential properties: the discrete gradient is the transposed of the divergence terms and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem, the steady and the transient Navier-Stokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the $L^2$-convergence of the components of the velocity, and, in the steady case, the weak $L^2$-convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov's theorem. The limit of this subsequence is then shown to be a weak solution of the Navier-Stokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and the nonlinear case.Comment: submitted September 0