Advanced Algorithms 3.0

Advanced Algorithms 3.0 - Theorem 17 ISIT-`2 2 P i.e...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Theorem 17 ISIT-`2 2 P. i.e., ISIT-`2 can be answered in polynomial time. Proof: Assume that there exists an embedding of f1; 2; : : : ; ng into fv1 = 0; v2; : : : ; vng. Consider one such embedding. Then, d2i; j = jjvi , vj jj2 = vi , vj vi , vj = vi2 , 2vi vj + vj2: But vi2 = d21; i and vj2 = d21; j which means that, vi vj = d 1; i + d 1; j , d i; j : 2 2 2 2 We now construct M = mij where, mij = d 1; i + d 1; j , d i; j : 2 2 2 2 Hence if M is not positive semi-de nite then there is no embedding into `2. If M is positive semide nite then we carry out a Cholesky decomposition on M to express M as M = V V T . From the rows of V we can obtain an embedding into Rn; `2. Theorem 18 ISIT-`1 is NP-complete. This theorem is given without proof. The reduction is from MAX CUT, since as we will see later there is a very close relationship between l1-embeddable metrics and cuts. We also omit the proof of the following theorem. Theorem 19 Let X Rn. X; `2 can be embedded into Rm; `1 for some m. The converse of this theorem is not true as can be seen from the metric space f0; 0; ,1; 0; 1; 0; 0; 1g; `1 . 8.1 Reducing multicommodity ow cut questions to embedding questions In this section, we relate Claim 20 Proof: and through the use of metrics. = min `1-embeddable metrics V; ` Approx-41 P u `x; y x;y Pk2E `xy ; t : f s i=1 i i i Given S , let 'x = 0 if x 62 S . 1 if x 2 S Let ` be the `1 metric on the line, i.e. `a; b = ja , bj. Then, X u S = uxy `x; y f S = x;y2E k X fi`si ; ti i=1 since `x; y = 1 if and only if x is separated from y by S and 0 otherwise. We can view any `1 embeddable metric ` as a combination of cuts. See gure 11 for the 2-dimensional case. 2 (1,3,4) 4 (1,3) 3 (1) 1 (1) (1,2) (1,2,3) Figure 11: Viewing an `1-metric as a combination of cuts. For any set S de ne a metric 1S by, 1S = 1 if x; y are separated by S 0 otherwise. Then we can write ` as, X `x; y = i 1S x; y ; i i where the i's are nonnegative. Hence, P P u S u S x;y2E uxy `x; y i Pk f `s ; t = P i f S i min f S : S i=1 i i i i i i Approx-42 Claim 21 = min `1 -embeddable metrics V; d P x;y2E uxy dx; y P f ds ; t : i i i i Note that by theorem 16 we actually minimize over all metrics. Proof: For any metricP let the volume of an edge x; y be uxy dx; y. The total volume d of the graph is x;y2E uxy dx; y. If we send a fraction of the demand then the amount of volume that we use is at least Pi fidsi; ti. Hence Pi fi dsi; ti P x;y2E uxy dx; y . We use the strong duality of linear programming. can be formulated as a linear program in several di erent ways. Here we use a formulation which works well for the purpose of this proof although it is quite impractical. Enumerate the paths from si to ti, let Pij be the j th such path and let xij be the ow on Pij . The linear program corresponding to multicommodity ow is, Max subject to: X fi , xij 0 X Xj xij ue i 2 f1; : : : ; kg e2E 0 xij 0 i j :e2P ij The dual of this linear program is: X Min ue`e subject to: k X i=1 e2E e2P hi 0 `e 0 ij fi hi 1 X `e , hi 0 8i; j The second constraint in the dual implies that hi is at most the shortest path length between si and ti with respect to `e . By strong duality if ` is an optimum Approx-43 solution to the dual then, = ue`e e2E P u` Pek2E f eh e P i=1 i ui di; j Pi;j2E dijs ; t ; k f i=1 i i i X where da; b represents the shortest path length with respect to `e . The rst inequality holds because Pk=1 fihi is constrained to be at least 1. i Linial, London, and Rabinovitch and Aumann and Rabani use the following strategy to bound and approximate the minimum multicommodity cut. 1. Using linear programming, nd and the corresponding metric d as given in claim 21. 