In this section, we introduce the multi-objectivek-Canadian traveller problem and per-form a worst-component competitive analysis. The Canadian traveller problem (ctp) is a variant of the shortest path problem and was initially introduced by Papadimitriou and Yannakakis (1991) in the following way:
Consider an undirected graph G = (V, E) with non-negative edge-weights w: E → R+ and distinguished nodes s∈V and t∈ V. The goal is to find a shortest path from s to t with respect to w. However, some of the edges may be blocked and an online algorithm only learns about the blockade of an edge when reaching one of its endpoints.
In the k-Canadian traveller problem (k-ctp), the number of blocked edges is bounded from above byk.
The k-ctp is firstly studied in the framework of competitive analysis by Westphal (2008). Here, a lower bound of 2k+ 1 for the competitive ratio of any deterministic algorithm is given and, additionally, the deterministic algorithmbacktrack achieving the optimal competitive ratio of2k+ 1, is presented. backtrackrepeatedly computes the shortest path inG, returns to swhen a blockade is reached, and erases the blocked edge fromG:
Algorithm 19: backtrack for the k-ctp(Westphal, 2008).
Data: GraphG= (V, E) with non-negative edge-weights w:E→R+, distinguished start and end nodess andt.
Result: Shortest pathp fromsto t.
1 whilet is not reached do
2 Compute shortest pathp with respect to wfrom sto tinG= (V, E)and traverse it.
3 if edge eon path p is blocked then
4 Return tosand set E←E\e.
s t
Figure 3.12: A worst case example for the multi-objective k-ctp.
Furthermore, a lower bound ofk+ 1 for any randomized algorithm for thek-ctpis proven. For node-disjoint paths, this lower bound is matched by a randomized version of backtrack given in (Bender and Westphal, 2013). The algorithm computes the k+ 1 shortest node-disjoint paths in the graph and chooses one of them according to an appropriately defined probability distribution. If the path is blocked, the procedure is repeated for a smaller set of paths.
We introduce a multi-objective version of the k-ctp by replacing the non-negative edge-weights w by a non-negative weight vector w : E → Rn+. The goal is to find a shortest path from s to t with respect to w. Note that the optimal offline solution is a set of efficient paths. This problem is denoted as themulti-objective k-ctp.
First of all, we observe that, without any further restrictions, there exist no compet-itive algorithms for the multi-objective k-ctp:
Theorem 3.6.1. There is no competitive randomized algorithm for the multi-objective k-ctp for any n≥2 and k≥1.
Proof. Consider the graph in Figure 3.12. An online player chooses the upper path from s to t with probability p1 and the lower path from s to t with probability p2, which characterizes all randomized algorithms for this instance. alg’s costs are then given by
alg=
if the lower edge is blocked, p1 2
if the upper edge is blocked.
Since opt is given by 1,0|
if the lower edge is blocked and 0,1|
otherwise, the competitive ratio is always unbounded in at least one component.
Thus, similar to the time series search problem investigated in Section 3.3, we assume that the weightw(p) of each pathp fromstot (without any cycles) is bounded in each component by 0 < m ≤ w(p)i ≤ M, for i = 1, . . . , n, and define ϕ = M/m as the fluctuation ratio of the weight of possible paths. This problem is denoted as the multi-objective k-ctp with bounded paths.
In the following, a competitive deterministic algorithm for the multi-objective k-ctp with bounded paths is presented, which is, basically, a multi-objective version of backtrack:
Algorithm 20: multi-backtrack.
Data: GraphG= (V, E) with non-negative edge-weights w:E →Rn+, distinguished start and end nodess andt.
Result: Shortest pathp fromsto t.
1 Calculate the set of efficient solutions P for shortest paths fromsto twith respect to the edge-weight vectorw.
2 Seti←0.
3 whilet is not reached do
4 Choose an efficient solution p∈ P that minimizes component (i mod n) + 1 and traverse it.
5 if edge eon path p is blocked then
6 Return tosand set E←E\e.
7 Recompute the set of efficient solutionsP inG= (V, E).
8 Seti←i+ 1.
Theorem 3.6.2. The worst-component competitive ratio of multi-backtrackfor the multi-objective k-ctpwith bounded paths is given by
Rf1(multi-backtrack) = 1 + 2
kϕ− k
n
(ϕ−1)
.
