Incompressibility of H-Free Edge Modification Problems: Towards a Dichotomy

Given a graph G and an integer k, the H-free Edge Editing problem is to find whether there exist at most k pairs of vertices in G such that changing the adjacency of the pairs in G results in a graph without any induced copy of H. The existence of polynomial kernels for H-free Edge Editing (that is, whether it is possible to reduce the size of the instance to k^O(1) in polynomial time) received significant attention in the parameterized complexity literature. Nontrivial polynomial kernels are known to exist for some graphs H with at most 4 vertices (e.g., path on 3 or 4 vertices, diamond, paw), but starting from 5 vertices, polynomial kernels are known only if H is either complete or empty. This suggests the conjecture that there is no other H with at least 5 vertices were H-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set ? of nine 5-vertex graphs such that if for every H ? ?, H-free Edge Editing is incompressible and the complexity assumption NP ? coNP/poly holds, then H-free Edge Editing is incompressible for every graph H with at least five vertices that is neither complete nor empty. That is, proving incompressibility for these nine graphs would give a complete classification of the kernelization complexity of H-free Edge Editing for every H with at least 5 vertices. We obtain similar result also for H-free Edge Deletion. Here the picture is more complicated due to the existence of another infinite family of graphs H where the problem is trivial (graphs with exactly one edge). We obtain a larger set ? of nineteen graphs whose incompressibility would give a complete classification of the kernelization complexity of H-free Edge Deletion for every graph H with at least 5 vertices. Analogous results follow also for the H-free Edge Completion problem by simple complementation

Incompressibility of H-free edge modification problems: Towards a dichotomy

Given a graph G and an integer k, the H-free Edge Editing problem is to find whether there exist at most k pairs of vertices in G such that changing the adjacency of the pairs in G results in a graph without any induced copy of H. Nontrivial polynomial kernels are known to exist for some graphs H with at most 4 vertices, but starting from 5 vertices, polynomial kernels are known only if H is either complete or empty. This suggests the conjecture that there is no other H with at least 5 vertices where H-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set of nine 5-vertex graphs such that if for every , H-free Edge Editing is incompressible and the complexity assumption holds, then H-free Edge Editing is incompressible for every graph H with at least five vertices that is neither complete nor empty. We obtain similar results also for H-free Edge Deletion/Completion

An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. We do this in the usual way away from the boundaries, by replacing the incompressibility condition on the velocity by a Poisson equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy (in L-infinity), for both the velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure

Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics

As two-dimensional fluid shells, lipid bilayer membranes resist bending and stretching but are unable to sustain shear stresses. This property gives membranes the ability to adopt dramatic shape changes. In this paper, a finite element model is developed to study static equilibrium mechanics of membranes. In particular, a viscous regularization method is proposed to stabilize tangential mesh deformations and improve the convergence rate of nonlinear solvers. The Augmented Lagrangian method is used to enforce global constraints on area and volume during membrane deformations. As a validation of the method, equilibrium shapes for a shape-phase diagram of lipid bilayer vesicle are calculated. These numerical techniques are also shown to be useful for simulations of three-dimensional large-deformation problems: the formation of tethers (long tube-like exetensions); and Ginzburg-Landau phase separation of a two-lipid-component vesicle. To deal with the large mesh distortions of the two-phase model, modification of vicous regularization is explored to achieve r-adaptive mesh optimization

A continuous-stress tetrahedron for finite strain problems

A finite-strain tetrahedron with continuous stresses is proposed and analyzed. The complete stress tensor is now a nodal tensor degree-of-freedom, in addition to displacement. Specifically, stress conjugate to the relative Green-Lagrange strain is used within the framework of the Hellinger-Reissner variational principle. This is an extension of the Dunham and Pister element to arbitrary constitutive laws and finite strain. To avoid the excessive continuity shortcoming, outer faces can have null stress vectors. The resulting formulation is related to the nonlocal approaches popularized as smoothed finite element formulations. In contrast with smoothed formulations, the interpolation and integration domain is retained. Sparsity is also identical to the classical mixed formulations. When compared with variational multiscale methods, there are no parameters. Very high accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being successfully solved. Although the ad-hoc factor is removed and performance is highly competitive, computational cost is high, as each tetrahedron has 36 degrees-of-freedom. Besides the inf-sup test, four benchmark examples are adopted, with exceptional results in bending and compression with finite strains