In the previous few lectures we have seen examples of LP-rounding, a method for obtaining approximation algorithms that involves solving a linear programming relaxation of the problem at hand and rounding the solution. In the last lecture we also discussed the basic theory of LP-duality. Today we will apply this theory to obtain a second LP-based technique for obtaining approximation algorithms...

We study a natural generalization of the problem of minimizing makespan on unrelated machines in which jobs may be split into parts. The different parts of a job can be (simultaneously) processed on different machines, but each part requires a setup time before it can be processed. First we show that a natural adaptation of the seminal approximation algorithm for unrelated machine scheduling [1...

This paper introduces to constructing problems of convex relaxations for nonconvex polynomial optimization problems. Branch-and-bound algorithms are convex relaxation based. The convex envelopes are of primary importance since they represent the uniformly best convex underestimators for nonconvex polynomials over some region. The reformulationlinearization technique (RLT) generates LP (linear p...

We study the effect of approximation matrices to semi-discrete relaxation schemes for the equations of one-dimensional elastodynamics. We consider a semi-discrete relaxation scheme and establish convergence using the Lp theory of compensated compactness. Then we study the convergence of an associated relaxation-diffusion system, inspired by the scheme. Numerical comparisons of fully-discrete sc...

Graphical models such as Markov random fields have been successfully applied to a wide variety of fields, from computer vision and natural language processing, to computational biology. Exact probabilistic inference is generally intractable in complex models having many dependencies between the variables. We present new approaches to approximate inference based on linear programming (LP) relaxa...

There are two fundamental approximation algorithm design techniques based on linear programming: (a) LP-relaxation and rounding, and (b) the primal-dual method. In this lecture note, we will discuss the former. The idea of LP-relaxation and rounding is quite simple. We first formulate an optimization problem as an integer program (IP), which is like a linear program (LP) with integer variables....

We focus on the reconstruction of binary functions from a small number of X-ray projections. The linear–programming (LP) relaxation to this combinatorial optimization problem due to Fishburn et al. is extended to objective functionals with quadratic smoothness priors. We show that the regularized LP–relaxation provides a good approximation and thus allows to bias the reconstruction towards solu...

This paper introduces algorithms for nonprojective parsing based on dual decomposition. We focus on parsing algorithms for nonprojective head automata, a generalization of head-automata models to non-projective structures. The dual decomposition algorithms are simple and efficient, relying on standard dynamic programming and minimum spanning tree algorithms. They provably solve an LP relaxation...

We consider the classical minimum Travelling Salesman Problem on the class of 3-edge-connected cubic graphs. More specifically we consider their (shortest path) metric completions. The well-known conjecture states that the subtour elimination LP relaxation on the min TSP yields a 4/3 approximation factor, yet the best known approximation factor is 3/2. The 3-edge-connected cubic graphs are inte...

The Feasibility Pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP relaxation feasible but fractional solutions, and integer but LP relaxation infeasible solutions. The process attempts to minimi...