1. IEOR March 121 The Complexity of Trade-offs Christos H. Papadimitriou UC Berkeley (JWWMY)

2. IEOR March 122 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 cost [S] =  i  S c i time [S] = max i  S t i quality [S] = 1   i  S (1  q i )

3. IEOR March 123 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?

4. IEOR March 124 quality M - cost Pareto curve solutions (sets S)

5. IEOR March 125 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

6. IEOR March 126 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)

7. IEOR March 127 idea:  -approximate Pareto curve Pareto curve: set of points (p 1,…,p M ) which collectively dominate all solutions  -approximate Pareto curve: set of points (p 1,…,p m ) such that ((1+  )p 1,…, ((1+  )p m ) collectively dominate all solutions

8. IEOR March 128 Theorem: There is always an  -approximate Pareto curve of polynomial size (in n and 1/  ) log(obj1) log(obj2) Proof: Plot objectives log-log Subdivide into (1 +  ) “cubes.” Retain one point per “cube” O((n/  ) k-1 ) points 1 + 

9. IEOR March 129 precursors: [Hansen 79] shortest paths [CJK 98] scheduling:  -aPc in polynomial time [Orlin and Safer 92] general definition, theory

10. IEOR March 1210 Theorem:  -aPc can be computed in polynomial time iff the following problem can be so solved: “Given an instance and b 1,…, b k, either: Find a solution x with obj i (x)  b i for all i, or Decide that there is no solution with obj i (x) > b i (1 +  ), for all i”

11. IEOR March 1211 Multiple linear objectives multiobjective shortest path multiobjective minimum spanning tree multiobjective minimum cost flow multiobjective matching multiobjective minimum cut

12. IEOR March 1212 Convex or discrete? is this point in the Pareto curve? is the convex combination of two solutions also a solution?

13. IEOR March 1213 multiobjective shortest path multiobjective minimum spanning tree multiobjective minimum cost flow multiobjective matching multiobjective minimum cut convex? 

14. IEOR March 1214 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  w i c i with all weights w i in the range [1, …, (1/  ) 2k ], and keep all undominated solutions.

15. IEOR March 1215 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?”

16. IEOR March 1216 multiobjective shortest path multiobjective minimum spanning tree multiobjective minimum cost flow multiobjective matching multiobjective minimum cut P RNC P P NP-hard (because the exact version of min cut is the same as the exact version of max cut…)

17. IEOR March 1217 PS: The web access problem can be approximated in O(n 2 /  ) 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]

18. IEOR March 1218 Open problems Faster algorithms? Other problems? Necessary and sufficient condition for discrete problems? The “sweet spot” problem: Find x such that (1 +  )obj i (x) > obj i (y) for all objectives and solutions y