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RARE APPROXIMATION RATIOS

This article discusses rare approximation ratios in NP-Hard problems and explores techniques for coping with the difficulty of approximation. It covers examples, frequent ratios, and lower and upper bounds for minimization and maximization problems. The text sheds light on special problems and techniques related to approximation ratios.

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RARE APPROXIMATION RATIOS

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  1. RARE APPROXIMATION RATIOS Guy Kortsarz Rutgers University Camden

  2. Approximation Ratios • NP-Hard problems • Coping with the difficulty: approximation • Minimization or maximization. • Approximation ratio (for minimization):

  3. A Generic Problem: Set-Cover SETS ELEMENTS A B

  4. Frequent Approximation Ratios • Constants. Example: • Max-3-SAT: Tight 8/7 ratio • Logarithmic for minimization problems: • Set-cover • PTAS (1 + ) for all  > 0 • Example: Euclidean TSP

  5. Frequent Ratios continued • Polynomial Ratios: • sqrt (n), n{1 - } • Example: • Clique: n{1 - } lower bound • Upper bound: • (n/log3n) (Halldorsson, Feige)

  6. Example: Constrained Satisfaction Problems • Given a collection of Boolean formulas, satisfy all constrains. Maximize # true variables. • Possible ratios: 1) Solvable in polynomial time 2) n 3) Constant 4) Unbounded • Due to Khanna, Sudan, Williamson

  7. "Natural" Problems • It is possible to artificially design problems to get any desired ratio • See for example the NP-complete column of D. Johnson: The many limits of approximation • If in set-cover we take the objective function to be sqrt(|S|) then the ratio is sqrt(ln n) • I discuss rare ratios that appeared as a natural consequence of the problem/techniques • This sheds light on special problems/techniques

  8. Rare Ratios: Example I • Until 2000 there was no MAXIMIZATION PROBLEM with log n threshold • Example: Domatic Number • Input: G (V, E) • Dominating set U: U N(U) = V

  9. The Domatic Number Problem • Given: G (V, E) • Find: V=V1V2 …. Vk so that Vidominating set (in G). • Goal: Maximize k • Example: A maximal independent set and its complement is dominating. k≥ 2

  10. A Simple Algorithm • Create bins • Throw every vertex into a bin at random • The expected number of neighbors of every v in bin i is 3 ln n • The probability that bin i has no neighbor of v:

  11. Domatic Number Continued • The number of bad events is n2 or less. • Each one has probability 1/n3 to hold • By the union bound size partition exists • Remark: + 1 is a trivial upper bound • This implies O(ln n) ratio

  12. Large Minimum Degree opt = 2

  13. More Lower and Upper Bounds • Feige, Halldorsson, Kortsarz, Srinivasan • The approximation is improved to O (log ) (LLL) • There is always /ln  solution (complex proof) • Can not be approximated within (1 - )  ln n for any constant  > 0

  14. Remarks on the Lower Bound • Lower Bound Method: 1R2P • Generalizes (or improves) the paper of Feige from 1996, (1 - )  ln n, lower bound for set-cover • Recycling solutions: One Set Cover implies many set-cover exist • Uses Zero-Knowledge techniques

  15. Perhaps log n for Maximization: Unique Set Cover

  16. Special Case: Every Element in B has Degree d • Choose every aA with probability 1/d • Hence, expected number of uniquely covered elements of B, a constant fraction • Hence, there always is a subset A’A that uniquely covers a fraction

  17. General Case: • Cluster the degrees into powers of 2: • There exists a cluster with  (|B| / log |A| ) vertices • Corollary: There always exists A’A that uniquely covers a 1 / log n fraction of B

  18. Lower Bounds • Demaine, Feige, Hajiaghayi, Salvatipour: • Hard to find complete bipartite graphs, Implies log n best possible • NP has no algorithm implies (log n) hard to approximate • Hard to refute random 3-sat instances, implies ( log n ) 1/3hard

  19. Polylogarithmic for Minimization • Group Steiner problem on trees: g1 g2 g3 g5 g4

  20. Integrality Gap Halperin, Kortsarz, Krauthgamer, Srinivasan,Wang g1,g2 g3,g4 g1,g3,g2 g2,g4 g1,g3 g4 g1,g2 g2

  21. Analysis: • The costs need to decrease by constant factor [HST] • The fractional value is the same at every level • Thus, if the height is H then the fractional is O(H) • The integral H2 log k (k is # groups) • (log k)2 gap • The same paper [HKKSW] gives O ( (log k)2 ) upper bound

  22. More Upper Bounds • Garg, Ravi, Konjevod : • O( (log n)2) using Linear Programming • Randomized rounding plus Jansen inequalities • Halperin, Krauthgamer: • Lower bound: (log k)2- • (log n / log log n)2 • “Hiding” a trapdor in the integrality gap construction

  23. Directed Steiner and Below • Directed Steiner:O( (log n)3) quasi-polynomial time and n for every  polynomial time [Charikar etal] • Special case: Group Steiner on general graphs: O( (log n)3) polynomial (reduction to trees using Bartal Trees) • In quasi-polynomial tine O( (log n)2) for general graphs [Chekuri, Pal] • Group Steiner trees: log2n / log log n, quasi-polynomial time [Chekuri, Even, Kortsarz]

