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Approximation Algorithm
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  1. Approximation Algorithm Instructor: YE, Deshi yedeshi@zju.edu.cn

  2. Dealing with Hard Problems • What to do if: • Divide and conquer • Dynamic programming • Greedy • Linear Programming/Network Flows • … • does not give a polynomial time algorithm?

  3. Dealing with Hard Problems • Solution I: Ignore the problem • Can’t do it ! There are thousands of problems for which we do not know polynomial time algorithms • For example: Traveling Salesman Problem (TSP) Set Cover

  4. Traveling Salesman Problem • Traveling SalesmanProblem (TSP) • Input: undirected graph with lengths on edges • Output: shortest cycle that visits each vertex exactly once • Best known algorithm: O(n 2n) time.

  5. The vertex-cover problem • A vertex cover of an undirected graph G = (V, E) is a subset V ' ⊆ V such that if (u, v) ∈ E, then u ∈ V ' or v ∈ V ' (or both). • A vertex cover for G is a set of vertices that covers all the edges in E. • As a decision problem, we define • VERTEX-COVER = {〈G, k〉 : graph G has a vertex cover of size k}. • Best known algorithm: O(kn + 1.274k)

  6. Dealing with Hard Problems • Exponential time algorithms for small inputs. E.g., (100/99)n time is not bad for n < 1000. • Polynomial time algorithms for some (e.g., average-case) inputs • Polynomial time algorithms for all inputs, but which return approximate solutions

  7. Approximation Algorithms • An algorithm A is ρ-approximate, if, on any inputof size n: • The cost CA of the solution produced by the algorithm, and • The cost COPT of the optimal solution are such that CA ≤ ρ COPT • We will see: • 2-approximation algorithm for TSP in the plane • 2-approximation algorithm for Vertex Cover

  8. Comments on Approximation • “CA ≤ ρ COPT” makes sense only for minimization problems • For maximization problems, replace by • COPT ≤ ρ CA • Additive approximation “CA ≤ ρ + COPT “ also makes sense, although difficult to achieve

  9. The Vertex-cover problem

  10. The vertex-cover problem • A vertex cover of an undirected graph G = (V, E) • is a subset V' ⊆ V such that if (u, v) ∈ E, then u ∈ V' or v ∈ V' (or both). • A vertex cover for G is a set of vertices that coversall the edges in E. • The goal is to find a vertex cover of minimum size in a given undirected graph G.

  11. Naive Algorithm • APPROX-VERTEX-COVER(G) • 1 C ← Ø • 2 E′ ← E[G] • 3 whileE′ ≠ Ø • 4 do let (u, v) be an arbitrary edge of E′ • 5 C ← C ∪ {u, v} • 6 remove from E′ every edge incident on either u or v • 7 returnC

  12. Illustration of Naive algorithm Edge bc is chosen Set C = {b, c} Input Edge ef is chosen Optimal solution {b, e, d} Naive algorithm C={b,c,d,e,f,g}

  13. Approximation 2 • Theorem. APPROX-VERTEX-COVER is a 2-approximation algorithm. • Pf.let A denote the set of edges that were picked by APPROX-VERTEX-COVER. • To cover the edges in A, any vertex cover, in particular, an optimal cover C* must include at least one endpoint of each edge in A. • No two edges in A share an endpoint. • Thus no two edges in A are covered by the same vertex from C*, and we have the lower bound C* ≥ |A| • On the other hand, the algorithm picks an edge for which neither of its endpoints is already in C. |C| = 2|A| Hence, |C| = 2|A| ≤ 2|C*|.

  14. Vertex cover: summary • No better constant-factor approximation is known!! • More precisely, minimum vertex cover is known to be approximable within (for a given |V|≥2) (ADM85) • but cannot be approximated within 7/6 (Hastad STOC97) for any sufficiently large vertex degree, Dinur Safra (STOC02)1.36067

  15. Vertex cover: summary • Eran Halperin, Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs, SIAM Journal on Computing, 31/5  (2002): 1608 - 1623 . • Tomokazu Imamura , Kazuo Iwama, Approximating vertex cover on dense graphs, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms 2005

  16. The Traveling Salesman Problem • Traveling SalesmanProblem (TSP) • Input: undirected graph G = (V, E) with edges cost c(u, v) associated with each edge (u, v) ∈ E • Output: shortest cycle that visits each vertex exactly once • Triangle inequality if for all vertices u, v, w ∈ V, • c(u, w) ≤ c(u, v) + c(v, w). u v w

  17. 2-approximation for TSP with triangle inequality • Compute MST T • An edge between any pair of points • Weight = distance between endpoints • Compute a tree-walk W of T • Each edge visited twice • Convert W into a cycle H using shortcuts

  18. Algorithm APPROX-TSP-TOUR(G, c) 1 select a vertex r ∈ V [G] to be a "root" vertex 2 compute a minimum spanning tree T for G from root r using MST-PRIM(G, c, r) 3 let L be the list of vertices visited in a preorder tree walk of T 4return the hamiltonian cycle H that visits the vertices in the order L

  19. Preorder Traversal • Preorder: (root-left-right) • Visit the root first; and then • traverse the left subtree; and then • traverse the right subtree. • Example: Order: A,B,C,D,E,F,G,H,I

  20. Illustration MST Tree walk W A full walk of the tree visits the vertices in the order a, b, c, b, h, b, a, d, e, f, e, g, e, d, a. preorder walk (Final solution H) OPT solution

  21. 2-approximation • Theorem. APPROX-TSP-TOUR is a polynomial-time 2-approximation algorithm for the traveling-salesman problem with the triangle inequality. • Pf.Let COPT be the optimal cycle • Cost(T) ≤ Cost(COPT) • Removing an edge from H gives a spanning tree, T is a spanning tree of minimum cost • Cost(W) = 2 Cost(T) • Each edge visited twice • Cost(H) ≤ Cost(W) • Triangle inequality • Cost(H) ≤ 2 Cost(COPT )

  22. Load Balancing • Input. m identical machines; n jobs, job j has processing time tj. • Job j must run contiguously on one machine. • A machine can process at most one job at a time. • Def. Let J(i) be the subset of jobs assigned to machine i. The • load of machine i is Li = j  J(i) tj. • Def. The makespan is the maximum load on any machine L = maxi Li. • Load balancing. Assign each job to a machine to minimize makespan.

  23. Load Balancing: List Scheduling • List-scheduling algorithm. • Consider n jobs in some fixed order. • Assign job j to machine whose load is smallest so far. • Implementation.O(n log n) using a priority queue. List-Scheduling(m, n, t1,t2,…,tn) { for i = 1 to m { Li  0 J(i)  } for j = 1 to n { i = argmink Lk J(i) J(i) {j} Li  Li + tj } } load on machine i jobs assigned to machine i machine i has smallest load assign job j to machine i update load of machine i

  24. Load Balancing: List Scheduling Analysis • Theorem. [Graham, 1966] Greedy algorithm is a (2-1/m)-approximation. • First worst-case analysis of an approximation algorithm. • Need to compare resulting solution with optimal makespan L*. • Lemma 1. The optimal makespan L*  maxj tj. • Pf. Some machine must process the most time-consuming job. ▪ • Lemma 2. The optimal makespan • Pf. • The total processing time is j tj . • One of m machines must do at least a 1/m fraction of total work. ▪

  25. Load Balancing: List Scheduling Analysis • Theorem. Greedy algorithm is a (2-1/m)-approximation. • Pf. Consider load Li of bottleneck machine i. • Let j be last job scheduled on machine i. • When job j assigned to machine i, i had smallest load. Its load before assignment is Li - tj  Li - tj  Lk for all 1  k  m. blue jobs scheduled before j machine i j 0 Li - tj L = Li

