Approximate Comb Routine

So far we have considered the problem of computing the smallest -convex Pareto set if the points are explicitly given in the input. In this section we consider the same problem for a multi-objective problem Π possessing a black box routine to access its solutions. We assume that either the Comb_δ routine or the GAPδroutine are available.

5.4.2.1 Lower Bound

Theorem 5.4.12. For d ≥ 3, any polynomial generic algorithm having oracle access to GAP_δ cannot bec-competitive for any constant c.

Proof. The lower bound holds a fortiori if only the Combδroutine is available. However, it is not clear that it holds when the primitives are exact.

We exploit the fact that GAPδ(b) is not uniquely defined for some points b. We construct two sets of points S and S⁰ inR³ such that GAP_δ cannot distinguish between them unless the error parameter δ becomes inverse exponential in the size of the input.

Given an  > 0 construct a set of points Q = {q₁, . . . , q_k} (points ordered left to right) in the same plane (having z = z0) such that the smallest -convex Pareto set of Q contains all its points CP (Q) = Q. This can clearly be done. Also consider the set Q⁰ = {q⁰₁, . . . , q_k⁰} where each q_i⁰has the same x and y coordinates as qiand whose z coordinate is by 1 smaller z⁰ = z₀− 1. Now choose a point p that lies an (1 + ) factor above the z = z0 plane and (1 + )-covers all the points of Q.

Moreover, we need x(q1) = min_q∈Qx(q) > (1 + )x(p) and y(q_k) = min_q∈Qy(q) > (1 + )y(p).

Clearly, this last condition implies that no convex combination of points in Q (or Q⁰) (1+)-covers p (because of the x and y coordinates). So, p must belong to any -convex Pareto set for the instances

S .

= {p} ∪ Q and S⁰ .

= {p} ∪ Q ∪ Q⁰. The smallest -convex Pareto set for the former set is {p}, while for the latter {p} ∪ Q⁰.

It is easy to see that if z0 = M is exponential in the size of the input and 1/, the primitive

cannot distinguish between the two cases. 

5.4.2.2 Upper Bound

As a consequence of the lower bound, for d ≥ 3 objectives we are forced to compute an ⁰-convex Pareto set, where ⁰ > , if we are to have a guarantee on its size. We show in this section that for any ⁰ > , we can get a constant factor approximation for d = 3 and a logarithmic approximation for any fixed d if we spend time proportional to 1/(⁰ − ). This positive result applies to all d-objective problems that possess a polynomial Combδroutine. We note that if the (stronger) GAPδ

routine is available, the corresponding guarantees can be improved by a factor of d.

Our main positive result for the case of general (fixed) d is the following theorem that applies to all d-objective problems that possess a polynomial Comb_δroutine:

Theorem 5.4.13. 1. For any ⁰>  there exists a polynomial time generic algorithm that computes an⁰-convex Pareto setQ such that |Q| ≤ O(d²log(dOPT_)) · OPT_.

2. Ford = 3, we can efficiently compute a constant factor approximation to OPT.

We first prove the following lemma that relates the approximability of problem Q_C,Rwith the problem in hand. Let  > 0 be a given rational number. For any ⁰ > , we can find a δ > 0 such that 1/ δ = O(1/(⁰− )) satisfying 1 + ⁰≥ (1 + )(1 + δ)².

Lemma 5.4.14. Suppose that there exists an r-factor approximation algorithm for QC,R. Then, for any⁰ > , we can compute an ⁰-convex Pareto setQ, such that |Q| ≤ drOPT_usingO((m/δ)^d) Combδcalls.

Proof. The generic algorithm proceeds in two phases; in the first phase, we compute a δ-convex Pareto set, by using the oblivious algorithm of Chapter 4 and in the second phase we post-process the points produced by the latter algorithm by using the r-approximation algorithm for Q_C,R as a black box.

So, suppose that R_δ is the output of the aforementioned algorithm; R_δ is a δ-convex Pareto set (without redundant points) for the (exponentially large or infinite) feasible space S and has size

polynomial in the size of the input and 1/δ. We apply the r-approximation algorithm for QC,R on input Rδto produce a set Q ⊆ Rδof solution points whose convex combinations (1 + )(1 + δ)-cover R_δ. It is easy to see that Q is an ⁰-convex Pareto set for the feasible space S. Indeed, for any solution point p ∈ S there exists a convex combination of points r1, r₂, . . . , r_lin Rδ that (1 + δ)-covers p. Also, each ri is (1 + )(1 + δ)-covered by some convex combination of points in Q.

Therefore, there exists a convex combination of points in Q that (1 + )(1 + δ)²-covers p.

We will now argue that |Q| ≤ drOPT. Let R^∗ denote a minimum cardinality subset of R_δ whose convex combinations (1 + )(1 + δ)-cover Rδ. By assumption we have that |Q| ≤ r|R^∗|.

