Lattice based extended formulations for integer linear equality systems

We study different extended formulations for the set $X = \{x\in\mathbb{Z}^n \mid Ax = Ax^0\}$ in order to tackle the feasibility problem for the set $X_+=X \cap \mathbb{Z}^n_+$. Here the goal is not to find an improved polyhedral relaxation of conv$(X_+)$, but rather to reformulate in such a way that the new variables introduced provide good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. For the case that $A$ has one row $a$ we analyze the reformulations in more detail. In particular, we determine the integer width of the extended formulations in the direction of the last coordinate, and derive a lower bound on the Frobenius number of $a$. We also suggest how a decomposition of the vector $a$ can be obtained that will provide a useful extended formulation. Our theoretical results are accompanied by a small computational study.Comment: uses packages amsmath and amssym

Lattice based extended formulations for integer linear equality systems

We study different extended formulations for the set $X = \{x \in Z^n \mid Ax = Ax^0\} in order to tackle the feasibility problem for the set$X^+ = X\cap Z^n_+$. Here the goal is not to find an improved polyhedral relaxation of conv$(X^+)$, but rather to reformulate in such a way that the new variables introduced provide good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. For the case that$A$has one row$a$we analyze the reformulations in more detail. In particular, we determine the integer width of the extended formulations in the direction of the last coordinate, and derive a lower bound on the Frobenius number of$a$. We also suggest how a decomposition of the vector$a$ can be obtained that will provide a useful extended formulation. Our theoretical results are accompanied by a small computational study

Robust Successive Compute-and-Forward over Multi-User Multi-Relay Networks

This paper develops efficient Compute-and-forward (CMF) schemes in multi-user multi-relay networks. To solve the rank failure problem in CMF setups and to achieve full diversity of the network, we introduce two novel CMF methods, namely, extended CMF and successive CMF. The former, having low complexity, is based on recovering multiple equations at relays. The latter utilizes successive interference cancellation (SIC) to enhance the system performance compared to the state-of-the-art schemes. Both methods can be utilized in a network with different number of users, relays, and relay antennas, with negligible feedback channels or signaling overhead. We derive new concise formulations and explicit framework for the successive CMF method as well as an approach to reduce its computational complexity. Our theoretical analysis and computer simulations demonstrate the superior performance of our proposed CMF methods over the conventional schemes. Furthermore, based on our simulation results, the successive CMF method yields additional signal-to-noise ratio gains and shows considerable robustness against channel estimation error, compared to the extended CMF method.Comment: 44 pages, 10 figures, 1 table, accepted to be published in IEEE Trans. on Vehicular Tec

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the Journal of the ACM. The earlier conference version in STOC'12 had the title "Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds