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RARE APPROXIMATION RATIOS Guy Kortsarz Rutgers University Camden Approximation Ratios NP-Hard problems Coping with the difficulty: approximation Minimization or maximization. Approximation ratio (for minimization): ALG ( I ) max I Inputs opt ( I ) A Generic Problem: Set-Cover SETS ELEMENTS A B 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 Frequent Ratios continued Polynomial Ratios: sqrt (n), n {1 - } Example: Clique: n {1 - } lower bound Upper bound: (n/log3n) (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 2) n 3) Constant 4) Unbounded Due to Khanna, Sudan, Williamson "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 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 The Domatic Number Problem (V, E) Given: G Find: V=V1 V2 …. Vk so that Vi dominating set (in G). Goal: Maximize k Example: A maximal independent set and its complement is dominating. k ≥ 2 A Simple Algorithm Create bins 3 ln n 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: 3 ln n 1 1 3 n 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 3 ln n Remark: + 1 is a trivial upper bound This implies O(ln n) ratio Large Minimum Degree opt = 2 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 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 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 d 1 1 1 1 Pr(Unique Neighbour for b) d 1 d d e 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 General Case: Cluster the degrees into powers of 2: i 1 D i {b B | 2 deg(b) 2 } i 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 Lower Bounds Demaine, Feige, Hajiaghayi, Salvatipour: Hard to find complete bipartite graphs, Implies log n best possible n NP has no 2 algorithm implies (log n) hard to approximate Hard to refute random 3-sat instances, implies ( log n ) 1/3 hard Polylogarithmic for Minimization Group Steiner problem on trees: g1 g2 g3 g5 g4 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 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 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 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: log2 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 Required: choose a subset U, |U| = k of the vertices Optimization criteria: Minimize max{dist (u,U )} uU A log* n Approximation Due to Vishwanathan Idea: k Lower Bound: log* n Due to: Chuzhoy, Guha, Halperin, Khanna, Kortsarz, Krauthgamer, J. Naor 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 Yes instance d-set-cover 3/d |A| enough to cover I Any (1-2/d)|A| subset covers at d-set-cover most (1-f(d)) fraction of B. No instance f(d)=(1/2) {poly d} A Hardness Result for Directed k-Center Compose the d-set-cover construction: d2 d1 di+1 = exp (di) 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 Complete partitions of graphs 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 log n sets Prove that with high probability its complete Complete Partitions Continued For non-regular graphs complex algorithm and proof. However log n possible Lower bound ( log n ) Uses the domatic number lower bound Complex analysis Gives ( log n ) lower bound for achromatic number 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. log n upper bound [M. Krivelevich, Z. Nutov, M. Salavatipour, R. Yuster]. log n lower bound. Salavatipour (private communication) More Between log n and O(1) Directed congestion minimization: O(log n / loglog n) upper bound [Raghavan and Thompson] (log n) 1- lower bound. [Andrews and Zhang] Min 2CNF deletion. log n upper bound [Agrawal etal]. Under the UNIQUE GAME CONJECTURE no constant ratio [Khot] More Between log n and O(1) Sparsest cut: log n upper bound [Arora, Rao and Vazirani] Under UGC no c loglog n ratio, constant c [Chawla etal] Point set width. log n upper bound [Varadarajan etal] (log n) lower bound [Varadarajan etal] 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 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] 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: Achromatic Number Achromatic Number of Trees The problem is hard on trees opt n 1 2 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 Poly-log Additive (tight): Radio Broadcast R1 R4 R2 R3 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}) A graph with radius = 1, opt = (log n)2 A construction by Alon, Bar-Noy, Lineal, Peleg P=(1/2){0.4log n} P=(1/2) {0.6log n} 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 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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posted: | 9/25/2012 |

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