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The Complexity of Trade-offs

The Complexity of Trade-offs. Christos H. Papadimitriou UC Berkeley (JWWMY). The web access problem [Etzioni et al , FOCS 1996]. n sources of information, 1, …, n for each one of them: cost c i , time t i , quality q i Choose a set S  {1,2,…, n }, with the best

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The Complexity of Trade-offs

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  1. The Complexity of Trade-offs Christos H. Papadimitriou UC Berkeley (JWWMY) IEOR March 12

  2. The web access problem [Etzioni et al, FOCS 1996] • n sources of information, 1, …, n • for each one of them: cost ci , time ti , quality qi Choose a set S {1,2,…,n}, with the best • cost [S] = i  S ci • time [S] = max i  S ti • quality [S] = 1 i  S (1 q i ) IEOR March 12

  3. Multiobjective optimization e.g., shortest path, minimum spanning tree, etc, or ad hoc problems, with k > 1 objectives in the Sequoia database system, users provide a desired time/$ trade-off What does it mean to solve such a problem? IEOR March 12

  4. quality Pareto curve solutions (sets S) M - cost IEOR March 12

  5. But… • number of undominated points is usually exponential • even for 2-objective shortest paths (and almost every other problem), it’s NP-hard even to find “the next point” : knapsack IEOR March 12

  6. so, the problem seems to lie outside the realm of • algorithmic analysis • many thousands of papers, dozens of books • over the past 50 years • algorithmic/computational issues ignored • otherwise, multiple criteria seen as a framework • for identifying the true single criterion • (e.g., goal programming) IEOR March 12

  7. idea: -approximate Pareto curve • Pareto curve: set of points (p1,…,pM) which collectively dominate all solutions • -approximate Pareto curve: set of points (p1,…,pm) such that ((1+)p1,…, ((1+)pm) collectively dominate all solutions IEOR March 12

  8. Theorem: There is always an -approximatePareto curve of polynomial size (in n and 1/ ) log(obj1) Proof: Plot objectives log-log Subdivide into (1 + ) “cubes.” 1 +  Retain one point per “cube” O((n/ )k-1) points log(obj2) IEOR March 12

  9. precursors: [Hansen 79] shortest paths [CJK 98] scheduling: -aPc in polynomial time [Orlin and Safer 92] general definition, theory IEOR March 12

  10. Theorem:-aPc can be computed in polynomial time iff the following problem can be so solved: • “Given an instance and b1 ,…, bk , either: • Find a solution x with obji(x) bi for all i, or • Decide that there is no solution with • obji(x) >bi (1 + ), for all i” IEOR March 12

  11. Multiple linear objectives • multiobjective shortest path • multiobjective minimum spanning tree • multiobjective minimum cost flow • multiobjective matching • multiobjective minimum cut IEOR March 12

  12. Convex or discrete? is the convex combination of two solutions also a solution? is this point in the Pareto curve? IEOR March 12

  13. convex? • multiobjective shortest path • multiobjective minimum spanning tree • multiobjective minimum cost flow • multiobjective matching • multiobjective minimum cut  IEOR March 12

  14. Theorem: A convex multiobjective problem is approximately solvable in polynomial time iff the single-objective problem is. • First proof: Ellipsoid, separation, duality • Second proof: Solve the single-objective problem approximately for all objectives of the form wi ci with all weights wi in the range [1, …, (1/ )2k], and keep all undominated solutions. IEOR March 12

  15. Discrete problems? Theorem: A discrete multiobjective problem can be approximated in polynomial time if the exact version can be solved in (pseudo)polynomial time Exact version: “Given an instance and an integer K in unary, is there a solution with cost exactly K?” IEOR March 12

  16. multiobjective shortest path • multiobjective minimum spanning tree • multiobjective minimum cost flow • multiobjective matching • multiobjective minimum cut P P P RNC NP-hard (because the exact version of min cut is the same as the exact version of max cut…) IEOR March 12

  17. PS: The web access problem can be approximated in O(n2/ ) time (dynamic programming for each time value) Ditto for the time/resources trade-off in the query optimization problem for the Sequoia database system [Stonebreaker et al. 95] [PY, to appear in PODS 01] IEOR March 12

  18. Open problems • Faster algorithms? • Other problems? • Necessary and sufficient condition for discrete problems? • The “sweet spot” problem: Find x such that (1 + )obji (x) > obji (y) for all objectives and solutions y IEOR March 12

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