  24. The Asymmetric k-Center Problem • Given: Directed graph G(V, E) and length l(e) on edges and a number k • Required: choose a subset U, |U| = k of the vertices • Optimization criteria: Minimize

  25. A log* n Approximation • Due to Vishwanathan • Idea: k

  26. Lower Bound: log* n • Due to: Chuzhoy, Guha, Halperin, Khanna, Kortsarz, Krauthgamer, J. Naor • Based on hardness for d-set-cover

  27. Simple Algorithm for d-Set-Cover • Choose all the neighbors of some bB and add them to the solution • The algorithm adds d elements to the solution • The optimum is reduced by 1 • An inductive proof gives d ratio

  28. Hardness: Based on d-Set Cover Hardness: d – 1 -  Dinur, Guruswami, Khot, Regev: Gap Reduction for d – Set - Cover 3/d |A| enough to cover Yes instance d-set-cover I Any (1-2/d)|A| subset covers at most (1-f(d)) fraction of B. f(d)=(1/2) {poly d} d-set-cover No instance

  29. A Hardness Result for Directed k-Center • Compose the d-set-cover construction: • di+1 = exp (di) d2 d1

  30. Analysis • Choose k = (V1/d1)- 1 • For a YES instance get dist =1 • For a NO instance: • We may assume all centers are at V1 • But the number of uncovered vertices remains larger than 0 • Approaches 0 at log (previous) speed • Gives log* n gap

  31. Complete partitions of graphs

  32. Approximation for d - Regular Graphs • sqrt(m/2) is an upper bound • Partition to sqrt(m/2) classes at random • There is an expected O(1) edges per sets • Merge randomly to groups of 3 sets • Prove that with high probability its complete

  33. Complete Partitions Continued • For non-regular graphs complex algorithm and proof. • However possible • Lower bound • Uses the domatic number lower bound • Complex analysis • Gives lower bound for achromatic number

  34. More Between log n and O(1) • Minimum congestion routing: • Given a collection of pairs (undirected graph) choose a path for each pair. Minimize the congestion: • Upper bound: O(log n / loglog n) . [Raghavan , Thompson] • Lower bound: (log log n) . [Andrews, Zhang] • Maximum cycle packing. • upper bound [M. Krivelevich, Z. Nutov, M. Salavatipour, R. Yuster]. • lower bound. Salavatipour (private communication)

  35. More Between log n and O(1) • Directed congestion minimization: • O(log n / loglog n) upper bound [Raghavan and Thompson] • (logn) 1-lower bound. [Andrews and Zhang] • Min 2CNF deletion. • upper bound [Agrawal etal]. • Under the UNIQUE GAME CONJECTURE no constant ratio [Khot]

  36. More Between log n and O(1) • Sparsest cut: • upper bound [Arora, Rao and Vazirani] • Under UGC no c loglog n ratio, constant c [Chawla etal] • Point set width. • upper bound [Varadarajan etal] • (log n)lower bound [Varadarajan etal]

  37. Additive Approximation Ratios • The cost of the solution returned is opt+ •  is called the additive approximation ratio • Much less common (or studied(?)) than multiplicative ratios

  38. New Result • Let G (V,E,c) be a graph that admits a spanning tree of cost at most c* and maximum degree at most d • Then, there exists a polynomial time algorithm that finds a spanning tree of cost at most c* and maximum degree d+2. Additive ratio 2 [Goemans, FOCS 2006]

  39. The Ultimate Approximation • Some problems admit +1 approximation • Known examples: • Coloring a planar graph • Chromatic index: coloring edges [Vizing] • Find spanning tree with minimum maximum degree [Furer Ragavachari] • Some less known +1 approximation:

  40. Achromatic Number

  41. Achromatic Number of Trees • The problem is hard on trees • Thus opt is bounded by roughly sqrt n • This bound is achievable within +1 (in polynomial time) • Similarly: Minimum Harmonious coloring of trees: +1 approximation

  42. Poly-log Additive (tight): Radio Broadcast R1 R2 R3 R4

  43. Upper and Lower Bounds • Since one can cover 1/log n uniquely, in O( (log n)2) rounds the other side of a Bipartite graph can be informed • Thus, in a BFS fashion: Radius(log n)2 • Best known [Kowalski, Pelc] : Radius+O(log n)2 • Lower bound [Elkin, Kortsarz] : For some constant c, opt + c (log n)2 not possible unless NP  DTIME(n{poly-log n})

  44. A graph with radius = 1, opt = (log n)2 A construction by Alon, Bar-Noy, Lineal, Peleg P=(1/2){0.4log n} P=(1/2) {0.6log n}

  45. Analysis • If we choose any subset of size 2j then the set of probability (½)j will be informed in log n rounds • Since there are 0.2 ln n sets, it will take O( (log n)2) • The difficulty: A size 2j does not affect the sets of p = (½)k, k > j • However, if k < j, size 2j causes collisions for k, hence is of little help

  46. Conclusion • No real conclusion • The NPC problem seems to admit little order if at all regarding approximation • The problems are ``unstable” • There does not seem to be a ``deep” reason these ratios are rare (because of techniques(?)) • Very good advances. • Still much we don’t understand in approximations

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