  26. Load Balancing: List Scheduling Analysis • Theorem. Greedy algorithm is a (2-1/m)- approximation. • Pf. Consider load Li of bottleneck machine i. • Let j be last job scheduled on machine i. • When job j assigned to machine i, i had smallest load. Its load before assignment is Li - tj  Li - tj  Lk for all 1  k  m. • Sum inequalities over all k and divide by m: • Now ▪ Lemma 2 Lemma 1 Lemma 1

  27. Load Balancing: List Scheduling Analysis • Q. Is our analysis tight? • A. Essentially yes. Indeed, LS algorithm has tight bound 2- 1/m • Ex: m machines, m(m-1) jobs length 1 jobs, one job of length m machine 2 idle machine 3 idle machine 4 idle m = 10 machine 5 idle machine 6 idle machine 7 idle machine 8 idle machine 9 idle machine 10 idle list scheduling makespan = 19

  28. Load Balancing: List Scheduling Analysis • Q. Is our analysis tight? • A. Essentially yes. Indeed, LS algorithm has tight bound 2- 1/m • Ex: m machines, m(m-1) jobs length 1 jobs, one job of length m m = 10 optimal makespan = 10

  29. Load Balancing on 2 Machines • Claim. Load balancing is hard even if only 2 machines. • Pf. NUMBER-PARTITIONING PLOAD-BALANCE. a b c d e f g length of job f Machine 1 a d f machine 1 yes machine 2 b Machine 2 c e g 0 Time L

  30. Load Balancing: LPT Rule • Longest processing time (LPT). Sort n jobs in descending order of processing time, and then run list scheduling algorithm. LPT-List-Scheduling(m, n, t1,t2,…,tn) { Sort jobs so that t1 ≥ t2 ≥…≥ tn for i = 1 to m { Li  0 J(i)  } for j = 1 to n { i = argmink Lk J(i) J(i) {j} Li  Li + tj } } load on machine i jobs assigned to machine i machine i has smallest load assign job j to machine i update load of machine i

  31. Load Balancing: LPT Rule • Observation. If at most m jobs, then list-scheduling is optimal. • Pf. Each job put on its own machine. ▪ • Lemma 3. If there are more than m jobs, L*  2tm+1. • Pf. • Consider first m+1 jobs t1, …, tm+1. • Since the ti's are in descending order, each takes at least tm+1 time. • There are m+1 jobs and m machines, so by pigeonhole principle, at least one machine gets two jobs. ▪ • Theorem. LPT rule is a 3/2 approximation algorithm. • Pf. Same basic approach as for list scheduling. • ▪ Lemma 3( by observation, can assume number of jobs > m )

  32. Load Balancing: LPT Rule • Q. Is our 3/2 analysis tight? • A. No. • Theorem. [Graham, 1969] LPT rule is a (4/3 – 1/(3m))-approximation. • Pf. More sophisticated analysis of same algorithm. • Q. Is Graham's (4/3 – 1/(3m))- analysis tight? • A. Essentially yes. • Ex: m machines, n = 2m+1 jobs, 2 jobs of length m+1, m+2, …, 2m-1 and three jobs of length m.

  33. LPT • Proof. Jobs are indexed t1 ≥ t2 ≥ … ≥ tn. • If n ≤ m, already optimal (one machine processes one job). • If n> 2m, then tn ≤ L*/3. Similar as the analysis of LS algorithm. • Suppose total 2m – h jobs, 0 ≤ h < m • Check that LPT is already optimal solution 1 h h+1 n h+2 n-1 Time

  34. Approximation Scheme • NP-complete problems allow polynomial-time approximation algorithms that can achieve increasingly smaller approximation ratios by using more and more computation time • Tradeoff between computation time and the quality of the approximation • For any fixed ∈>0, An approximation scheme for an optimization problem is an (1 + ∈)-approximation algorithm.

  35. PTAS and FPTAS • We say that an approximation scheme is a polynomial-time approximation scheme (PTAS) if for any fixed ∈ > 0, the scheme runs in time polynomial in the size n of its input instance. • Example: O(n2/∈). • an approximation scheme is a fully polynomial-time approximation scheme (FPTAS) if it is an approximation scheme and its running time is polynomial both in 1/∈ and in the size n of the input instance • Example: O((1/∈)2n3).

