Q2. What is the goal of the median problem?
Given a bound D > 0, the goal of the (discrete) median problem is to choose vertices as medians so that the sum of distances from each vertex to its nearest median is no more than D and the number of medians chosen is minimized.
Q3. What is the deterministic approximation algorithm for the median problem in metric spaces?
There exists a deterministic approximation algorithm for the median problem in metric spaces that, given any > 0, outputs a median set U satisfyingX j2V min i2U cij 2(1 +
Q4. What is the proof of the triangle inequality?
Since Vi; is bounded by a ball of diameter 2 (1+1= ) bCi in d-dimensional Euclideanspace, there exists a median set Ui of size at most (2 1) d such that for all j 2 Vi; the authors havemin `2Ui cj` (1 + 1= ) bCi: For each j 2 Vi Vi; , there exists j0 2 Vi; such that cjj0 (1 + 1= ) bCj.
Q5. what is the proof of the theorem?
Let vertex i be a median selected by Algorithm M and let Vi; Vi be a subset of vertices such that a vertex j 2 Vi is in Vi; if and only if bCj bCi.
Q6. What is the proof of the symmetry and the triangle inequality?
For each i 2 V and > 0, the authors haveX j2Vi byj X j2Vi bxij > 1 + ;where Vi is the neighborhood of vertex i. Proof : Suppose Pj2Vi bxij =(1 + ).
Q7. What is the deterministic approximation algorithm for the median problem?
Let us consider a complete (directed or undirected) graph G = (V;E) on n vertices, with vertex set V = f1; . . . ; ng, edge set E V V , and nonnegative distance cij associated with edges.
Q8. What is the proof of the Theorem 1?
Without loss of generality, the authors assume Vi is selected before Vj. Since cij0 (1 + 1= ) bCi and cjj0 (1 + 1= ) bCj, by (R2) the authors must have j 2 Vi. Hence, Algorithm M will delete
Q9. what is the proof of the median problem?
The median problem can be formulated as a 0-1 integer linear program of minimizingnX j=1 yj (3)subject tonX i=1 nX j=1 cijxij D (4)nX j=1 xij = 1; i = 1; . . . ; n; (5)xij yj; i; j = 1; . . . ; n; (6)xij; yj 2 f0; 1g; i; j = 1; . . . ; n; (7)where yj = 1 if and only if j is chosen as a median, xij = 1 if and only if yj = 1 and i is assigned to j, and D > 0 is a given bound on the total distance.
Q10. What is the proof of the Theorem 2?
by Lemma 3 the number of sets (medians) selected is less than s1=(1 + ) = (1 + )s:By Lemma 5 the authors haveX j2V min i2U cij 2(1 + 1= ) X j2V bCj 2(1 + 1= )D:In this section the authors give the proof of Theorem 2.
Q11. What is the way to get the median?
1= )D (1)andjU j < (1 + )s; (2)where s is the optimal size of median sets with a total distance at most D.The greedy approximation algorithm in [LiV] gives a better cost bound using more medians; the right-hand sides of 1 and 2 are replaced by (1+1= )D and (1 + )s(lnn+ 1), respectively.
Q12. What is the last inequality from Lemma 6?
By symmetry and the triangle inequality, the authors havecji(j) cjj0 + cj0i(j) (1 + 1= ) bCj + (1 + 1= ) bCi(j) 2(1 + 1= ) bCj:The last inequality follows from Lemma 4.2Lemma 6