Proof. In the worst case, the first k paths chosen by alg are blocked by the adversary and alg reaches tvia the (k+ 1)-st path. Denote the costs of the paths by w(pj), for j= 1, . . . , k+ 1. Then, the costs of alg are given by
alg=w(pk+1) + 2 Xk j=1
w(pj). (3.30)
For some efficient solutionx∈opt[I], we then have alg(3.30)= opt(x) + 2
Xk j=1
w(pj), (3.31)
sincealg always computes the current set of efficient solutions and, consequently, there exists an efficient offline solution x ∈ opt[I] such that w(pk+1)i = opt(x)i, for i = 1, . . . , n. The competitive ratio with respect to f1 of alg is now given by
Rf1(multi-backtrack)≤ max
i=1,...,n
(
opt(x)i+ 2Pk
j=1w(pj)i opt(x)i
)
, (3.32)
where w(pj)i is the i-th component of the weight of path pj. The path pj chosen by alg is selected from the set of (currently available) efficient paths such that component i0 := (j mod n) + 1is minimized (see Step 4 of Algorithm 20). Thus, for every efficient offline solution ˜x, we haveopt(˜x)i0 ≥w(pj)i0 and, in particular,
The worst-component competitive ratio of multi-backtrack is best possible, as shown by the following theorem:
Theorem 3.6.3. No deterministic algorithm for the multi-objectivek-ctpwith bounded paths can achieve a smaller worst-component competitive ratio than
Rf1(multi-backtrack) = 1 + 2
with m at the1 + (j mod n)-th position. Note that the weight for each path p from s to tis bounded by m≤w(p)i≤M,i= 1, . . . , n.
For any deterministic algorithm alg, the first k choices are blocked, forcing the online player to return to s. The last remaining possibility cannot be blocked by the adversary. Assume that the successful path has cost m in component j0, and M in all other components. Since there is only one path left, the efficient solution chosen by the adversary denoted by xis the path ultimately chosen by the online player.
s t
Figure 3.13: A worst case example for the multi-objective k-ctpwith bounded paths.
Ifn > k, all other paths previously chosen byalghave costM in component j0 (see the definition ofwj). Otherwise, ifn≤k, there are at mostk
n
paths previously chosen by algthat have cost min componentj. Therefore, we have
i=1,...,nmax
Note that, forn > k, the worst-component competitive ratio ofmulti-backtrackis given by1 + 2kϕand, forn= 1,multi-backtrackcoincides with the single-objective algorithm backtrack. Furthermore, Theorem 3.6.2 considers the competitive ratio rather than the strong competitive ratio of multi-backtrack. In the course of the proof of Theorem 3.6.2 the choice of a specific efficient solution is crucial in order to obtain the stated competitive ratio. If all efficient solutions are taken into account, the competitive ratio with respect tof1 of multi-backtrackincreases:
Theorem 3.6.4. The strong competitive ratio with respect to f1(c) = maxi=1,...,nci of multi-backtrackfor the multi-objective k-ctp with bounded paths is given by
Rfs1(multi-backtrack) =ϕ+ 2
Proof. Consider (3.31) in the proof of Theorem 3.6.2. Here, we chose the efficient solution x ∈ opt[I] such that opt(x)i = w(pk+1)i for i = 1, . . . , n, i.e., the efficient solution coincides with the last path of the online player. If the efficient solutionxchosen by the
adversary differs from the path chosen by the online player in component i0, the ratio of the weight of the adversary’s and the online player’s path in component i0 is only bounded by ϕ. Analogous to the proof of Theorem 3.6.2, the strong competitive ratio with respect to f1 is then given by
Rfs1(multi-backtrack) = max
The worst case instance given in Theorem 3.6.3 can be extended in order to provide a lower bound for the strong competitive ratio with respect to f1 for any deterministic algorithm for the multi-objectivek-ctpwith bounded paths:
Theorem 3.6.5. No deterministic algorithm for the multi-objectivek-ctpwith bounded paths can achieve a strong competitive ratio smaller than
Rfs1(multi-backtrack) =ϕ+ 2
Proof. Consider the graph in Figure 3.13 and add another path fromstotconsisting of an edge with cost wk+2 and a subsequent edge with cost0. The weight vector wk+2 is defined analogously to the weight vectorswj in the proof of Theorem 3.6.3.
The adversary can always choose between two remaining paths after blocking the first kpaths and obviously selects the one not chosen by the online player. By the same argumentation as in the proof of Theorem 3.6.3, we then have
i=1,...,nmax