  36. The Subset Sum • Input. A pair (S, t), where S is a set {x1, x2, ..., xn} of positive integers and t is a positive integer • Output. A subset S′ of S • Goal. Maximize the sum of S′ but its value is not larger than t.

  37. An exponential-time exact algorithm • If L is a list of positive integers and x is another positive integer, • then we let L + x denote the list of integers derived from L by increasing each element of L by x. • For example, if L = 〈1, 2, 3, 5, 9〉, then L + 2 = 〈3, 4, 5, 7, 11〉. • We also use this notation for sets, so that S + x = {s + x : s ∈ S}.

  38. Exact algorithm • MERGE-LISTS(L, L′): returns the sorted list that is the merge of its two sorted input lists L and L′ with duplicate values removed. • EXACT-SUBSET-SUM(S, t) • 1 n ← |S| • 2 L0 ← 〈0〉 • 3 fori ← 1 ton • 4 doLi ← MERGE-LISTS(Li-1, Li-1 + xi) • 5 remove from Li every element that is greater than t • 6 return the largest element in Ln

  39. Example • For example, if S = {1, 4, 5}, then • P1 ={0, 1} , • P2 ={0, 1, 4, 5} , • P3 ={0, 1, 4, 5, 6, 9, 10} . • Given the identity • Since the length of Li can be as much as 2i, it is an exponential-time algorithm .

  40. The Subset-sum problem: FPTAS • Trimming or rounding: if two values in L are close to each other, then for the purpose of finding an approximate solution there is no reason to maintain both of them explicitly. • Let δ such that 0 < δ < 1. • L′ is the result of trimming L, for every element y that was removed from L, there is an element z still in L′ that approximates y, that is

  41. Example • For example, if δ = 0.1 and • L = 〈10, 11, 12, 15, 20, 21, 22, 23, 24, 29〉, • then we can trim L to obtain • L′ = 〈10, 12, 15, 20, 23, 29〉, • TRIM(L, δ) • 1 m ← |L| • 2 L′ ← 〈y1〉 • 3 last ← y1 • 4 fori ← 2 tom • 5 do ifyi > last · (1 + δ) ▹ yi ≥ last because L is sorted • 6 then append yi onto the end of L′ • 7 last ← yi • 8 returnL′

  42. (1+ ∈)-Approximation algorithm • APPROX-SUBSET-SUM(S, t, ∈) • 1 n ← |S| • 2 L0 ← 〈0〉 • 3 fori ← 1 ton • 4 doLi ← MERGE-LISTS(Li-1, Li-1 + xi) • 5 Li ← TRIM(Li, ∈/2n) • 6 remove from Li every element that is greater than t • 7 let z* be the largest value in Ln • 8 returnz*

  43. FPTAS • Theorem.APPROX-SUBSET-SUM is a fully polynomial-time approximation scheme for the subset-sum problem. • Pf. The operations of trimming Li in line 5 and removing from Li every element that is greater than t maintain the property that every element of Li is also a member of Pi. Therefore, the value z* returned in line 8 is indeed the sum of some subset of S.

  44. Pf. Con. • Pf. Let y*∈ Pn denote an optimal solution to the subset-sum problem. we know that z* ≤ y*. We need to show that y*/z* ≤ 1 + ∈. • By induction on i, it can be shown that for every element y in Pi that is at most t, there is a z ∈ Li such that • Thus, there is a z ∈ Ln , such that

  45. Pf. Con. • And thus, • Since there is a z ∈ Ln • Hence,

  46. Pf. Con. • To show FPTAS, we need to bound Li. • After trimming, successive elements z and z′ of Li must have the relationship z′/z > 1+∈/2n • Each list, therefore, contains the value 0, possibly the value 1, and up to ⌊log1+∈/2nt⌋ additional values