Combinatorial optimization explained

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects,[1] where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.

Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science.

Applications

Applications of combinatorial optimization include, but are not limited to:

Methods

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization. A considerable amount of it is unified by the theory of linear programming. Some examples of combinatorial optimization problems that are covered by this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.

For NP-complete discrete optimization problems, current research literature includes the following topics:

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Widely applicable approaches include branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman (decision) problem,[6] this is expected unless P=NP.

For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure

m0

. For example, if there is a graph

G

which contains vertices

u

and

v

, an optimization problem might be "find a path from

u

to

v

that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from

u

to

v

that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

The field of approximation algorithms deals with algorithms to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is then more naturally characterized as an optimization problem.

NP optimization problem

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

y\inf(x)

is polynomially bounded in the size of the given instance

x

,

\{x\midx\inI\}

and

\{(x,y)\midy\inf(x)\}

can be recognized in polynomial time, and

m

is polynomial-time computable.

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

NPO is divided into the following subclasses according to their approximability:

1/c

of the optimal cost (for maximization problems). In Hromkovič's book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains MAX-SAT and metric TSP.

An NPO problem is called polynomially bounded (PB) if, for every instance

x

and for every solution

y\inf(x)

, the measure

m(x,y)

is bounded by a polynomial function of the size of

x

. The class NPOPB is the class of NPO problems that are polynomially-bounded.

Specific problems

See also

References

External links

Notes and References

  1. .
  2. 10.1007/s10288-007-0047-3. Combinatorial optimization and Green Logistics. 4OR. 5. 2. 99–116. 2007. Sbihi. Abdelkader. Eglese. Richard W.. 207070217. 2019-12-26. 2019-12-26. https://web.archive.org/web/20191226162210/https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf. live.
  3. 10.1016/j.omega.2015.01.006. Sustainable supply chain network design: An optimization-oriented review. Omega. 54. 11–32. 2015. Eskandarpour. Majid. Dejax. Pierre. Miemczyk. Joe. Péton. Olivier. 2019-12-26. 2019-12-26. https://web.archive.org/web/20191226162207/https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%2520review%2520R2.pdf. live.
  4. 10.1016/j.advwatres.2018.10.002. Estimating fluid flow rates through fracture networks using combinatorial optimization. Advances in Water Resources. 2018. Hobé. Alex. Vogler. Daniel. Seybold. Martin P.. Ebigbo. Anozie. Settgast. Randolph R.. Saar. Martin O.. 122. 85–97. 1801.08321. 2018AdWR..122...85H. 119476042. 2020-09-16. 2020-08-21. https://web.archive.org/web/20200821114559/https://www.sciencedirect.com/science/article/abs/pii/S0309170818300666. live.
  5. .
  6. Web site: Approximation-TSP. 2022-02-17. 2022-03-01. https://web.archive.org/web/20220301220806/https://www.csd.uoc.gr/~hy583/papers/ch11.pdf. live.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Combinatorial optimization".

Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2024, A B Cryer, All Rights Reserved. Cookie policy.