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    "title": "Dinitz: a capacity-independent polynomial maximum-flow method using shortest-path referents",
    "summary": "Dinitz organized shortest augmenting paths into a maintained layered referent, deleting saturated arcs and dead ends until the shortest source-to-sink distance increases, obtaining an O(n²m) arithmetic-operation maximum-flow bound.\n\nA maximum-flow algorithm must deliver as much flow as possible without exceeding capacities or losing flow at intermediate vertices. Arbitrary augmenting-path choices can take a number of steps tied to capacity magnitudes and can even fail to terminate with general real capacities. Dinitz’s 1970 article chooses shortest residual augmenting paths and maintains their union, called a referent. Each augmentation deletes at least one arc from the current shortest-path structure; recursively removing dead ends preserves the useful part. Once that structure is empty, the next shortest-path distance increases. There are fewer than n distance levels and only finitely many deletions at each level. The article bounds arithmetic/machine operations by O(n²m), independently of numerical capacity magnitudes. This is an exact-arithmetic model claim, not a bound on arbitrary-precision bit costs. The widely taught BFS-plus-DFS blocking-flow implementation is related but has a later contribution: the author’s retrospective credits Shimon Even and Alon Itai with that modification, using Karzanov’s blocking-flow concept. The current record assesses the original maintained-referent method rather than silently attributing every later implementation to 1970.",
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    "topic": "Network optimization and maximum flow",
    "year": 1970,
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        "title": "E.A.Dinic, Algorithm for Solution of a Problem of Maximum Flow in a Network with Power Estimation —1970 English translation",
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      "sources": [
        {
          "citation": "E.A.Dinic, Algorithm for Solution of a Problem of Maximum Flow in a Network with Power Estimation —1970 English translation",
          "url": "https://www.cs.bgu.ac.il/~dinitz/D70.pdf"
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      "statement": "# Dinitz: a capacity-independent polynomial maximum-flow method using shortest-path referents\n\nDinitz organized shortest augmenting paths into a maintained layered referent, deleting saturated arcs and dead ends until the shortest source-to-sink distance increases, obtaining an O(n²m) arithmetic-operation maximum-flow bound.\n\nA maximum-flow algorithm must deliver as much flow as possible without exceeding capacities or losing flow at intermediate vertices. Arbitrary augmenting-path choices can take a number of steps tied to capacity magnitudes and can even fail to terminate with general real capacities. Dinitz’s 1970 article chooses shortest residual augmenting paths and maintains their union, called a referent. Each augmentation deletes at least one arc from the current shortest-path structure; recursively removing dead ends preserves the useful part. Once that structure is empty, the next shortest-path distance increases. There are fewer than n distance levels and only finitely many deletions at each level. The article bounds arithmetic/machine operations by O(n²m), independently of numerical capacity magnitudes. This is an exact-arithmetic model claim, not a bound on arbitrary-precision bit costs. The widely taught BFS-plus-DFS blocking-flow implementation is related but has a later contribution: the author’s retrospective credits Shimon Even and Alon Itai with that modification, using Karzanov’s blocking-flow concept. The current record assesses the original maintained-referent method rather than silently attributing every later implementation to 1970.\n\n## Scientific scope\n\nThe original method uses nonnegative capacities, opposite residual arcs and shortest-path referent maintenance. Its O(n^2 m) bound is an arithmetic/machine-operation count, not arbitrary-precision bit complexity or a measured modern runtime. Later DFS and blocking-flow modifications are credited separately. Finite graph tests do not independently certify the original amortized proof for every real-capacity network.\n\n## People and systems\n\nYefim Dinitz\n\n\n\n## Original sources\n\n- [E.A.Dinic, Algorithm for Solution of a Problem of Maximum Flow in a Network with Power Estimation —1970 English translation](https://www.cs.bgu.ac.il/~dinitz/D70.pdf)\n- [Yefim Dinitz, The Original Version and Even’s Version — author retrospective manuscript](https://www.cs.bgu.ac.il/~dinitz/Papers/Dinitz_alg.pdf)\n- [Yefim Dinitz institutional selected publications: historical E.A.Dinic spelling and1970 article](https://www.cs.bgu.ac.il/~dinitz/pub.html)\n",
      "summary": "Dinitz organized shortest augmenting paths into a maintained layered referent, deleting saturated arcs and dead ends until the shortest source-to-sink distance increases, obtaining an O(n²m) arithmetic-operation maximum-flow bound.\n\nA maximum-flow algorithm must deliver as much flow as possible without exceeding capacities or losing flow at intermediate vertices. Arbitrary augmenting-path choices can take a number of steps tied to capacity magnitudes and can even fail to terminate with general real capacities. Dinitz’s 1970 article chooses shortest residual augmenting paths and maintains their union, called a referent. Each augmentation deletes at least one arc from the current shortest-path structure; recursively removing dead ends preserves the useful part. Once that structure is empty, the next shortest-path distance increases. There are fewer than n distance levels and only finitely many deletions at each level. The article bounds arithmetic/machine operations by O(n²m), independently of numerical capacity magnitudes. This is an exact-arithmetic model claim, not a bound on arbitrary-precision bit costs. The widely taught BFS-plus-DFS blocking-flow implementation is related but has a later contribution: the author’s retrospective credits Shimon Even and Alon Itai with that modification, using Karzanov’s blocking-flow concept. The current record assesses the original maintained-referent method rather than silently attributing every later implementation to 1970.",
      "title": "Dinitz: a capacity-independent polynomial maximum-flow method using shortest-path referents"
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