CS 264 : Beyond Worst - Case Analysis Lecture # 14 : Smoothed Analysis of Pareto Curves ∗

Our next application of smoothed analysis is to Pareto curves. We begin by explaining one of several reasons why you should care about the size of Pareto curves: they govern the running time of some interesting algorithms. In the Knapsack problem, the input comprises n items, each with a positive value vi and a positive weight (or size) wi. Also given is a knapsack capacity W . The goal is choose a subset S ⊆ {1, 2, . . . , n} of the items that maximizes the total value ∑ i∈S vi, subject to the chosen items fitting in the knapsack (i.e., ∑ i∈S wi ≤ W ). When you were first introduced to this problem, you may have been told some hokey story about a burglar trying to escape with the best loot — but it really is a fundamental problem, relevant whenever you’re trying to make optimal use of a limited resource. We say that a subset S ⊆ {1, 2, . . . , n} dominates T ⊆ {1, 2, . . . , n} if: (i) the total value of S is at least that of T ( ∑ i∈S vi ≥ ∑ i∈T vi); (ii) the total weight of S is at most that of T ( ∑ i∈S wi ≤ ∑ i∈T wi); and (iii) at least one of these two inequalities is strict. If S dominates T then it renders T moot — the solution T can be safely pruned without regret. The Pareto curve of a Knapsack instance is the set of all undominated solutions; see Figure 1. Geometrically, a point is dominated if and only if there is another point “to the northwest” of it. You’ve seen this concept before, in Lecture #2, when we studied the 2D Maxima problem. The Pareto curve corresponds, after a reflection about the y-axis, to the set of maxima of the point set with one point for each subset of {1, 2, . . . , n}, the two coordinates of a point being the total weight and total value of S. Here is a Knapsack algorithm that you may not have seen before (due to [5]).