scholarly journals Learning to Run Heuristics in Tree Search

2017 ◽  
Author(s):  
Elias B. Khalil ◽  
Bistra Dilkina ◽  
George L. Nemhauser ◽  
Shabbir Ahmed ◽  
Yufen Shao
Keyword(s):  
Integer Programming ◽  
State Of The Art ◽  
Independent Set ◽  
Search Tree ◽  
Mixed Integer ◽  
Tree Search ◽  
Trial And Error ◽  
Benchmark Instances ◽  
Primal Heuristics ◽  
Improved Performance

``Primal heuristics'' are a key contributor to the improved performance of exact branch-and-bound solvers for combinatorial optimization and integer programming. Perhaps the most crucial question concerning primal heuristics is that of at which nodes they should run, to which the typical answer is via hard-coded rules or fixed solver parameters tuned, offline, by trial-and-error. Alternatively, a heuristic should be run when it is most likely to succeed, based on the problem instance's characteristics, the state of the search, etc. In this work, we study the problem of deciding at which node a heuristic should be run, such that the overall (primal) performance of the solver is optimized. To our knowledge, this is the first attempt at formalizing and systematically addressing this problem. Central to our approach is the use of Machine Learning (ML) for predicting whether a heuristic will succeed at a given node. We give a theoretical framework for analyzing this decision-making process in a simplified setting, propose a ML approach for modeling heuristic success likelihood, and design practical rules that leverage the ML models to dynamically decide whether to run a heuristic at each node of the search tree. Experimentally, our approach improves the primal performance of a state-of-the-art Mixed Integer Programming solver by up to 6% on a set of benchmark instances, and by up to 60% on a family of hard Independent Set instances.

scholarly journals Conflict-Driven Heuristics for Mixed Integer Programming

2020 ◽  
Author(s):  
Jakob Witzig ◽  
Ambros Gleixner
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
State Of The Art ◽  
Computational Study ◽  
Search Tree ◽  
General Purpose ◽  
Conflict Analysis ◽  
Mixed Integer ◽  
Linear Programs ◽  
Feasible Solutions

Two essential ingredients of modern mixed-integer programming solvers are diving heuristics, which simulate a partial depth-first search in a branch-and-bound tree, and conflict analysis, which learns valid constraints from infeasible subproblems. So far, these techniques have mostly been studied independently: primal heuristics for finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. In this paper, we pose the question of whether and how the orthogonal goals of proving infeasibility and generating improving solutions can be pursued in a combined manner such that a state-of-the-art solver can benefit. To do so, we integrate both concepts in two different ways. First, we develop a diving heuristic that simultaneously targets the generation of valid conflict constraints from the Farkas dual and the generation of improving solutions. We show that, in the primal, this is equivalent to the optimistic strategy of diving toward the best bound with respect to the objective function. Second, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classic coefficient diving. In a detailed computational study, both methods are evaluated on the basis of an implementation in the source-open-solver SCIP. The experimental results underline the potential of combining both diving heuristics and conflict analysis. Summary of Contribution. This original article concerns the advancement of exact general-purpose algorithms for solving one of the largest and most prominent problem classes in optimization, mixed-integer linear programs. It demonstrates how methods for conflict analysis that learn from infeasible subproblems can be combined successfully with diving heuristics that aim at finding primal solutions. For two newly designed diving heuristics, this paper features a thoroughly computational study regarding their impact on the overall performance of a state-of-the-art MIP solver.

scholarly journals Clearing Kidney Exchanges via Graph Neural Network Guided Tree Search (Student Abstract)

2020 ◽  
Vol 34 (10) ◽  
pp. 13989-13990
Author(s):  
Zeyu Zhao ◽  
John P. Dickerson
Keyword(s):  
Neural Network ◽  
Independent Set ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Tree Search ◽  
Scalable Algorithms ◽  
Monte Carlo Tree Search ◽  
Kidney Exchange ◽  
Stage Renal Disease

