RARE APPROXIMATION RATIOS

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