branch and bound

The goal of a branch-and-bound algorithm is to find a value {{mvar|x}} that maximizes or minimizes the value of a real-valued function {{math|f(x)}}, called an objective function, among some set {{mvar|S}} of admissible, or candidate solutions. The set {{mvar|S}} is called the search space, or feasible region. The rest of this section assumes that minimization of {{math|f(x)}} is desired; this assumption comes without loss of generality, since one can find the maximum value of {{math|f(x)}} by finding the minimum of {{math|g(x) {{=}} −f(x)}}. A B&B algorithm operates according to two principles:

• It recursively splits the search space into smaller spaces, then minimizing {{math|f(x)}} on these smaller spaces; the splitting is called branching.
• Branching alone would amount to brute-force enumeration of candidate solutions and testing them all. To improve on the performance of brute-force search, a B&B algorithm keeps track of bounds on the minimum that it is trying to find, and uses these bounds to "prune" the search space, eliminating candidate solutions that it can prove will not contain an optimal solution.

Turning these principles into a concrete algorithm for a specific optimization problem requires some kind of data structure that represents sets of candidate solutions. Such a representation is called an instance of the problem. Denote the set of candidate solutions of an instance {{mvar|I}} by {{mvar|S_I}}. The instance representation has to come with three operations:

• {{math|branch(I)}} produces two or more instances that each represent a subset of {{mvar|S_I}}. (Typically, the subsets are disjoint to prevent the algorithm from visiting the same candidate solution twice, but this is not required. However, an optimal solution among {{mvar|S_I}} must be contained in at least one of the subsets.{{cite encyclopedia |first1=David A. |last1=Bader |first2=William E. |last2=Hart |first3=Cynthia A. |last3=Phillips |author3-link=Cynthia A. Phillips |title=Parallel Algorithm Design for Branch and Bound |editor-first=H. J. |editor-last=Greenberg |encyclopedia=Tutorials on Emerging Methodologies and Applications in Operations Research |publisher=Kluwer Academic Press |year=2004 |url=http://www.cc.gatech.edu/~bader/papers/ParallelBranchBound.pdf |access-date=2015-09-16 |archive-url=https://web.archive.org/web/20170813145917/http://www.cc.gatech.edu/~bader/papers/ParallelBranchBound.pdf |archive-date=2017-08-13 |url-status=dead }})
• {{math|bound(I)}} computes a lower bound on the value of any candidate solution in the space represented by {{mvar|I}}, that is, {{math|bound(I) ≤ f(x)}} for all {{mvar|x}} in {{mvar|S_I}}.
• {{math|solution(I)}} determines whether {{mvar|I}} represents a single candidate solution. (Optionally, if it does not, the operation may choose to return some feasible solution from among {{mvar|S_I}}.{{r|bader}}) If {{math|solution(I)}} returns a solution then {{math|f(solution(I))}} provides an upper bound for the optimal objective value over the whole space of feasible solutions.

Using these operations, a B&B algorithm performs a top-down recursive search through the tree of instances formed by the branch operation. Upon visiting an instance {{mvar|I}}, it checks whether {{math|bound(I)}} is equal or greater than the current upper bound; if so, {{mvar|I}} may be safely discarded from the search and the recursion stops. This pruning step is usually implemented by maintaining a global variable that records the minimum upper bound seen among all instances examined so far.

The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function {{mvar|f}}.{{cite tech report |first=Jens |last=Clausen |title=Branch and Bound Algorithms—Principles and Examples |year=1999 |publisher=University of Copenhagen |url=http://www.diku.dk/OLD/undervisning/2003e/datV-optimer/JensClausenNoter.pdf |access-date=2014-08-13 |archive-url=https://web.archive.org/web/20150923214803/http://www.diku.dk/OLD/undervisning/2003e/datV-optimer/JensClausenNoter.pdf |archive-date=2015-09-23 |url-status=dead }} To obtain an actual algorithm from this, one requires a bounding function {{math|bound}}, that computes lower bounds of {{mvar|f}} on nodes of the search tree, as well as a problem-specific branching rule. As such, the generic algorithm presented here is a higher-order function.

