We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least.87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of 1/2 for MAX CUT and 3/4 or MAX 2SAT. Slight extensions of our analysis lead to a.79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a.758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of 1/4 and 3/4, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

/pdf/improved-approximation-algorithms-for-maximum-cut-and-3utsb8qos4.pdf

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

In this article, we have provided general, comprehensive coverage of the SDR technique, from its practical deployments and scope of applicability to key theoretical results. We have also showcased several representative applications, namely MIMO detection, B? shimming in MRI, and sensor network localization. Another important application, namely downlink transmit beamforming, is described in [1]. Due to space limitations, we are unable to cover many other beautiful applications of the SDR technique, although we have done our best to illustrate the key intuitive ideas that resulted in those applications. We hope that this introductory article will serve as a good starting point for readers who would like to apply the SDR technique to their applications, and to locate specific references either in applications or theory.

/pdf/semidefinite-relaxation-of-quadratic-optimization-problems-5ayn06v8bl.pdf

Semidefinite Relaxation of Quadratic Optimization Problems

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible, but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n x n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog(n).

/pdf/the-power-of-convex-relaxation-near-optimal-matrix-50c64annwg.pdf

The Power of Convex Relaxation: Near-Optimal Matrix Completion

1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.

https://hal.archives-ouvertes.fr/hal-00609687/document

Real Algebraic Geometry

Convex sets at large Faces and extreme points Convex sets in topological vector spaces Polarity, duality and linear programming Convex bodies and ellipsoids Faces of polytopes Lattices and convex bodies Lattice points and polyhedra Bibliography Index.

/pdf/a-course-in-convexity-3yhfww4y9x.pdf

A Course in Convexity

We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d-dimensional Euclidean space. Previously such algorithms were known for dimensions d = 1, 2, 3, and 4 only.

A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed

We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations to classical and higher-dimensional Dedekind sums, complexity of the Presburger arithmetic, efficient computations with rational functions, and others. Although the main slant is algorithmic, structural results are discussed, such as relations to the general theory of valuations on polyhedra and connections with the theory of toric varieties. The paper surveys known results and presents some new results and connections.

An Algorithmic Theory of Lattice Points in Polyhedra

A weighted graph is calledd-realizable if its vertices can be chosen ind-dimensional Euclidean space so that the Euclidean distance between every pair of adjacent vertices is equal to the prescribed weight. We prove that if a weighted graph withk edges isd-realizable for somed, then it isd-realizable for $$d = \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ (this bound is sharp in the worst case). We prove that for a graphG withn vertices andk edges and for a dimensiond the image of the so-called rigidity map ?dn??k is a convex set in ?k provided $$d \geqslant \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]$$ . These results are obtained as corollaries of a general convexity theorem for quadratic maps which also extends the Toeplitz-Hausdorff theorem. The main ingredients of the proof are the duality for linear programming in the space of quadratic forms and the "corank formula" for the strata of singular quadratic forms.

https://link.springer.com/content/pdf/10.1007%2FBF02574037.pdf

Problems of distance geometry and convex properties of quadratic maps

We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d-dimensional Euclidean space. Previously such algorithms were known for dimensions d=1,2,3, and 4 only. >

Alexander Barvinok

Papers

A Course in Convexity

A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed

An Algorithmic Theory of Lattice Points in Polyhedra

Problems of distance geometry and convex properties of quadratic maps

A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed