.
Again, since can be chosen arbitrarily small, alg cannot achieve a finite competitive ratio with respect to a reasonable function f, i.e., a function f that maps a vector with at least one unbounded component to ∞.
3.8 Yao’s Principle for Multi-Objective Online Problems
Yao’s principle is a method for obtaining lower bounds on the competitive ratio of any randomized online algorithm. In order to obtain a lower bound on the competitive ratio of any randomized online algorithm with this method, it is sufficient to choose a probability distributionqover request sequences and then consider the ratio of the expected optimal offline value and the expected value of any deterministic online algorithm, where the expectation is taken with respect toq. An upper bound on this ratio then gives a lower bound on the competitive ratio of any randomized online algorithm.
Instead of considering all possible randomized online algorithms, it suffices to con-sider the expected value of any deterministic online algorithm and, hence, this method often facilitates the search for lower bounds on the competitive ratio of any randomized online algorithm. In this section, we extend Yao’s principle to multi-objective online algo-rithms, i.e., we present a method for obtaining lower bounds on the (strong) competitive ratio of any multi-objective randomized online algorithm. As for single-objective online algorithms, the cases of profit maximization and cost minimization online problems are treated separately. We start with profit maximization problems:
Theorem 3.8.1. Let algbe any randomized objective online algorithm for a multi-objective online maximization problem and let{algi :i∈ I} denote the finite set of all deterministic multi-objective online algorithms for this problem. Letq be any probability distribution over the finite set of possible request sequences{σj :j∈ J }. Letf :Rn→R+
be a monotonically increasing function and, for j∈ J, let xj be an efficient solution with respect to request sequence σj, i.e., xj ∈opt[σj]. Then,
Rfs(alg)≥f(c), where c= c1, . . . , cn|
∈Rn is given by
ck = min
i
Eq(j)
opt xj
k
Eq(j)
algi(σj)k, for k= 1, . . . , n.
For the worst-component competitive ratio Rfs1(alg), where f1(c) = maxi=1,...,nci, we additionally have
Rfs1(alg)≥min
i
1 Eq(j)
h
maxkalgi(σj)k
opt(xj)k
i.
Proof. First of all, we prove that any randomized multi-objective online algorithm alg is at mostc-competitive with c= c1, . . . , cn|
∈Rn given by ck= min
i
Eq(j)
opt(xj)k
Eq(j)[algi(σj)k] for k= 1, . . . , n.
Since f is a monotonically increasing function and the strong competitive ratio is given by the infimum over the set of all valuesf(c)such thatalgis stronglyc-competitive, we then have Rfs(alg) ≥ f(c). The proof given below differs only slightly from the proof of Yao’s principle for single-objective online problems, cf. (Borodin and El-Yaniv, 1998, pp. 117-118, Theorem 8.3).
Consider the constant vector ˜c = ˜c1, . . . ,c˜n|
∈ Rn. For k = 1, . . . , n, define the two-person zero-sum gameGk(˜c)between the online player and the adversary. The payoff function hk(i, j) for the online player is given by
hk(i, j) = ˜ck·algi(σj)k−opt(xj)k, k= 1, . . . , n, (3.38) i.e., hk(i, j) gives the payoff for the online player choosing strategy i against strategy j of the adversary. Note thatalgi(σj)k denotes thek-th component of the solution of the deterministic multi-objective online algorithmalgi with respect to request sequenceσj andopt(xj)kdenotes thek-th component of the efficient offline solutionxj with respect to request sequenceσj. Due to (Borodin and El-Yaniv, 1998, p.112, Theorem 8.2), every finite two-person zero-sum game has a value V(˜ck). Furthermore, due to (Borodin and El-Yaniv, 1998, p.113, Lemma 8.2), V(˜ck) is given by
V(˜ck) = max
p(i) min
j Ep(i)[hk(i, j)], k= 1, . . . , n, (3.39)
wherep(i)is a mixed strategy for the online player. If V(˜ck)<0, for k= 1, . . . , n, then, due to (3.38), the best randomized online algorithm is not strongly ˜c-competitive. For k= 1, . . . , n, suppose that
Due to Yao’s inequality, cf. (Borodin and El-Yaniv, 1998, p.113), we hence have 0>max
i Eq(j)[hk(i, j)]≥max
p(i) min
j Eq(j)[hk(i, j)](3.39)= V(˜ck).
Therefore,alg is at mostc-competitive and since f is a monotonically increasing func-tion, we haveRfs(alg)≥f(c).
Now, we consider the worst-component competitive ratio, i.e.,f1(c) = maxk=1,...,nck, and the payoff function
h(i, j) = max
k
algi(σj)k
opt(xj)j. (3.40)
Consider some distributionq(j)over request sequences and somexj ∈opt[σj]forj ∈ J. By applying Yao’s inequality (cf. (Borodin and El-Yaniv, 1998, p.113)), we get
maxp(i) Eq(j) Here, (3.41) is due to the convexity of the maximum and, in (3.42),maxkis just replaced by mink. Since the optimal strong competitive ratio for maximization problems with respect tof1 is given by
For cost minimization problems, the lower bounds are given in the same line. How-ever, the proof differs slightly from the one given for profit maximization problems.