1. Using a heuristic, find a solution {{mvar|x_h}} to the optimization problem. Store its value, {{math|B {{=}} f(x_h)}}. (If no heuristic is available, set {{mvar|B}} to infinity.) {{mvar|B}} will denote the best solution found so far, and will be used as an upper bound on candidate solutions.
2. Initialize a queue to hold a partial solution with none of the variables of the problem assigned.
3. Loop until the queue is empty:
4. Take a node {{mvar|N}} off the queue.
5. If {{mvar|N}} represents a single candidate solution {{mvar|x}} and {{math|f(x) < B}}, then {{mvar|x}} is the best solution so far. Record it and set {{math|B ← f(x)}}.
6. Else, branch on {{mvar|N}} to produce new nodes {{mvar|N_i}}. For each of these:
7. If {{math|bound(N_i) > B}}, do nothing; since the lower bound on this node is greater than the upper bound of the problem, it will never lead to the optimal solution, and can be discarded.
8. Else, store {{mvar|N_i}} on the queue.

Several different queue data structures can be used. This FIFO queue-based implementation yields a breadth-first search. A stack (LIFO queue) will yield a depth-first algorithm. A best-first branch and bound algorithm can be obtained by using a priority queue that sorts nodes on their lower bound.

Examples of best-first search algorithms with this premise are Dijkstra's algorithm and its descendant A* search. The depth-first variant is recommended when no good heuristic is available for producing an initial solution, because it quickly produces full solutions, and therefore upper bounds.{{cite book |last1=Mehlhorn |first1=Kurt |author-link1=Kurt Mehlhorn |first2=Peter |last2=Sanders|author2-link=Peter Sanders (computer scientist) |title=Algorithms and Data Structures: The Basic Toolbox |publisher=Springer |year=2008 |page=249 |url=http://people.mpi-inf.mpg.de/~mehlhorn/ftp/Toolbox/GenericMethods.pdf}}

A C++-like pseudocode implementation of the above is:

// C++-like implementation of branch and bound,

// assuming the objective function f is to be minimized

CombinatorialSolution branch_and_bound_solve(

CombinatorialProblem problem,

ObjectiveFunction objective_function /*f*/,

BoundingFunction lower_bound_function /*bound*/)

{

// Step 1 above.

double problem_upper_bound = std::numeric_limits::infinity; // = B

CombinatorialSolution heuristic_solution = heuristic_solve(problem); // x_h

problem_upper_bound = objective_function(heuristic_solution); // B = f(x_h)

CombinatorialSolution current_optimum = heuristic_solution;

// Step 2 above

queue candidate_queue;

// problem-specific queue initialization

candidate_queue = populate_candidates(problem);

while (!candidate_queue.empty()) { // Step 3 above

// Step 3.1

CandidateSolutionTree node = candidate_queue.pop();

// "node" represents N above

if (node.represents_single_candidate()) { // Step 3.2

if (objective_function(node.candidate()) < problem_upper_bound) {

current_optimum = node.candidate();

problem_upper_bound = objective_function(current_optimum);

}

// else, node is a single candidate which is not optimum

}

else { // Step 3.3: node represents a branch of candidate solutions

// "child_branch" represents N_i above

for (auto&& child_branch : node.candidate_nodes) {

if (lower_bound_function(child_branch) <= problem_upper_bound) {

candidate_queue.enqueue(child_branch); // Step 3.3.2

}

// otherwise, bound(N_i) > B so we prune the branch; step 3.3.1

}

}

}

return current_optimum;

}

In the above pseudocode, the functions heuristic_solve and populate_candidates called as subroutines must be provided as applicable to the problem. The functions {{mvar|f}} (objective_function) and {{math|bound}} (lower_bound_function) are treated as function objects as written, and could correspond to lambda expressions, function pointers and other types of callable objects in the C++ programming language.