2. Embed d into Rm; `1 with distortion c. Let ` be the resulting metric. By claim 20 this shows that c since, P P x;y x;y P 2E uxy `x; y c P 2E uxy dx; y = c : k f `s ; t k f ds ; t i=1 i i i i=1 i i i In order to approximate the minimum multicommodity cut, we can use the proof of claim 20 to decompose ` into cuts. If S is the best cut among them then, P u S x;y2E uxy `x; y : Pk f `s ; t f S i=1 i i i Our remaining two questions are: How do we get an embedding of d into `? Equivalently, how can we embed `1 into `1. What is c? 8.2 Embedding metrics into `1 The following theorem is due to Bourgain. Theorem 22 For all metrics d on n points, there exists an embedding of d into `1 which satis es: dx; y jjx , yjj1 Olog ndx; y: Approx-44 Let k range over f1; 2; 4; 8; : : : ; 2j ; : : : ; 2pg where p = blog nc. Hence we have p + 1 = Olog n di erent for k. Now choose nk sets of size k. At rst values take all sets of size k, i.e., nk = n . Introduce a coordinate for every such set. This k implies that points are mapped into a space of dimension Pp=0 n2 2n . For a set A j of size k the corresponding coordinate of a point x is, dx; A nk where dx; A = minz2A dx; z and is a constant which we shall determine later. Suppose that dx; A = dx; s and dy; A = dy; t, where s and t are in A. Then, dx; A , dy; A = dx; s , dy; t dx; t , dy; t dx; y: Exchanging the roles of x and y, we deduce that jdx; A , dy; Aj dx; y. Hence, X jjx , yjj1 = n jdx; A , dy; Aj A jAj X 1 n dx; y j Proof: = p + 1dx; y = Olog ndx; y: We now want to prove that jjx , yjj1 dx; y. Fix two points x and y and de ne, B x; r = fz : dx; z rg; B x; r = fz : dx; z rg; 0 = 0; t t t = inf fr : jB x; rj 2 ; jB y; rj 2 g: A jAj Let ` be the least index such that ` dx;y . Rede ne ` so that it is equal to dx;y . 4 4 Observe that for all t either jBx; tj 2t or jBy; tj 2t. Since B x; `,1 B y; `,1 = ; we have 2`,1 +2`,1 n ` p. Now x k = 2j where p,1 j p,` and let t = p , j thus, 1 t l. By our observation we can assume without loss of generality that jBx; tj 2t . Let A be a set of size k and consider the following two conditions. 1. A Bx; t = ;. 2. A B y; t,1 6= ;. If 1. and 2. hold then dx; A t and dy; A t,1 and so jdx; A , dy; Aj t , t,1 . Let Rk = fA : jAj = k and A satis es conditions 1. and 2.g Approx-45 Lemma 23 For some constant 1 independent of k, there are at least of size k which satisfy conditions 1. and 2, i.e. jRk j n . From this lemma we derive, p,1 X X jjx , yjj1 jdx; A , dy; Aj R nk j =p,`;k=2 p,1 X nk nk p,j , p,j,1 j =p,`;k=2 p,1 X p,j , p,j,1 = k j k j n k sets = j =p,` ` = 4 dx; y: Hence if we choose = 4 then we have jjx , yjj1 dx; y. We now have to prove lemma 23. Proof of lemma 23: Since jBx; tj 2t, jB y; t,1j 2t,1 and we are considering all sets of size k the following is a restatement of the lemma: Given disjoint sets P and Q with a = jP j 2t and b = jQj 2t,1, if E is the event that a uniformly selected A misses P and intersects Q then Pr E 1 . We calculate this probability as follows: n,a n,a,b , Pr E = k n k !n k !n k = nn, aa , k, n! , nn, aa, bb , k, n! , ! ! , , ! ! a a k = n , a n ,,, 1 n ,,, ++ 1 , n n 1 n k 1 n , a, bn , a, b , 1 n , a ,b , k + 1 k n a n , 1 n , a + 1 a 1 , = 1, n 1, n,1 n , k + 1 ,! ! ! a+ b 1 , a + b 1 , a + b : 1, n n,1 n,k +1 a As an approximation this can be made formal, we replace 1 , n,j by e,a=n, and a 1 , n+b by e,a+b=n. Thus, ,j Pr E e, , e, e, 1 , e, : ak n k a+bk n ak n bk n Approx-46 This for example shows that if a; b and k are all n then this probability is a constant, which may seem a bit paradoxical. Using our bounds on a and b, we get , 1 , e, Pr E e, 1 , e, e, , , , e,1 1 , e, : = e 1,e ak n bk n p 2t 2j n 2t 1 2j n 2p n 2p 1 n 1 4 ,1 and the proof is complete. We now choose e,1 1 , e, Bourgain's proof is not quite algorithmic since the dimension is exponential. Linial, London and Rabinovitch just sample uniformly with nk = Olog n and show that with high probability the embedding has the required properties. This follows from a Cherno bound. We have thus shown that the distortion c can be chosen to be Olog n. We can do even better by proving the following variant to Bourguain's theorem. 