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| Cost-scaling algorithm | |
|---|---|
| Name | Cost-scaling algorithm |
| Field | Computer science |
| Invented by | Andrew V. Goldberg, Robert E. Tarjan |
| First published | 1987 |
| Applications | Network flow, combinatorial optimization, operations research |
Cost-scaling algorithm The cost-scaling algorithm is a family of algorithms for solving the minimum-cost flow problem and related combinatorial optimization problems. It transforms cost parameters progressively and uses combinatorial primitives to maintain feasible flows while reducing reduced costs, providing strong worst-case performance and practical efficiency in large-scale instances.
The cost-scaling algorithm addresses the minimum-cost flow problem, connecting to landmark results by Andrew V. Goldberg, Robert E. Tarjan, and work at institutions like AT&T Bell Laboratories, Princeton University, and Stanford University. It builds on classical studies such as the Ford–Fulkerson algorithm, the Edmonds–Karp algorithm, and the Successive Shortest Path algorithm, and relates to frameworks developed in texts from Richard M. Karp and Jack Edmonds. The method is relevant to practical systems developed by organizations including IBM, Microsoft Research, Google, and Hewlett-Packard and to theoretical advances by scholars like Michael L. Fredman and Daniel Spielman.
A representative cost-scaling scheme maintains a flow and a price vector (potentials) on vertices, using an epsilon-scaling parameter similar to approaches in Dijkstra's algorithm refinements and scaling techniques seen in Gomory–Hu tree methods. Each phase reduces epsilon, performing augmentations along admissible arcs determined by reduced costs, invoking data structures reminiscent of those in Fibonacci heap implementations and exploiting invariants from Tarjan's amortized analysis. Subroutines use shortest-path computations akin to Bellman–Ford algorithm or modified Dijkstra's algorithm with potentials, and augment along residual networks akin to procedures in Push–relabel algorithm and Dinic's algorithm. The design parallels cost-reduction strategies from the Hungarian algorithm for assignment problems and leverages duality concepts related to classical results by John von Neumann and Leonid Kantorovich.
Theoretical bounds for cost-scaling variants often match or improve upon earlier algorithms: Goldberg and Tarjan provided strongly polynomial or pseudo-polynomial guarantees comparable to results in Karp's algorithmic complexity studies and later refinements by Ullman and Megiddo. Worst-case running times are typically near O(n^2 m log U) for integer costs with scaling parameter U, with variants achieving near O(n m log n) under additional assumptions, echoing improvements found in Fredman and Tarjan analyses. Practical performance comparisons involve benchmarks used by groups such as NEOS Server and algorithm collections at DIMACS, and are reported in empirical studies from research centers including MIT and CMU.
Multiple variants exist: epsilon-scaling, cost-scaling with blocking flows, and combined cost/flow-scaling hybrids influenced by techniques from Klein's algorithm and adaptations used in IBM ILOG CPLEX and Gurobi solvers. Implementations differ in residual graph maintenance, priority queue choices (e.g., binary heap, Fibonacci heap), and global relabeling schedules inspired by strategies in Push–relabel algorithm codebases. Engineering adaptations draw on software libraries from LEDA and CGAL and are employed in systems developed at Bell Labs, Microsoft Research, and the University of Washington.
Cost-scaling algorithms are widely applied in transportation planning for projects like Interstate Highway System modeling and urban transit optimization used by agencies associated with Port Authority of New York and New Jersey or Transport for London. They underpin assignment and matching problems in labor markets studied by Alvin E. Roth and Lloyd Shapley type markets, logistics for firms such as UPS and FedEx, and telecommunication routing in networks managed by companies like AT&T and Verizon Communications. Scientific applications occur in computational biology at institutions including Broad Institute and Sanger Institute, and in energy grid optimization involving organizations like National Grid plc and California Independent System Operator.
The cost-scaling paradigm emerged alongside breakthroughs in network flow theory from researchers at Bell Labs and universities including Princeton University and Stanford University. Foundational contributions by Goldberg and Tarjan built on classical predecessors such as Edmonds–Karp and Hochbaum; subsequent related work includes scaling ideas from Megiddo and combinatorial refinements by Gomory and Hu. Later connections were drawn to interior-point methods popularized by N. Z. Shor and Karmarkar, and to algorithm engineering traditions advanced at DIMACS and by practitioners at IBM and Microsoft Research.
Practical deployments tune scaling schedules, choose priority queue implementations (e.g., binary heap vs Fibonacci heap), and integrate heuristics such as global relabeling and gap relabeling drawn from Push–relabel algorithm literature. Memory layout and low-level optimizations mirror engineering practices from projects at Google and Facebook and follow performance profiling methodologies used at Intel and AMD. For large-scale instances, parallel and distributed adaptations have been explored in collaborations involving Lawrence Berkeley National Laboratory and Sandia National Laboratories, and hybrid approaches combine cost-scaling with linear programming solvers from CPLEX and Gurobi for commercial-grade workflows.
Category:Network flow algorithms