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The Matroid Median Problem
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  1. The Matroid Median Problem Viswanath Nagarajan IBM Research Joint with R. Krishnaswamy, A. Kumar, Y. Sabharwal, B. Saha

  2. k-Median Problem • Set of locations in a metric space (V,d) • Symmetric, triangle inequality • Place k facilities such that sum of connection costs (to nearest facility) is minimized: minFµV, |F|·ku2V d(u,F)

  3. k-Median Results • poly(log n) approx via tree embeddings [B ’96] • LP rounding O(1)-approx [CGST ’99] • Lagrangian relaxation + primal dual [JV ’01] • Local search with p-exchanges [AGKMMP ’04] • best known ratio 3+² • Hardness of approximation ¼ 1.46 [GK ’98]

  4. Red-Blue Median • Facilities are of two different types • Partition V into red and blue sets • Separate bounds kr and kb on facilities • Recently introduced [HKK ’10] • Motivated by Content Distribution Networks • T facility-types (RB Median is T=2) • O(1)-approximation ratio via Local Search kr=3kb=2

  5. Matroid Median • Given matroid M on ground-set V • Locate facilities F that are independent in M • Minimize connection cost • Recap matroid M=(V, Iµ2V) • A,B2 I and |A|<|B| )9 e 2 BnA : A[{e} 2 I • Substantial generalization of RB Median • The CDN application with T facility-types reduces to partition matroid constraint B A e k3=1 k4=2 k1=2 k2=3

  6. Talk Outline Thm: 16-approximation for Matroid Median • Bad example for Local Search • LP relaxation • Phase I : sparsification • Phase II: reformulation

  7. Local Search? • Partition matroid with T parts • T-1 exchange local search • Swap up to T-1 facilities in each step • Unlikely to work beyond T=O(1) Eg. T=5 Uniform metric on T+1 Clients n=mT+1 OPT = 1 (small fac.) LOPT = m (big fac.) m 1 m m m m locality gap (n/T)

  8. LP relaxation min uv d(u,v) ¢xuv s.t. vxuv = 1 8 u 2 V xuv·yv8 u,v 2 V v2Syv· r(S) 8 Sµ V x, y¸ 0. connection constraints matroid rank constraints xuv u v y 2 M facilities clients

  9. Solving the LP • Exponential number of rank constraints • Use separation oracle: minSµV r(S) - v2S yv • An instance of submodular minimization • Also more efficient algorithms to separate over the matroid polytope [C ’84] • Solvable in poly-time via Ellipsoid algorithm

  10. Idea for approach(1) • Problem non-trivial even if metric is a tree  • Even O(log n)-approximation not obvious • What’s easier than a tree? • Suppose input is special star-like instance One root facility (can help any client) Others are private facilities (help only 1 client) client 2 client 1 client 3 root facility

  11. Idea for approach(2) • Recall LP variables • yj : facility opening (in matroid polytope) • xij : connection • For any client i, private j2 P(i) WMA xij = yj • Connection constraintj xij = 1 • So xir = 1 - j2P(i) xij = 1 - j2P(i) yj • Can eliminate all connection variables ! client i r private facilities P(i)

  12. Idea for approach(3) • Reformulate the LP min i[j2P(i) dij¢ yj + dir¢(1- j2P(i) yj) ] s.t. j2P(i) yj· 1, 8 clients i y 2 M • This is just an instance of intersection of M with partition matroid from P(i)s  xij xir To ensure xir¸ 0 matroid constraint

  13. Idea for approach(4) • Start with LP optimum (x,y) of arbitrary matroid median instance • Phase I: Use (x,y) to form clusters of disjoint star-like instances • Phase II: Resolve the new star-LP • (x,y) itself restricted to the stars not integral • Show that new LP is integral • ¼ matroid intersection

  14. Phase I: sparsify LP solution

  15. Outline • Modify LP connections x in four steps • Similar to [CGST ’99] • Key: no change in facility variables y • Need to ensure y remains in matroid polytope • Not true in [CGST ’99] • Require some more (technical) work

  16. Step 1: cluster clients • Lu = v duv¢xuv, contribution of u to LP obj. • B(u) is local ball of u • vertices within distance 2¢Lufrom u • Order clients u in increasing Lu • Pick maximal disjoint set of local balls • T are the chosen clients • Move each client to T-client close to it Loss in obj · 4¢ LP* (additive) 3 4 3 4 5 5 1 1 6 6 2 2

  17. Obs on step 1 • Local balls of T clients are disjoint • y-value inside any local ball ¸ ½ • Markov inequality • Restrict to clients T (now weighted) • For any p,q2T : d(p,q) ¸ 2¢(LPp + LPq) • well separated clients y¸½ separated T balls

  18. More obs on step 1 • Supposey-value in each T’s local ball ¸ 1 • Then instance of matroid intersection: • Matroid M and partition from local-ball(T) • Resolving suitable LP ) integral soln • Will need intersection with `laminar’ constraints, not just partition matroid

  19. Step 2: private facilities • Ensure that each facility in some T-ball or helps at most one client (ie. private) • Break connections from all except closest client 1 to facility j • Reconnect to facilities in B(1), y-value ¸ ½ • Total reconnection for any client · ½ j 3 1 Constant factor loss in obj 2

  20. Step 3: uniform objective • Each connection from client p to any facility in B(q) will pay same objective d(p,q) • Since p,q well separated d(p,q) · O(1)¢ d(p,j) • For any j 2 B(q) • Constant factor loss in obj p q

  21. Step 4: building stars • WMA each client i 2 T connected to • Its private facilities P(i), OR • Its closest other client k2T, ie. facility in B(k) • Set of `outer’ connections ¼ directed tree • Unique out-edge from each client • Lem: Can modify outer connection to `star’ Constant factor loss in obj

  22. The star structure • One pseudo-root { r, r’ } • Every other client connected to either r or r’ • All LP-connections x are from client i to: • private facility j2P(i), obj d(i,j) OR • facility in B(k) with k2{ r, r’ }, uniform obj d(i,k) r r’ i

  23. Phase II: using star • Will drop all the connection x-variables • WMA xij = yj for j2P(i) private facilities • Total outer connection=1 - j2P(i) xij =1 - j2P(i) yj • Each outer-connection pays same obj d(i,r) • Want property (in integral soln) that P(i)=;) there is a recourse connection to r • Do not quite ensure this, but…

  24. Phase II contd. • Add constraint that y(P(r)) + y(P(r’)) ¸ 1 • Indeed feasible for (x,y) since each local ball has y-value ¸ ½ • This ensures (in integral soln) that P(i)=;) there is a recourse connection to r or r’ • Lose another constant factor in obj

  25. Phase II: new LP • Apply constraints for each star to get LP min i[j2P(i) dij¢ yj + d(i,r(i))¢(1- j2P(i) yj)] s.t. j2P(i) yj· 1, 8 clients i y(P(r)) + y(P(r’)) ¸ 1, 8 p-root {r, r’} y 2 M Lem: Integral polytope (via proof similar to matroid intersection) laminar constraints matroid constraint

  26. Summarize • Using LP solution and metric properties reduce to star-like instances • Formulate new LP for star-like instances, with only facility variables • New LP is integral

  27. Other Results • O(1)-approximation for prize-collecting version of matroid median • Knapsack Median problem (knapsack constraint on open facilities) • Give bi-criteria approx, violate budget by wmax • Can we get true O(1)-approx? • Handle other constraints in k-median?

  28. Thank You