1 4 Theorem 24 Let d be a metric on a set V of n points. Suppose that T V and jT j = k. Then there exists an embedding of d into `1 which satis es: jjx , yjj1 Olog kdx; y 8x; y 2 V: jjx , yjj1 dx; y 8x; y 2 T: In order to prove this theorem we restrict the metric to T and then embed the restricted metric. If we look at the entire vertex set V then the rst part of the original proof still works. This new theorem is enough to show that Olog k and to approximate the multicommodity cut to within Olog k. This result is best possible in the sense that we can have = log k. References 1 S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof veri cation and hardness of approximation problems. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 14 23, 1992. 2 S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 2 13, 1992. 3 Y. Aumann and Y. Rabani. An Olog k approximate min-cut max- ow theorem and approximation algorithm. Manuscript, 1994. 4 R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2:198 203, 1981. Approx-47 5 M. Bellare and M. Sudan. Improved non-approximability results. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 184 193, 1994. 6 N. Christo des. Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA, 1976. 7 W. F. de la Vega and G. S. Luecker. Bin packing can be solved within 1 + in linear time. Combinatorica, 14, 1981. 8 J. Edmonds. Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards B, 69B:125 130, 1965. 9 R. Fagin. Generalized rst-order spectra, and polynomial-time recognizable sets. In R. Karp, editor, Complexity of Computations. AMS, 1974. 10 H. N. Gabow. Data structures for weighted matching and nearest common ancestors with linking. In Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms, pages 434 443, 1990. 11 H. N. Gabow, M. X. Goemans, and D. P. Williamson. An e cient approximation algorithm for the survivable network design problem. In Proceedings of the Third MPS Conference on Integer Programming and Combinatorial Optimization, pages 57 74, 1993. 12 H. N. Gabow and R. E. Tarjan. Faster scaling algorithms for general graph matching problems. Technical Report CU-CS-432-89, University of Colorado, Boulder, 1989. 13 M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. In Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 307 316, 1992. 14 M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satis ability problems using semide nite programming. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 422 431, 1994. 15 A. Haken and M. Luby. Steepest descent can take exponential time for symmetric connection networks. Complex Systems, 2:191 196, 1988. 16 D. Hochbaum. Approximation algorithms for set covering and vertex cover problems. SIAM Journal on Computing, 11:555 556, 1982. Approx-48 17 D. Hochbaum and D. Shmoys. Using dual approximation algorithms for scheduling problems: theoretical and practical results. Journal of the ACM, 341, Jan. 1987. 18 N. Karmarkar and R. Karp. An e cient approximation scheme for the onedimensional bin-packing problem. In Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, 1982. 19 T. Leighton and S. Rao. An approximate max- ow min-cut theorem for uniform multicommodity ow problems with applications to approximation algorithms. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science, pages 422 431, 1988. 20 N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994. 21 C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43:425 440, 1991. 22 S. Poljak. Integer linear programs and local search for max-cut. Preprint, 1993. 23 P. Raghavan. Probabilistic construction of deterministic algorithms: approximating packing integer programs. Journal of Computer and System Sciences, 37:130 143, 1988. 24 P. Raghavan and C. D. Thompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica, 7:365 374, 1987. Approx-49 Approx-50 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online