Alexander I. Barvinok

Problems of distance geometry and convex properties of quadratic maps

AbstractA 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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hzaiabg2da9maadmaabaWaaeWaaeaadaGc% aaqaaiabiIda4iaa-TgacqGHRaWkcqaIXaqmaSqabaGccqGHsislcq% aIXaqmaiaawIcacaGLPaaacqGGVaWlcqaIYaGmaiaawUfacaGLDbaa% aaa!47D5!

Two algorithmic results for the traveling salesman problem

d = \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]

Computing the Ehrhart quasi-polynomial of a rational simplex

(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 ℝdn→ℝk is a convex set in ℝk provided % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaaceiGaa8hzaiabgwMiZoaadmaabaWaaeWaaeaadaGc% aaqaaiabiIda4iaa-TgacqGHRaWkcqaIXaqmaSqabaGccqGHsislcq% aIXaqmaiaawIcacaGLPaaacqGGVaWlcqaIYaGmaiaawUfacaGLDbaa% aaa!4895!

The maximum traveling salesman problem under polyhedral norms

d \geqslant \left[ {\left( {\sqrt {8k + 1} - 1} \right)/2} \right]

The geometric maximum traveling salesman problem

. 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.

Computing Mixed Discriminants, Mixed Volumes, and Permanents ⁄

Abstract. We prove that for any fixed d the generating function of the projectionof the set of integer points in a rational d-dimensional polytope can be computed inpolynomial time. As a corollary, we deduce that various interesting sets of latticepoints, notably integer semigroups and (minimal) Hilbert bases of rational cones,have short rational generating functions provided certain parameters (the dimensionand the number of generators) are fixed. It follows then that many computationalproblems for such sets (for example, finding the number of positive integers notrepresentable as a non-negative integer combination of given coprime positive integersa 1 ,... ,a d ) admit polynomial time algorithms. We also discuss a related problem ofcomputing the Hilbert series of a ring generated by monomials. 1. Introduction and Main ResultsOur main motivation is the following question which goes back to Frobenius andSylvester.(1.1) The Frobenius Problem. Let a 1 ,... ,a d be positive coprime integers andletS =nµ