Integer Programs and LP Relaxations • Consider an integer program (IP). Total unimodularity • An integer matrix A is called Totally Unimodular (TUM) if every square submatrix of A has determinant 0 , 1 or - 1 Examples: A = 0 - 1 1 1 1 - 1 is TUM since each of its 9 square submatrices has determinant 0 8. Integer linear programming and totally unimodular matrices. so that the total cost will be minimized. Finding a shortest path is thus a special case of dynamic programming. Exercises. 1.1. Solve the dynamic programming problem in Application 1.2 with Dijkstra's method. Mathematical Programs. • Linear Program (LP). Can be efficiently solved e.g., by Ellipsoid Method. • Integer Program (IP). For example, if constraint matrix is Totally Unimodular then IP and LP are equivalent. • We'll see examples of these approaches. View Integer Program Research Papers on Academia.edu for free. A (0, ±1) matrix A is restricted unimodular if every matrix obtained from A by setting to zero any subset of its entries is totally unimodular. Method 3: Totally Unimodular Matrices Assuming all entries of A and b are integral When does has an integral solution x? By Cramer's rule See the geometric interpretation of linear programming. How to prove a linear program gives integer optimal solutions? • Prove that every vertex solution is Chapter 2 Integer Programming. Paragraph 1 Total Unimodularity. - A matrix A is called totally unimodular (TU), iff the determinants of all submatrices of A are either -1, 0, or 1. • Theorem. - A polytope P={ x | Ax=b, x ? 0 } with A TU and b integer has only integer basic solutions. In this paper we introduce a family of integer polytopes and characterize them in terms of forbidden submatrices. The two extreme cases in this family arise when the constraint matrix is totally unimodular and balanced, respectively. The unimodularity of the problem of integer linear programming in which the bounding matrix is a Petrie matrix is demonstrated. Corollary 2. A Petrie matrix is unimodular (for qr4). From this it follows in turn that when qJ-1, j-l,Z,,n, the integral linear programming problem can be solved like the Pure Integer Programming Problem Network Formulation of the Relaxed. Further, it is shown that the group problem associated with the fixed-charge transportation prob­ lem may be viewed as a multiparametrie integer programming problem hav­ ing a totally unimodular constraint matrix. 3 Programming and Data Structures. The next chapters of this rst part develop the basic notions: how to make symbolic or numerical computations in analysis, how to work with vectors or matrices, write programs, deal with data lists, produce graphics, etc. 15.083/6.859 Integer Programming & Conibinatorial Optinization Recitation 3. Kwong-Meng Teo. lst October 2004. A is totally unimodular if and only if (x[a ? Ax ? b. ? x ?. is integral for all integer vectors a, b, 1, u for which the feasible set is not empty integer linear programming, Seymours decomposition theorem for totally unimodular matrices, and the theory of total dual integrality. 19.1 19.2 19.3 19.4 Total unimodularity and optimization, 266 More characterizations of total unimodularity, 269 The basic examples: network matrices, 272 integer linear programming, Seymours decomposition theorem for totally unimodular matrices, and the theory of total dual integrality. 19.1 19.2 19.3 19.4 Total unimodularity and optimizati