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Figure from:Multi-covering problems and their variants

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In the remainder of this chapter, we outline the ideas necessary to establish union of the k covers. Because server subsets in F are disjoint, this union yields a the approximation ratio of the algorithm above, using properties of the partitioning

Table 3 In the remainder of this chapter, we outline the ideas necessary to establish union of the k covers. Because server subsets in F are disjoint, this union yields a the approximation ratio of the algorithm above, using properties of the partitioning

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Figure 1.1: Clients are filled (blue) circles, servers are (red) crosses, and the labels represent inter-point distances, where r € RU{0},¢€ > 0. Ifa feasible solution includes the ball centered on the server in the middle, then an improvement to that solution can only be made if the local search algorithm considers all balls of radius € centered on the remaining servers in a single improvement step. this approach (of considering the balls in a certain order determined by their size)
Figure 1.1: Clients are filled (blue) circles, servers are (red) crosses, and the labels represent inter-point distances, where r € RU{0},¢€ > 0. Ifa feasible solution includes the ball centered on the server in the middle, then an improvement to that solution can only be made if the local search algorithm considers all balls of radius € centered on the remaining servers in a single improvement step. this approach (of considering the balls in a certain order determined by their size)
instance, a 2-net is simply an independent set that is maximal by inclusion. path in G having at most r — 1 edges. An r-net is a fairly well-known concept; for
instance, a 2-net is simply an independent set that is maximal by inclusion. path in G having at most r — 1 edges. An r-net is a fairly well-known concept; for
Figure 3.1: An example of OuterCover of level 2. The (blue) circles represent clients in X, the (red) squares represent servers in Y. The disk covering each client in X has a radius that is at least as large is its distance to its second nearest server in Y.
Figure 3.1: An example of OuterCover of level 2. The (blue) circles represent clients in X, the (red) squares represent servers in Y. The disk covering each client in X has a radius that is at least as large is its distance to its second nearest server in Y.
Figure 6.1: Illustration for the proof of Claim 6.2. For the client « € X, the dashed edges correspond to a path a in G; from x to x. Here, y: € Ni(xz) A Ni(x1), yo € Ni(a1) ON; (Z), and y € Y;N N;(z). The ball 6(y, pi(y)) serves . Here, « happens to be z. Note that we can get from y to x using 5 edges of the figure, and from y to y using 2 edges. Therefore, expanding the ball at y by a factor of 6 will cover both x and y. (In this example, even a factor of 5 suffices.)
Figure 6.1: Illustration for the proof of Claim 6.2. For the client « € X, the dashed edges correspond to a path a in G; from x to x. Here, y: € Ni(xz) A Ni(x1), yo € Ni(a1) ON; (Z), and y € Y;N N;(z). The ball 6(y, pi(y)) serves . Here, « happens to be z. Note that we can get from y to x using 5 edges of the figure, and from y to y using 2 edges. Therefore, expanding the ball at y by a factor of 6 will cover both x and y. (In this example, even a factor of 5 suffices.)
and any available server from NN(x,,1) to YY only if K(a,) > 2i — 1 Assuming that servers are available when the algorithm looks for them, we
and any available server from NN(x,,1) to YY only if K(a,) > 2i — 1 Assuming that servers are available when the algorithm looks for them, we
number of servers used. Now, The third inequality follows due to the fact that the set of outer covers {11,72,°+* 51k. 0; of level 7, for each 1+ 1 <2 < k with the desired upper bounds on the cost and the
number of servers used. Now, The third inequality follows due to the fact that the set of outer covers {11,72,°+* 51k. 0; of level 7, for each 1+ 1 <2 < k with the desired upper bounds on the cost and the