IEOR March 12 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

M - cost

Pareto curve

solutions (sets S)

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 -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 +

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 this point in the Pareto curve?

is the convex combination of two solutions also a solution?

IEOR March 12 13

• multiobjective shortest path

• multiobjective minimum spanning tree

• multiobjective minimum cost flow

• multiobjective matching

• multiobjective minimum cut

convex?

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

PRNC

PP

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 timevalue)

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