RARE APPROXIMATION RATIOS Guy Kortsarz Rutgers University Camden

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

A Generic Problem: Set-Cover A SETS ELEMENTS B

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

Frequent Ratios continued Polynomial Ratios: Polynomial Ratios: sqrt (n), n {1 -  } sqrt (n), n {1 -  } Example: Example: Clique: n {1 -  } lower bound Clique: n {1 -  } lower bound Upper bound: Upper bound:  (n/log 3 n) (Halldorsson, Feige)  (n/log 3 n) (Halldorsson, Feige)

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

" Natural" Problems It is possible to artificially design problems to get any desired ratio 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 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) 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 I discuss rare ratios that appeared as a natural consequence of the problem/techniques This sheds light on special problems/techniques This sheds light on special problems/techniques

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

The Domatic Number Problem Given: G (V, E) Given: G (V, E) Find: V=V 1  V 2  ….  V k Find: V=V 1  V 2  ….  V k so that V i dominating set (in G). so that V i dominating set (in G). Goal: Maximize k Goal: Maximize k Example: A maximal independent set Example: A maximal independent set and its complement is dominating. k ≥ 2 and its complement is dominating. k ≥ 2

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

Domatic Number Continued The number of bad events is n 2 or less. The number of bad events is n 2 or less. Each one has probability 1/n 3 to hold Each one has probability 1/n 3 to hold By the union bound size partition exists By the union bound size partition exists Remark:  + 1 is a trivial upper bound Remark:  + 1 is a trivial upper bound This implies O(ln n) ratio This implies O(ln n) ratio

Large Minimum Degree opt = 2

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

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

Perhaps log n for Maximization: Unique Set Cover

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

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

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

Polylogarithmic for Minimization Group Steiner problem on trees: Group Steiner problem on trees: g1 g2g3 g4 g5

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

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

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

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

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

A log* n Approximation Due to Vishwanathan Due to Vishwanathan Idea: Idea: k

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

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

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

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

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

Complete partitions of graphs Complete partitions of graphs

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

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

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

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

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

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

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 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] 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]

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

Achromatic Number

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

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

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

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}

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

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