When $\backslash mathbf\{x\}$ is a vector of $\backslash mathbb\{R\}^n$, branch and bound algorithms can be combined with interval analysis{{cite book|last1=Moore|first1=R. E.|

title=Interval Analysis|

year=1966|publisher=Prentice-Hall|

location=Englewood Cliff, New Jersey|isbn=0-13-476853-1}}

and contractor techniques in order to provide guaranteed enclosures of the global minimum.

{{cite book|last1=Jaulin|first1=L.|last2=Kieffer|first2=M.|last3=Didrit|first3=O.|last4=Walter|first4=E.|

title=Applied Interval Analysis|year=2001|publisher=Springer|location=Berlin|isbn=1-85233-219-0}}

{{cite book|last=Hansen|first=E.R.|

title=Global Optimization using Interval Analysis|year=1992|

publisher=Marcel Dekker|location=New York}}

{{Prose|section|date=February 2023}}

This approach is used for a number of NP-hard problems:

• Integer programming
• Nonlinear programming
• Travelling salesman problem (TSP){{cite journal |last1=Little |first1=John D. C. |last2=Murty |first2=Katta G. |last3=Sweeney |first3=Dura W. |last4=Karel |first4=Caroline |title=An algorithm for the traveling salesman problem |journal=Operations Research |volume=11 |issue=6 |year=1963 |pages=972–989 |doi=10.1287/opre.11.6.972 |url=http://dspace.mit.edu/bitstream/handle/1721.1/46828/algorithmfortrav00litt.pdf|hdl=1721.1/46828 |hdl-access=free }}{{cite book |author1-link=Richard W. Conway|author2-link=William L. Maxwell|first1=Richard Walter |last1=Conway |first2=William L. |last2=Maxwell |first3=Louis W. |last3=Miller |year=2003 |title=Theory of Scheduling |url=https://archive.org/details/theoryofscheduli0000conw |url-access=registration |publisher=Courier Dover Publications |pages=[https://archive.org/details/theoryofscheduli0000conw/page/56 56–61]|isbn=978-0-486-42817-8 }}
• Quadratic assignment problem (QAP)
• Maximum satisfiability problem (MAX-SAT)
• Nearest neighbor search{{cite journal |last1=Fukunaga |first1=Keinosuke |first2=Patrenahalli M. |last2=Narendra |title=A branch and bound algorithm for computing {{mvar|k}}-nearest neighbors |journal=IEEE Transactions on Computers |year=1975 |issue=7 |pages=750–753|doi=10.1109/t-c.1975.224297 |s2cid=5941649 }} (by Keinosuke Fukunaga)
• Flow shop scheduling
• Cutting stock problem
• Computational phylogenetics
• Set inversion
• Parameter estimation
• 0/1 knapsack problem
• Set cover problem
• Feature selection in machine learning{{cite journal |title=A branch and bound algorithm for feature subset selection |last1=Narendra |first1=Patrenahalli M. |last2=Fukunaga |first2=K. |journal=IEEE Transactions on Computers |volume=C-26 |issue=9 |year=1977 |pages=917–922 |doi=10.1109/TC.1977.1674939 |s2cid=26204315 |url=http://www.computer.org/csdl/trans/tc/1977/09/01674939.pdf}}{{cite arXiv |last1=Hazimeh |first1=Hussein| last2=Mazumder |first2=Rahul |last3=Saab |first3=Ali |eprint=2004.06152 |title=Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization |date=2020|class=stat.CO }}
• Structured prediction in computer vision{{Cite journal | first1 = Sebastian | last1 = Nowozin | first2 = Christoph H. | last2 = Lampert | title = Structured Learning and Prediction in Computer Vision | journal = Foundations and Trends in Computer Graphics and Vision | volume = 6 | issue = 3–4 | year = 2011 | pages = 185–365 | doi = 10.1561/0600000033 | isbn = 978-1-60198-457-9| citeseerx = 10.1.1.636.2651 }}{{rp|267–276}}
• Arc routing problem, including Chinese Postman problem
• Talent Scheduling, scenes shooting arrangement problem

