Combinatorial optimization

Combinatorial optimization
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Name	Combinatorial optimization
Field	Mathematics; Computer Science; Operations Research
Introduced	20th century

Combinatorial optimization is a field in Mathematics and Computer science that studies optimization of discrete structures such as graphs, networks, and sets. It connects to topics in Operations research, Theoretical computer science, and Discrete mathematics, and has historical ties to problems studied by figures affiliated with institutions like Bell Labs, MIT, and Princeton University. Research in the area spans contributions from awardees of the Turing Award, recipients of the Gödel Prize, and members of academies such as the National Academy of Sciences.

Overview

The field considers problems defined on combinatorial objects like graphs, sets, and matroids arising in settings associated with practitioners from Bell Labs, Bell Telephone Laboratories, and departments at Stanford University and Carnegie Mellon University. Classic instances include tasks tied to historical projects at RAND Corporation and algorithmic breakthroughs associated with scholars awarded the Turing Award and the Knuth Prize. Methodological threads link to results published in journals of the American Mathematical Society, contributions by researchers at IBM Research, and collaborations involving universities such as University of California, Berkeley and Harvard University.

Problem types and models

Central problem families include the Traveling salesman problem (TSP), the Knapsack problem, the Minimum spanning tree problem, the Maximum flow problem, the maximum matching problem, and the Set cover problem. Models used encompass Integer programming, Linear programming, matroids, and constraint satisfaction problems as studied by teams at INRIA, ETH Zurich, and University of Cambridge. Specialized formulations reference structures like Hypergraph, Bipartite graph, and constructs from work at Los Alamos National Laboratory and projects funded by agencies such as the National Science Foundation.

Algorithms and methods

Algorithmic paradigms include exact techniques like branch-and-bound and cutting planes popularized in contexts involving IBM, heuristic and metaheuristic strategies like simulated annealing inspired by work of researchers connected to Bell Labs and Los Alamos National Laboratory, and approximation algorithms developed in the tradition of scholars affiliated with MIT and Stanford University. Key algorithmic milestones reference the Ellipsoid method, Simplex algorithm, and combinatorial algorithms such as the Edmonds' blossom algorithm for matching and the Dijkstra's algorithm for shortest paths. Methods also incorporate randomized algorithms linked to results associated with the ACM and derandomization techniques explored by authors connected to Microsoft Research and the Institute for Advanced Study.

Complexity and computational issues

Complexity classifications draw on the framework of P versus NP and hardness results for problems like the Traveling salesman problem and the Set cover problem proven via reductions related to seminal work by researchers connected to the Princeton University and the University of Waterloo. Approximation hardness and inapproximability bounds reference concepts developed alongside the Probabilistically checkable proofs (PCP) theorem and contributions associated with the Clay Mathematics Institute Millennium problems community. Resource-bounded computation and parameterized complexity tie into programs at Carnegie Mellon University and research supported by the European Research Council.

Applications

Applications span route planning challenges used by firms such as UPS and FedEx, network design problems relevant to research at AT&T and Cisco Systems, and scheduling tasks in industries connected to Boeing and General Motors. In bioinformatics, techniques are applied to sequence assembly and phylogenetics in collaborations involving Broad Institute and Wellcome Trust Sanger Institute. Financial optimization and portfolio construction draw on models used at Goldman Sachs and JPMorgan Chase, while logistics and supply chain problems involve deployments by organizations such as Walmart and Amazon.

Software and benchmarking

Prominent software ecosystems include solvers like CPLEX and Gurobi developed by commercial teams, open-source projects such as COIN-OR and GLPK, and research platforms from groups at Google and Microsoft Research. Benchmarking suites and challenge instances are curated by conferences and workshops organized by entities like the SIAM and the ACM and hosted datasets originating from institutions including DIMACS and competitions sponsored by the European Union. Performance evaluation often references leaderboards maintained by consortia involving INFORMS and research labs at ETH Zurich.

Category:Optimization