Amitabh Basu

Maximal lattice-free convex sets in linear subspaces

We consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovasz characterizing maximal lattice-free convex sets in Rn.

Distributed localization using noisy distance and angle information

Localization is an important and extensively studied problem in ad-hoc wireless sensor networks. Given the connectivity graph of the sensor nodes,along with additional local information (e.g. distances, angles, orientations etc.), the goal is to reconstruct the global geometry of the network. In this paper, we study the problem of localization with noisy distance and angle information. With no noise at all, the localization problem with both angle (with orientation) and distance information is trivial. However, in the presence of even a small amount of noise, we prove that the localization problem is NP hard.Localization with accurate distance information and relative angle information is also hard. These hardness results motivate our study of approximation schemes. We relax the non-convex constraints to approximating convex constraints and propose linear programs (LP) for two formulations of the resulting localization problem, which we call the weak deployment and strong deployment problems.These two formulations give upper and lower bounds on the location uncertainty respectively: No sensor is located outside its weak deployment region, and each sensor can be anywhere in its strong deployment region without violating the approximate distance and angle constraints. Though LP-based algorithms are usually solved by centralized methods, we propose distributed, iterative methods, which are provably convergent to the centralized algorithm solutions. We give simulation results for the distributed algorithms, evaluating the convergence rate, dependence on measurement noises,and robustness to link dynamics.

On the relative strength of split, triangle and quadrilateral cuts

Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.

Minimal Inequalities for an Infinite Relaxation of Integer Programs

We show that maximal

Security Types Preserving Compilation

S

A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation

-free convex sets are polyhedra when

A counterexample to a conjecture of Gomory and Johnson

S

Experiments with Two-Row Cuts from Degenerate Tableaux

is the set of integral points in some rational polyhedron of

Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. I. The One-Dimensional Case

\mathbb{R}^n

Intersection Cuts with Infinite Split Rank

. This result extends a theorem of Lovasz characterizing maximal lattice-free convex sets. Our theorem has implications in integer programming. In particular, we show that maximal