Branch-and-bound may also be a base of various heuristics. For example, one may wish to stop branching when the gap between the upper and lower bounds becomes smaller than a certain threshold. This is used when the solution is "good enough for practical purposes" and can greatly reduce the computations required. This type of solution is particularly applicable when the cost function used is noisy or is the result of statistical estimates and so is not known precisely but rather only known to lie within a range of values with a specific probability.{{Citation needed|date=September 2015}}

Nau et al. present a generalization of branch and bound that also subsumes the A*, B* and alpha-beta search algorithms.{{cite journal |last1=Nau |first1=Dana S. |first2=Vipin |last2=Kumar |first3=Laveen |last3=Kanal |title=General branch and bound, and its relation to A∗ and AO∗ |journal=Artificial Intelligence |volume=23 |issue=1 |year=1984 |pages=29–58 |url=https://www.cs.umd.edu/~nau/papers/nau1984general.pdf | doi = 10.1016/0004-3702(84)90004-3 }}

Branch and bound can be used to solve this problem

Maximize $Z=5x\_1+6x\_2$ with these constraints

$x\_1+x\_2\backslash leq\; 50$

$4x\_1+7x\_2\backslash leq280$

$x\_1,\; x\_2\backslash geq0$

$x\_1$ and $x\_2$ are integers.

The first step is to relax the integer constraint. We have two extreme points for the first equation that form a line: $\backslash begin\{bmatrix\}\; x\_1\; \backslash \backslash \; x\_2\; \backslash end\{bmatrix\}=\backslash begin\{bmatrix\}50\; \backslash \backslash 0\backslash end\{bmatrix\}$ and $\backslash begin\{bmatrix\}0\; \backslash \backslash 50\backslash end\{bmatrix\}$. We can form the second line with the vector points $\backslash begin\{bmatrix\}0\backslash \backslash 40\backslash end\{bmatrix\}$ and $\backslash begin\{bmatrix\}\; 70\backslash \backslash 0\backslash end\{bmatrix\}$.

File:Branch and Bound optimization example with linear constraints in BricsCad Ultimate Academic Edition.png

The third point is $\backslash begin\{bmatrix\}0\backslash \backslash 0\backslash end\{bmatrix\}$. This is a convex hull region so the solution lies on one of the vertices of the region. We can find the intersection using row reduction, which is $\backslash begin\{bmatrix\}70/3\backslash \backslash 80/3\backslash end\{bmatrix\}$, or $\backslash begin\{bmatrix\}\; 23.333\backslash \backslash 26.667\backslash end\{bmatrix\}$ with a value of 276.667. We test the other endpoints by sweeping the line over the region and find this is the maximum over the reals.

We choose the variable with the maximum fractional part, in this case $x\_2$ becomes the parameter for the branch and bound method. We branch to $x\_2\backslash leq26$ and obtain 276 @ $\backslash langle\; 24,26\backslash rangle$. We have reached an integer solution so we move to the other branch $x\_2\backslash geq27$. We obtain 275.75 @$\backslash langle\; 22.75,\; 27\backslash rangle$. We have a decimal so we branch $x\_1$ to $x\_1\backslash leq22$ and we find 274.571 @$\backslash langle\; 22,27.4286\backslash rangle$. We try the other branch $x\_1\backslash geq23$ and there are no feasible solutions. Therefore, the maximum is 276 with $x\_1\backslash longmapsto\; 24$ and $x\_2\backslash longmapsto\; 26$.

• Backtracking
• Branch-and-cut, a hybrid between branch-and-bound and the cutting plane methods that is used extensively for solving integer linear programs.
• Evolutionary algorithm
• Alpha–beta pruning

{{Reflist|30em}}

• [http://sourceforge.net/projects/lipside/ LiPS] – Free easy-to-use GUI program intended for solving linear, integer and goal programming problems.
• [https://projects.coin-or.org/Cbc Cbc] – (Coin-or branch and cut) is an open-source mixed integer programming solver written in C++.

{{Optimization algorithms|combinatorial|state=expanded}}

{{DEFAULTSORT:Branch And Bound}}

Category:Optimization algorithms and methods

Category:Combinatorial optimization