The proof is complete by the following claim:

Claim 5.4.15. |R^∗| ≤ d · OPT_

Proof. Let CP_be any -convex Pareto set for S. It suffices to show that there exists an (1 + )(1 + δ)-convex cover C for Rδof cardinality at most d · |CP|. Recall that Rδis a δ-convex Pareto set for S (having no redundant points). Thus, for any solution point s ∈ CP_ ⊆ S, there exists a convex combination of points in Rδthat (1 + δ)-covers s. C is constructed as follows: For each s ∈ CP

pick a set of at most d points in R_δ whose convex combinations (1 + δ)-cover s. It is clear that

|C| ≤ d|CP_| and that C is an (1 + )(1 + δ)-convex cover for S (and thus also for Rδ) - using

points only from R_δ. 



Recall from Section 5.4.1 that problem QC,Rcan be approximated within a factor of O(d log(dOPT)) for general d and within a constant factor for d = 3. Theorem 5.4.13 follows by combining this fact with Lemma 5.4.14.

We remark that there is a direct analogue of the above lemma (without the extra factor of d) for the case that the GAPδroutine is available. In this case, we can save a factor of d by computing a δ-Pareto set in the first step followed by the approximation algorithm for problem QD.

Chapter 6

The Chord Algorithm

6.1 Introduction

The Chord algorithm is a popular, simple method for the succinct approximation of planar curves, which is widely used, under different names, in a variety of areas, such as, bi-objective and para-metric optimization, computational geometry, and graphics. We analyze the performance of the chord algorithm, as compared to the optimal approximation that achieves a desired accuracy with the minimum number of points. We prove sharp upper and lower bounds, both in the worst case and average case setting.

We now briefly describe the algorithm. Let f₁, f₂ be the two objectives (say minimization objectives for concreteness), and let P be the (unknown) convex Pareto curve. First optimize f1, and f₂separately (i.e. call Comb for the weight tuples (1, 0) and (0, 1)) to compute the leftmost and rightmost points a, b of the curve P . The segment (a, b) is a first approximation to P ; its quality is determined by a point q ∈ P that is least well covered by the segment. It is easy to see that this worst point q is a point of the Pareto curve P that minimizes the linear combination f2+ λf1, where λ is the absolute value of the slope of (a, b), i.e. it is a point of P with a supporting line parallel to the ‘chord’ (a, b). Compute such a worst point q; if the error is ≤ , then terminate, otherwise add q to the set S to form an approximate set {a, q, b} and recurse on the two intervals (a, q) and (q, b). In Section 6.2 we give a more detailed formal description (for example, in some cases we can determine from previous information that the maximum possible error in an interval is ≤  and do not need to call Comb).

In this chapter we analyze the performance of the Chord algorithm, both in the worst case and in the average case setting. We define the performance ratio to be the ratio between the number of Comb calls performed by the Chord algorithm and the minimum number of points required to obtain an -convex Pareto set for the given instance (see Section 6.2)¹. We provide sharp upper and lower bounds on the performance (competitive) ratio of the Chord algorithm, both in the worst case and in the average case setting. Consider a bi-objective problem where the objective functions take values in [2^−m, 2^m]. We prove that the worst-case performance ratio of the Chord algorithm for computing an -convex Pareto set is Θ(log m+log log(1/)^m+log(1/) ). The upper bound implies in particular that for problems with polynomially computable objective functions and a polynomial-time (exact or approximate) Comb routine, the Chord algorithm runs in polynomial time in the input size and 1/. We show furthermore that there is no algorithm with constant performance ratio. In particular, every algorithm (even randomized) has performance ratio at least Ω(log m + log log(1/)).

Similar results hold for the approximation of convex curves with respect to the Hausdorff dis-tance. That is, the performance ratio of the Chord algorithm for approximating a convex curve of length L within Hausdorff distance , is Θ(log log(L/)^log(L/) ). Furthermore, every algorithm has worst-case performance ratio at least Ω(log log(L/)).

We also analyze the expected performance of the Chord algorithm for some natural probability distributions. Given that the algorithm is used in practice in various contexts with good performance, and since worst-case instances are often pathological and extreme, it is interesting to analyze the average case performance of the algorithm. Indeed, we show that the performance on the average is exponentially better. Note that Chord is a simple natural greedy algorithm, and is not tuned to any particular distribution. We consider instances generated by a class of product distributions that are “approximately” uniform and prove that the expected performance ratio of the Chord algorithm is Θ(log m + log log(1/)) (upper and lower bound). Again similar results hold for the Hausdorff distance.

Related Work. There is extensive work on multiobjective optimization, as well as on approxima-tion of curves in various contexts. We have discussed already the main related references. The problem addressed by the Chord algorithm fits within the general framework of determining the shape by probing [CY]. Most of the work in this area concerns the exact reconstruction, and the

1We note that this notion of performance is different than the ones used in the previous chapters.

analytical works on approximation (e.g., [LB, Ro, YG]) compute only the worst-case cost of the algorithm in terms of  (showing bounds of the form O(pL/)). There does not seem to be any prior work comparing the cost of the algorithm to the optimal cost for the instance at hand, i.e., the approximation ratio, which is the usual way of measuring the performance of approximation algorithms.

Organization. The rest of the chapter is organized as follows. Section 6.2 describes the model and states our main results, Section 6.3 concerns the worst-case analysis, and Section 6.4 the average-case analysis.