Theorem 3.8.2. Let algbe any randomized objective online algorithm for a multi-objective online minimization problem and let {algi :i∈ I} denote the finite set of all deterministic multi-objective online algorithms for this problem. Let q be any probability distribution over the finite set of possible request sequences{σj :j ∈ J }. Letf :Rn→R+
be a monotonically increasing function and, for j∈ J, let xj be an efficient solution with respect to request sequence σj, i.e., xj ∈opt[σj]. Then,
and applying Yao’s inequality for cost minimization problems. The remaining steps can be performed analogously to Theorem 3.8.1
For the worst-component competitive ratio, i.e.,f1(c) = mini=1,...,nci, we define the payoff function
h(i, j) = min
k
algi(σj)k
opt(xj)j. (3.44)
Consider some distributionq(j)over request sequences and somexj ∈opt[σj]forj∈ J. By applying Yao’s inequality for cost minimization problems, we get
mini Eq(j)
Here, (3.45) is due to the concavity of the minimum, and, in (3.46)minkis just replaced by maxk. Since the optimal strong competitive ratio with respect tof1for cost minimization problems is given by
The lower bounds obtained in Theorem 3.8.1 and Theorem 3.8.2 for the strong com-petitive ratio depend on the efficient offline solution chosen for each sequence. If all efficient offline solutions for each sequence are taken into account, a lower bound on the competitive ratioRf(alg) is obtained.
In the following example, we show how to apply Theorem 3.8.2 to a multi-objective online minimization problem in order to obtain a lower bound on the (strong) competitive ratio for any randomized algorithm.
Example 3.8.1. Consider the bi-objective ski rental problem as introduced in Section 3.4 and set B = 2 and C = 1, i.e., buying skis costs 2,1|
and renting skis costs 1,1| . Now, we choose the distribution q over the request sequences (the number of skiing days) as q(1) = 1/2 and q(3) = 1/2, i.e, with probability 1/2 there is one skiing day and with probability 1/2 there are three skiing days. For n= 1, the only efficient offline solution is given by renting skis with costs 1,1|
, since buying skis would cost 2,1|
. For n= 3, the only efficient solution is given by buying skis with costs 2,1|
, since renting skis
Let algi denote the deterministic online algorithm that rents skis i−1 times and buys skis on thei-th skiing day. The expected value with respect to q of algi is then given by
Eq(j)[algi(σj)] =
Therefore, we have
mini
Eq(j)[algi(σj)]
Eq(j)[opt(xj)] = 4/3
1
, and, by Theorem 3.8.2,
Rfs1(alg)≥max 4
3,1
= 4 3.
Consequently, the strong competitive ratio with respect to f1(c) = maxi=1,...,nci for any randomized online algorithm for the bi-objective ski rental problem is at least 4/3. Since there is only one efficient solution in both cases (n= 1 andn= 3), this holds also for the competitive ratio for any randomized online algorithm. Considering the competitive ratio with respect to f2 or f3, we get the following lower bounds on the (strong) competitive ratio of any randomized algorithm: for f2(c) = n1Pn
i=1ci, we have a lower bound of 7/6
and, for f3(c) = pQn n
i=1ci, we have a lower bound ofp
4/3.
3.9 Conclusion and Future Research
In this chapter, we introduced a general framework for the competitive analysis of multi-objective online problems which expands the known theory of competitive analysis for online problems in a straightforward manner. In the course of this chapter, we gave several examples for the application of competitive analysis for multi-objective online problems and demonstrated that the analysis of multi-objective online problems by means of the introduced notions of competitiveness yields reasonable results which are closely related to the single-objective algorithms and their competitive ratios. Furthermore, we discussed relations between multi-objective online problems and the corresponding single-objective online problems and extended Yao’s principle to multi-objective online problems.
The concept of competitive analysis for multi-objective online problems seems highly promising and provides further insight into the nature of online problems. Questions for future research include a more in-depth analysis of the problems proposed in this chapter such as randomized algorithms for the multi-objective k-ctp, and the analysis of multi-objective counterparts of other well-known online problems such as scheduling problems. Additionally, the definition of the competitive ratio given in this work serves as a basis for further extensions such as a vector of competitive ratios with respect to different functions: for example, if the analyst of the online problem wants the worst component and the average of the components to be reasonably small at the same time, the vector of both competitive ratios could be analyzed in the sense of multi-objective optimization.
4
The Linear Search Problem with Turn Costs
4.1 Introduction
The linear search problem is an optimal search problem independently introduced by Bell-man (1963) and Beck (1964) in which an immobile object is located on the real line according to a probability distribution. A searcher starts from the origin and tries to find the object in minimum expected time. It is assumed that the searcher cannot see the object until she reaches the point at which the object is located. Originally, it is also assumed that the location of the object is given by a known probability distribution.
However, in this work we consider the problem without knowledge about the probability distribution, leading to a basic online problem which is then analyzed by competitive analysis. The optimal competitive ratio for deterministic algorithms solving the linear search problem is 9, as first shown by Beck and Newman (1970). The optimal strategy for the searcher is to alternate between going to the right and to the left, doubling the step size in each iteration.
In (Demaine et al., 2006), the linear search problem (in the context of competitive analysis) is expanded by turn costs, i.e., each time the searcher changes direction a cost of dis incurred. The authors consider the sum of searching time and turn cost as objective function and present an algorithm which guarantees a solution smaller than9·opt+ 2d.
Note that, for deterministic algorithms, a minimum cost of d is required, regardless of opt, by placing the object arbitrarily close to the origin on the side not chosen by the searcher to start with. The additive term 2d is minimal subject to the (optimal) com-petitive ratio 9. As mentioned in (Demaine et al., 2006), it may be desirable to improve the additive term, while allowing an increase in the competitive ratio. The determining tradeoff curve is obtained experimentally, but it is not characterized analytically.
In this chapter, we close this gap by presenting an analytical characterization of the tradeoff curve between the competitive ratio and the additive term for the linear search problem with turn costs. We apply the analysis of the linear search problem presented in (Demaine et al., 2006) to an arbitrary competitive factorc˜and determine the minimal additive factor analytically.
111
start x1
x2 x3
x4
x2+ǫ
first turn searcher finds the object
×
Figure 4.1: Path of the searcher for an object placed atx2+.