Kidney exchange is an organized barter market that allows patients with end-stage renal disease to trade willing donors—and thus kidneys—with other patient-donor pairs. The central clearing problem is to find an arrangement of swaps that maximizes the number of transplants. It is known to be NP-hard in almost all cases. Most existing approaches have modeled this problem as a mixed integer program (MIP), using classical branch-and-price-based tree search techniques to optimize. In this paper, we frame the clearing problem as a Maximum Weighted Independent Set (MWIS) problem, and use a Graph Neural Network guided Monte Carlo Tree Search to find a solution. Our initial results show that this approach outperforms baseline (non-optimal but scalable) algorithms. We believe that a learning-based optimization algorithm can improve upon existing approaches to the kidney exchange clearing problem.

scholarly journals Solving the Set Packing Problem via a Maximum Weighted Independent Set Heuristic

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ruizhi Li ◽  
Yupan Wang ◽  
Shuli Hu ◽  
Jianhua Jiang ◽  
Dantong Ouyang ◽  
...  
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problem ◽  
State Of The Art ◽  
Independent Set ◽  
Packing Problem ◽  
Combinatorial Optimization Problem ◽  
Set Packing ◽  
Benchmark Instances ◽  
Set Packing Problem

The set packing problem (SPP) is a significant NP-hard combinatorial optimization problem with extensive applications. In this paper, we encode the set packing problem as the maximum weighted independent set (MWIS) problem and solve the encoded problem with an efficient algorithm designed to the MWIS problem. We compare the independent set-based method with the state-of-the-art algorithms for the set packing problem on the 64 standard benchmark instances. The experimental results show that the independent set-based method is superior to the existing algorithms in terms of the quality of the solutions and running time obtained the solutions.

scholarly journals Estimating the size of search trees by sampling with domain knowledge

2017 ◽  
Author(s):  
Gleb Belov ◽  
Samuel Esler ◽  
Dylan Fernando ◽  
Pierre Le Bodic ◽  
George L. Nemhauser
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Branch And Bound ◽  
Domain Knowledge ◽  
Estimation Error ◽  
Search Tree ◽  
Branch And Bound Algorithm ◽  
Mixed Integer ◽  
Search Trees ◽  
Sampling Algorithm

We show how recently-defined abstract models of the Branch-and-Bound algorithm can be used to obtain information on how the nodes are distributed in B&B search trees. This can be directly exploited in the form of probabilities in a sampling algorithm given by Knuth that estimates the size of a search tree. This method reduces the offline estimation error by a factor of two on search trees from Mixed-Integer Programming instances.

scholarly journals An exploratory computational analysis of dual degeneracy in mixed-integer programming

2020 ◽  
Vol 8 (3-4) ◽  
pp. 241-261 ◽  
Author(s):  
Gerald Gamrath ◽  
Timo Berthold ◽  
Domenico Salvagnin
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Computational Study ◽  
Solution Process ◽  
Mixed Integer ◽  
Different Types ◽  
Primal Heuristics ◽  
Individual Variables ◽  
The Individual ◽  
Constraint Ratio

Abstract Dual degeneracy, i.e., the presence of multiple optimal bases to a linear programming (LP) problem, heavily affects the solution process of mixed integer programming (MIP) solvers. Different optimal bases lead to different cuts being generated, different branching decisions being taken and different solutions being found by primal heuristics. Nevertheless, only a few methods have been published that either avoid or exploit dual degeneracy. The aim of the present paper is to conduct a thorough computational study on the presence of dual degeneracy for the instances of well-known public MIP instance collections. How many instances are affected by dual degeneracy? How degenerate are the affected models? How does branching affect degeneracy: Does it increase or decrease by fixing variables? Can we identify different types of degenerate MIPs? As a tool to answer these questions, we introduce a new measure for dual degeneracy: the variable–constraint ratio of the optimal face. It provides an estimate for the likelihood that a basic variable can be pivoted out of the basis. Furthermore, we study how the so-called cloud intervals—the projections of the optimal face of the LP relaxations onto the individual variables—evolve during tree search and the implications for reducing the set of branching candidates.

scholarly journals Two-row and two-column mixed-integer presolve using hashing-based pairing methods

