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| Held–Karp bound | |
|---|---|
| Name | Held–Karp bound |
| Authors | Richard M. Karp; Michael Held |
| Introduced | 1962 |
| Field | Combinatorial optimization |
| Related | Travelling Salesman Problem, Linear programming, Branch and bound |
Held–Karp bound
The Held–Karp bound is a lower bound technique for the Travelling Salesman Problem first proposed by Michael Held and Richard M. Karp. It combines ideas from Lagrange multiplier methods, minimum spanning tree, and linear programming relaxations to produce strong provable bounds used in exact and approximate algorithms. The bound plays a central role in studies by researchers associated with institutions such as Bell Labs, MIT, UC Berkeley, and research groups influencing work at AT&T Bell Laboratories and IBM Research.
The Held–Karp bound is defined as the optimal value of a particular relaxation of the Traveling Salesman Problem that enforces degree constraints via dual variables associated with vertices. The statement of the bound originates from the Held–Karp 1962 formulation which uses a 1-tree relaxation together with Lagrangian penalties; the resulting bound equals the maximum, over a space of dual variables, of the cost of the cheapest feasible structure adjusted by penalties. Early expositions appeared in surveys connected to conferences such as the Symposium on Theory of Computing and journals linked to the IEEE and SIAM communities.
Mathematically, the bound is obtained by considering the complete graph K_n on n labeled vertices and relaxing the integer constraints of the classical assignment problem or subtour elimination form of the Traveling Salesman Problem. Introduce dual variables (often called potentials) λ_i associated with each vertex i; form a modified edge weight c'_{ij} = c_{ij} + λ_i + λ_j. Compute a minimum-cost 1-tree on K_n under weights c'_{ij} and subtract 2∑_i λ_i to obtain the Lagrangian value. The Held–Karp value is the supremum of these Lagrangian values over all choices of λ = (λ_1,...,λ_n). This optimization can be seen as a dual of a linear program closely related to the subtour elimination linear program and is connected to classical results by Dantzig, Fulkerson, and Johnson.
Algorithmic computation of the bound typically alternates between computing minimum spanning tree or minimum 1-tree structures and updating dual variables using subgradient or bundle methods inspired by work at Bell Labs and optimization labs at Stanford University and Carnegie Mellon University. Practical implementations use iterative schemes such as the subgradient optimization algorithm, cutting-plane methods akin to those in Gomory cutting-plane theory, or interior-point methods developed in Mathematical Programming research. Held–Karp bound computations are integrated into branch-and-bound solvers used by teams at Concorde TSP Solver projects, and improvements have been reported by groups at Microsoft Research, Google Research, and academic groups at Princeton University and ETH Zurich.
The Held–Karp bound is provably at most the cost of an optimal TSP tour and at least half the cost in metric instances under the triangle inequality; stronger inequalities have been established in metric spaces studied by researchers at University of Waterloo and University of Rome. It equals the optimum of the subtour-elimination linear programming relaxation in many formulations and therefore inherits polyhedral properties studied by Wolsey and Padberg in the theory of combinatorial optimization. Tightness results, integrality gap bounds, and worst-case instances have been analyzed in literature from Princeton and Cornell, with connections to probabilistic analyses by groups including Google Brain affiliates and statistical treatments by researchers at Columbia University.
Practitioners use the Held–Karp bound as a preprocessing and pruning tool in exact solvers for routing, logistics, and scheduling problems tackled by teams at UPS, FedEx, DHL, and national laboratories such as Los Alamos National Laboratory. It also serves as a benchmark in approximation algorithm research at institutions like Harvard University and Yale University and in empirical studies comparing heuristics developed by startups and centers at UC Irvine and Georgia Tech. The bound is applied within hybrid methods combining metaheuristics from INRIA research groups with deterministic solvers employed in industrial projects by Siemens and General Electric.
Extensions include strengthened Lagrangian relaxations incorporating blossom inequalities from Edmonds and branch-and-cut hybrids developed in projects at DIMACS and NEOS Server collaborations. Related bounds arise from semidefinite programming relaxations explored by researchers at Princeton University and EPFL, polyhedral bounds studied by Padberg and Rinaldi, and probabilistic bounds in random-graph models investigated by groups at University of Chicago and University of British Columbia. Contemporary work links the Held–Karp approach to modern convex relaxations used in machine learning collaborations between Stanford and DeepMind labs.