On a generalization of the master cyclic group polyhedron

  title={On a generalization of the master cyclic group polyhedron},
  author={Sanjeeb Dash and Ricardo Fukasawa and Oktay G{\"u}nl{\"u}k},
  journal={Mathematical Programming},
AbstractWe study the master equality polyhedron (MEP) which generalizes the master cyclic group polyhedron (MCGP) and the master knapsack polyhedron (MKP). We present an explicit characterization of the polar of the nontrivial facet-defining inequalities for MEP. This result generalizes similar results for the MCGP by Gomory (1969) and for the MKP by Araóz (1974). Furthermore, this characterization gives a polynomial time algorithm for separating an arbitrary point from MEP. We describe how… 

The master equality polyhedron with multiple rows

This paper defines the notion of a polaroid, a set containing all nontrivial facet defining inequalities, and shows how to use linear programming (LP) to efficiently solve the separation problem for the MEP when the polaroid has a compact polyhedral description.

The Master Equality Polyhedron: Two-Slope Facets and Separation Algorithm

This thesis presents our findings about the Master Equality Polyhedron (MEP), an extension of Gomory’s Master Group Polyhedron. We prove a theorem analogous to Gomory and Johnson’s two-slope theorem

Mixed Integer Rounding Cuts and Master Group Polyhedra

  • S. Dash
  • Mathematics
    Combinatorial Optimization - Methods and Applications
  • 2011
This survey surveys recent research on mixed-integer rounding (MIR) inequalities and a generalization, namely the two-step MIR inequalities defined by Dash and Gunluk (2006), and gives a short proof of the well-known fact that the MIR closure of a polyhedral set is a polyhedron.

The Group-Theoretic Approach in Mixed Integer Programming

An overview of the mathematical foundations and recent theoretical and computational advances in the study of the grouptheoretic approach in mixed integer programming and fundamental results about the structure of group relaxations are discussed.

Branch‐and‐cut and hybrid local search for the multi‐level capacitated minimum spanning tree problem

A branch‐and‐cut algorithm is developed that performs moves by solving to optimality subproblems corresponding to partial solutions of the multilevel capacitated minimum spanning tree problem and results show improved best known (UBs) for almost all instances that could not be solved to optimability.

A robust branch‐cut‐and‐price algorithm for the heterogeneous fleet vehicle routing problem

This article presents a robust branch‐cut‐and‐price algorithm for the heterogeneous fleet vehicle routing problem (HFVRP), vehicles may have distinct capacities and costs. The columns in the

Robust Branch-Cut-and-Price Algorithms for Vehicle Routing Problems

This chapter presents techniques for constructing robust Branch-Cut-and-Price algorithms on a number of Vehicle Routing Problem variants and summarizing older research on the topic.

Vectors in a box

The Steinitz lemma is used to prove an upper bound of τ(d) ≤ dd+o(d), and based on a construction of Alon and Vũ, a lower bound is obtained, contributing to understanding the master equality polyhedron with multiple rows defined by Dash et al.



Mixed-Integer Cuts from Cyclic Groups

The lower bound value obtained when adding (implicitly) all the interpolated subadditive cuts that can be derived from the individual rows of an optimal LP tableau is computed, thus approximating the optimization over the so-called Gomory's Corner polyhedron.

Valid inequalities based on the interpolation procedure

It is shown that MIR based inequalities dominate inequalities generated by the interpolation procedure in some important cases and that the Gomory mixed-integer cut is likely to dominate any inequality generated by this procedure in a certain probabilistic sense.

Some continuous functions related to corner polyhedra, II

The group problem on the unit interval is developed, with and without continuous variables, and a class of functions is shown to give extreme valid inequalities for P-+(I, u0) and for certain subsetsU ofI.

On the strength of Gomory mixed-integer cuts as group cuts

This paper numerically shows that once GMI cuts from different rows of the optimal simplex tableau are added to the formulation, all other group cuts from the same tableau rows are satisfied.

Facets of the Knapsack Polytope From Minimal Covers

In this paper we give easily computable best upper and lower bounds on the coefficients of facets of the knapsack polytope associated with minimal covers. For some coefficients the upper bounds are

Sequence Independent Lifting for Mixed-Integer Programming

It is seen that nonlinearity in lifting problems is resolved easily with superadditive lifting functions, which may pave the way for efficient applications of lifting with general integer variables.

Valid inequalities based on simple mixed-integer sets

F facets of the convex hull of mixed-integer sets with two and three variables are used to derive valid inequalities for integer sets defined by a single equation.

Cyclic group and knapsack facets

The master cyclic group problem and several master knapsack problems are defined, and several classes of facet-defining inequalities for each problem are given, as well as a set of mappings that take facets from one type of master polyhedra to another.

T-space and cutting planes

It is shown how knowledge about T-space translates directly into cutting planes for general integer programming problems, and a variety of constructions for T- space facets are given, all of which translate intocutting planes, and continuous families of facets are introduced.

Some polyhedra related to combinatorial problems