2020 ◽  
Vol 8 (3-4) ◽  
pp. 205-240
Author(s):  
Patrick Gemander ◽  
Wei-Kun Chen ◽  
Dieter Weninger ◽  
Leona Gottwald ◽  
Ambros Gleixner ◽  
...  
Keyword(s):  
Integer Programming ◽  
Linear Complexity ◽  
State Of The Art ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Set Packing ◽  
Large Array ◽  
Model Formulation ◽  
Reduction Techniques ◽  
The Impact

Abstract In state-of-the-art mixed-integer programming solvers, a large array of reduction techniques are applied to simplify the problem and strengthen the model formulation before starting the actual branch-and-cut phase. Despite their mathematical simplicity, these methods can have significant impact on the solvability of a given problem. However, a crucial property for employing presolve techniques successfully is their speed. Hence, most methods inspect constraints or variables individually in order to guarantee linear complexity. In this paper, we present new hashing-based pairing mechanisms that help to overcome known performance limitations of more powerful presolve techniques that consider pairs of rows or columns. Additionally, we develop an enhancement to one of these presolve techniques by exploiting the presence of set-packing structures on binary variables in order to strengthen the resulting reductions without increasing runtime. We analyze the impact of these methods on the MIPLIB 2017 benchmark set based on an implementation in the MIP solver SCIP.

scholarly journals FAST AND REGULARIZED RECONSTRUCTION OF BUILDING FAÇADES FROM STREET-VIEW IMAGES USING BINARY INTEGER PROGRAMMING

2020 ◽  
Vol V-2-2020 ◽  
pp. 365-371
Author(s):  
H. Hu ◽  
L. Wang ◽  
M. Zhang ◽  
Y. Ding ◽  
Q. Zhu
Keyword(s):  
Integer Programming ◽  
State Of The Art ◽  
A Priori ◽  
Mixed Integer ◽  
Binary Integer Programming ◽  
Building Facades ◽  
Interactive Environment ◽  
Real Value ◽  
Bounding Boxes ◽  
Additional Constraints

Abstract. Regularized arrangement of primitives on building façades to aligned locations and consistent sizes is important towards structured reconstruction of urban environment. Mixed integer linear programing was used to solve the problem, however, it is extremely time consuming even for state-of-the-art commercial solvers. Aiming to alleviate this issue, we cast the problem into binary integer programming, which omits the requirements for real value parameters and is more efficient to be solved. Firstly, the bounding boxes of the primitives are detected using the YOLOv3 architecture in real-time. Secondly, the coordinates of the upper left corners and the sizes of the bounding boxes are automatically clustered in a binary integer programming optimization, which jointly considers the geometric fitness, regularity and additional constraints; this step does not require a priori knowledge, such as the number of clusters or pre-defined grammars. Finally, the regularized bounding boxes can be directly used to guide the façade reconstruction in an interactive environment. Experimental evaluations have revealed that the accuracies for the extraction of primitives are above 0.82, which is sufficient for the following 3D reconstruction. The proposed approach only takes about 10% to 20% of the runtime than previous approach and reduces the diversity of the bounding boxes to about 20% to 50%.

scholarly journals Clairvoyant Restarts in Branch-and-Bound Search Using Online Tree-Size Estimation

2019 ◽  
Vol 33 ◽  
pp. 1427-1434 ◽  
Author(s):  
Daniel Anderson ◽  
Gregor Hendel ◽  
Pierre Le Bodic ◽  
Merlin Viernickel
Keyword(s):  
Branch And Bound ◽  
State Of The Art ◽  
Search Tree ◽  
Tree Size ◽  
Large Set ◽  
Size Estimation ◽  
Current Estimate ◽  
Benchmark Instances ◽  
Branch And Bound Search ◽  
Restart Strategy

We propose a simple and general online method to measure the search progress within the Branch-and-Bound algorithm, from which we estimate the size of the remaining search tree. We then show how this information can help solvers algorithmically at runtime by designing a restart strategy for MixedInteger Programming (MIP) solvers that decides whether to restart the search based on the current estimate of the number of remaining nodes in the tree. We refer to this type of algorithm as clairvoyant. Our clairvoyant restart strategy outperforms a state-of-the-art solver on a large set of publicly available MIP benchmark instances. It is implemented in the MIP solver SCIP and will be available in future releases.

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