An ILP Solver for Multi-label MRFs with Connectivity Constraints

Integer Linear Programming (ILP) formulations of Markov random fields (MRFs) models with global connectivity priors were investigated previously in computer vision, e.g., \cite{globalinter,globalconn}. In these works, only Linear Programing (LP) relaxations \cite{globalinter,globalconn} or simplified versions \cite{graphcutbase} of the problem were solved. This paper investigates the ILP of multi-label MRF with exact connectivity priors via a branch-and-cut method, which provably finds globally optimal solutions. The method enforces connectivity priors iteratively by a cutting plane method, and provides feasible solutions with a guarantee on sub-optimality even if we terminate it earlier. The proposed ILP can be applied as a post-processing method on top of any existing multi-label segmentation approach. As it provides globally optimal solution, it can be used off-line to generate ground-truth labeling, which serves as quality check for any fast on-line algorithm. Furthermore, it can be used to generate ground-truth proposals for weakly supervised segmentation. We demonstrate the power and usefulness of our model by several experiments on the BSDS500 and PASCAL image dataset, as well as on medical images with trained probability maps.Comment: 19 page

Disjunctive Inequalities: Applications and Extensions

A general optimization problem can be expressed in the form min{cx: x ∈ S}, (1) where x ∈ R n is the vector of decision variables, c ∈ R n is a linear objective function and S ⊂ R n is the set of feasible solutions of (1). Because S is generall

Lattice reformulation cuts

Here we consider the question whether the lattice reformulation of a linear integer program can be used to produce effective cutting planes. In particular, we aim at deriving split cuts that cut off more of the integrality gap than Gomory mixed-integer (GMI) inequalities generated from LP-tableaus, while being less computationally demanding than generating the split closure. We consider integer programs (IPs) in the form max{ Ax=b x =Zn+}, where the reformulation takes the form max\{cx +cQ> -xu u =Zn-m Z n - m\}, where Q is an n (n - m) integer matrix. Working on an optimal LP-tableau in the u -space allows us to generate n - m GMIs in addition to the m GMIs associated with the optimal tableau in the x space. These provide new cuts that can be seen as GMIs associated to n - m nonelementary split directions associated with the reformulation matrix \Q . On the other hand it turns out that the corner polyhedra associated to an LP basis and the GMI or split closures are the same whether working in the x or u spaces. Our theoretical derivations are accompanied by an illustrative computational study. The computations show that the effectiveness of the cuts generated by this approach depends on the quality of the reformulation obtained by the reduced basis algorithm used to generate Q and that it is worthwhile to generate several rounds of such cuts. However, the effectiveness of the cuts deteriorates as the number of constraints is increased

Performance Variability in Mixed-Integer Programming

The performance of mixed-integer programming solvers is subject to some unexpected variability that appears, for example, when changing from one computing platform to another, when permuting rows and/or columns of a model, when adding seemingly neutral changes to the solution process, etc. This phenomenon has been observed for decades, but only recently has it started to be methodologically analyzed with the two possible aims of either reducing or exploiting it, ideally both. In this tutorial we discuss the roots of performance variability, we provide useful tips to recognize it, and we point out some severe misinterpretations that might be generated by not performing/analyzing benchmark results carefully. Finally, we report on the most recent attempts to gain from variability

Enhanced Mixed Integer Programming Techniques and Routing Problems

Mixed integer programming is up today one of the most widely used techniques for dealing with hard optimization problems. On the one side, many practical optimization problems arising from real-world applications (such as, e.g., scheduling, project planning, transportation, telecommunications, economics and finance, timetabling, etc) can be easily and effectively formulated as Mixed Integer linear Programs (MIPs). On the other hand, 50 and more years of intensive research has dramatically improved on the capability of the current generation of MIP solvers to tackle hard problems in practice. However, many questions are still open and not fully understood, and the mixed integer programming community is still more than active in trying to answer some of these questions. As a consequence, a huge number of papers are continuously developed and new intriguing questions arise every year. When dealing with MIPs, we have to distinguish between two different scenarios. The first one happens when we are asked to handle a general MIP and we cannot assume any special structure for the given problem. In this case, a Linear Programming (LP) relaxation and some integrality requirements are all we have for tackling the problem, and we are ``forced" to use some general purpose techniques. The second one happens when mixed integer programming is used to address a somehow structured problem. In this context, polyhedral analysis and other theoretical and practical considerations are typically exploited to devise some special purpose techniques. This thesis tries to give some insights in both the above mentioned situations. The first part of the work is focused on general purpose cutting planes, which are probably the key ingredient behind the success of the current generation of MIP solvers. Chapter 1 presents a quick overview of the main ingredients of a branch-and-cut algorithm, while Chapter 2 recalls some results from the literature in the context of disjunctive cuts and their connections with Gomory mixed integer cuts. Chapter 3 presents a theoretical and computational investigation of disjunctive cuts. In particular, we analyze the connections between different normalization conditions (i.e., conditions to truncate the cone associated with disjunctive cutting planes) and other crucial aspects as cut rank, cut density and cut strength. We give a theoretical characterization of weak rays of the disjunctive cone that lead to dominated cuts, and propose a practical method to possibly strengthen those cuts arising from such weak extremal solution. Further, we point out how redundant constraints can affect the quality of the generated disjunctive cuts, and discuss possible ways to cope with them. Finally, Chapter 4 presents some preliminary ideas in the context of multiple-row cuts. Very recently, a series of papers have brought the attention to the possibility of generating cuts using more than one row of the simplex tableau at a time. Several interesting theoretical results have been presented in this direction, often revisiting and recalling other important results discovered more than 40 years ago. However, is not clear at all how these results can be exploited in practice. As stated, the chapter is a still work-in-progress and simply presents a possible way for generating two-row cuts from the simplex tableau arising from lattice-free triangles and some preliminary computational results. The second part of the thesis is instead focused on the heuristic and exact exploitation of integer programming techniques for hard combinatorial optimization problems in the context of routing applications. Chapters 5 and 6 present an integer linear programming local search algorithm for Vehicle Routing Problems (VRPs). The overall procedure follows a general destroy-and-repair paradigm (i.e., the current solution is first randomly destroyed and then repaired in the attempt of finding a new improved solution) where a class of exponential neighborhoods are iteratively explored by heuristically solving an integer programming formulation through a general purpose MIP solver. Chapters 7 and 8 deal with exact branch-and-cut methods. Chapter 7 presents an extended formulation for the Traveling Salesman Problem with Time Windows (TSPTW), a generalization of the well known TSP where each node must be visited within a given time window. The polyhedral approaches proposed for this problem in the literature typically follow the one which has been proven to be extremely effective in the classical TSP context. Here we present an overall (quite) general idea which is based on a relaxed discretization of time windows. Such an idea leads to a stronger formulation and to stronger valid inequalities which are then separated within the classical branch-and-cut framework. Finally, Chapter 8 addresses the branch-and-cut in the context of Generalized Minimum Spanning Tree Problems (GMSTPs) (i.e., a class of NP-hard generalizations of the classical minimum spanning tree problem). In this chapter, we show how some basic ideas (and, in particular, the usage of general purpose cutting planes) can be useful to improve on branch-and-cut methods proposed in the literature

Improving branch-and-cut performance by random sampling

Abstract We discuss the variability in the performance of multiple runs of branchand-cut mixed integer linear programming solvers, and we concentrate on the one deriving from the use of different optimal bases of the linear programming relaxations. We propose a new algorithm exploiting more than one of those bases and we show that different versions of the algorithm can be used to stabilize and improve the performance of the solver