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    "title": "Goldberg–Tarjan: computing maximum flow with preflows and local push–relabel operations",
    "summary": "Goldberg and Tarjan developed a maximum-flow method that temporarily permits excess flow at intermediate vertices, then uses local pushes and valid height labels to restore conservation and obtain a maximum feasible flow.\n\nA network can represent a transport system, a matching problem or another optimization task. Maximum flow asks how much can reach a sink while respecting edge capacities and conserving flow at intermediate vertices. Goldberg and Tarjan relaxed the conservation condition during computation: a preflow may temporarily leave excess at a vertex. A valid residual-distance labeling guides local pushes; when no push is possible, a relabel operation raises the vertex height. For the full-flow algorithm, termination removes intermediate excess and yields a maximum feasible flow. The generic bound is O(n²m) basic operations. Particular FIFO or maximum-distance implementations have O(n³) bounds; a dynamic-tree variant has the paper’s O(nm log(n²/m)) bound. These are distinct implementations, not interchangeable claims. The paper also treats parallel and distributed execution under explicit machine and message assumptions. It credits Karzanov’s earlier preflow idea and several inherited flow and dynamic-tree techniques. A minimum-cut/value-only variant may stop before restoring a complete feasible flow; that variant has a different output contract. The historical article does not establish current practical superiority over every other algorithm.",
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        "description": "The method maintains a preflow and admissible height labels on a finite capacitated network. Intermediate excess is permitted; termination and feasibility need the prescribed push/relabel conditions. Variant-specific arithmetic complexity bounds are not arbitrary-precision or real-world throughput measurements. Parallel and later optimized implementations require their own guarantees.",
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      "statement": "# Goldberg–Tarjan: computing maximum flow with preflows and local push–relabel operations\n\nGoldberg and Tarjan developed a maximum-flow method that temporarily permits excess flow at intermediate vertices, then uses local pushes and valid height labels to restore conservation and obtain a maximum feasible flow.\n\nA network can represent a transport system, a matching problem or another optimization task. Maximum flow asks how much can reach a sink while respecting edge capacities and conserving flow at intermediate vertices. Goldberg and Tarjan relaxed the conservation condition during computation: a preflow may temporarily leave excess at a vertex. A valid residual-distance labeling guides local pushes; when no push is possible, a relabel operation raises the vertex height. For the full-flow algorithm, termination removes intermediate excess and yields a maximum feasible flow. The generic bound is O(n²m) basic operations. Particular FIFO or maximum-distance implementations have O(n³) bounds; a dynamic-tree variant has the paper’s O(nm log(n²/m)) bound. These are distinct implementations, not interchangeable claims. The paper also treats parallel and distributed execution under explicit machine and message assumptions. It credits Karzanov’s earlier preflow idea and several inherited flow and dynamic-tree techniques. A minimum-cut/value-only variant may stop before restoring a complete feasible flow; that variant has a different output contract. The historical article does not establish current practical superiority over every other algorithm.\n\n## Scientific scope\n\nThe method maintains a preflow and admissible height labels on a finite capacitated network. Intermediate excess is permitted; termination and feasibility need the prescribed push/relabel conditions. Variant-specific arithmetic complexity bounds are not arbitrary-precision or real-world throughput measurements. Parallel and later optimized implementations require their own guarantees.\n\n## People and systems\n\nAndrew V. Goldberg; Robert E. Tarjan\n\n\n\n## Original sources\n\n- [A.V.Goldberg and R.E.Tarjan, A New Approach to the Maximum-Flow Problem (1988) — complete journal article](https://www.lrde.epita.fr/~jchazalo/SHARE/goldberg-tarjan-88-maximum-flow.pdf)\n",
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