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Hardness of Approximation

Hardness of Approximation. Introduction. Objectives: To show several approximation problems are NP-hard Overview: Reminder: How to show inapproximability? Probabilistic Checkable Proofs Hardness of approximation for clique. Optimization Problems. Consider an optimization problem P :.

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Hardness of Approximation

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  1. Hardness of Approximation Complexity

  2. Introduction • Objectives: • To show several approximation problems are NP-hard • Overview: • Reminder: How to show inapproximability? • Probabilistic Checkable Proofs • Hardness of approximation for clique Complexity

  3. Optimization Problems Consider an optimization problem P: Example: all graphs instances: x1,x2,x3,… all cliques in that graph feasible solutions the clique’s size (max) optimization measure Complexity

  4. x1 x2 x3 x4 Each Instance Has an Optimal Solution OPT Complexity

  5. xi Approximation (Max Version) OPT Complexity

  6. xi A B gap How To Show Hardness of Approximation? Hardness of distinguishing far off instances  Hardness of approximation OPT Complexity

  7. Gap Problems (Max Version) • Instance: … • Problem: to distinguish between the following two cases: The maximal solution  B The maximal solution ≤ A YES NO Complexity

  8. Formally: Claim: If the [A,B]-gap version of a problem is NP-hard, then that problem is NP-hard to approximate within factor B/A. Complexity

  9. Formally: Proof: Suppose there is an approximation algorithm that outputs C so that C/C*≤B/A A proper distinguisher: * If CB, return ‘YES’ * Otherwise return ‘NO’ Complexity

  10. Proof Since C*≥AC/B, (1) If C>B(we answer ‘YES’), then necessarily C*>A(the correct answer cannot be ‘NO’). (2) If C*≤A (the correct answer is ‘NO’), then necessarily C≤B (we answer ‘NO’) Complexity

  11. Idea • We’ve shown “standard” problems are NP-hard by reductions from 3SAT. • We want to prove gap-problems are NP-hard, • Why won’t we prove some canonical gap-problem is NP-hard and reduce from it? • If a reduction reduces one gap-problem to another we refer to it as approximation-preserving Complexity

  12. Gap-3SAT[] Instance: a set of clauses {c1,…,cm} over variables v1,…,vn. Problem:to distinguish between the following two cases: There exists an assignment which satisfies all clauses. No assignment can satisfy more than 7/8+ of the clauses. YES NO Complexity

  13. Gap-3SAT: Example ( x1  x2  x3 ) ( x1  x2  x2 ) ( x1  x2  x3 ) ( x1  x2  x2 ) (x1  x2  x3 ) ( x3  x3  x3 )  = { x1  F ; x2 T ; x3 F } satisfies 5/6 of the clauses Complexity

  14. Why 7/8? Claim: For any set of clauses with exactly three independent literals, there always exists an assignment which satisfies at least 7/8. Complexity

  15. x1 x2 x3 xn The Probabilistic Method Proof: Consider a random assignment. . . . Complexity

  16. 1. Find the Expectation Let Yi be the random variable indicating the outcome of the i-th clause. For any 1im,E[Yi]=0·1/8+1·7/8=7/8 E[ Yi] =  E[Yi] = 7/8m Complexity

  17. 2. Conclude Existence Expectedly, the number of clauses satisfied is 7/8m. Thus, there exists an assignment which satisfies at least that many.  Complexity

  18. PCP (Without Proof) Theorem (PCP): For any >0, Gap-3SAT[] is NP-hard. This is tight! Gap-3SAT[0] is polynomial time decidable Complexity

  19. Approximation Preservation A B • YES • YES • don’t care • don’t care • NO • NO Complexity

  20. Hardness of Approximation • Do the reductions we’ve seen also work for the gap versions? • We’ll revisit the CLIQUE example. Complexity

  21. CLIQUE Construction a vertex for each literal a part for each clause edge indicates consistency . . . Complexity

  22. Approximation Preservation • If there is an assignment which satisfies all clauses, there is a clique of size m. • If there is a clique of size (7/8+)m, there is an assignment which satisfies more than 7/8+ of the clauses. Complexity

  23. Gap-CLIQUE (Ver1) The following problem is NP-hard for any >0: Instance: a graph G=(V,E) composed of m independent sets of size 3. Problem:to distinguish between: There’s a clique of size m Every clique is of size at most (7/8+)m YES NO Complexity

  24. Corollary Theorem: for any >0, CLIQUE is hard to approximate within a factor of 1/(7/8+) Complexity

  25. Amplification • The bigger the gap is, the better the hardness result. • We’ll see how a gap can be amplified. Complexity

  26. ... ... . . . Amplification Given an instance of the Gap-CLIQUE problem and a constant k: A part for every k vertices edge indicates consistency vertex for each Boolean assignment Complexity

  27. Boolean assignments • A Boolean assignment over k vertices {v1,…,vk} is a function A:{v1,…,vk}{0,1}. • Think about it as if it indicates whether each vertex belongs to the clique. Complexity

  28. Good Assignments Complexity

  29. . . . . . . . . . Consistency • Two assignments are inconsistent, when they give the same vertex different truth-values. n Complexity

  30. Consistency • They are also inconsistent, if they both assign 1 to two vertices not connected by an edge. non-edge Complexity

  31. Correctness Complexity

  32. Chromatic Number • Instance: a graph G=(V,E). • Problem: To minimize k, so that there exists a function f:V{1,…,k}, for which (u,v)E  f(u)f(v) Complexity

  33. Chromatic Number Complexity

  34. Chromatic Number Observation: Each color group is an independent set Complexity

  35. Clique Cover Number (CCN) • Instance: a graph G=(V,E). • Problem: To minimize k, so that there exists a function f:V{1,…,k}, for which (u,v)E  f(u)=f(v) Complexity

  36. Clique Cover Number (CCN) Complexity

  37. . . . Reduction Idea CLIQUE CCN . . . . . . m • cyclic shift-morphic • clique preserving q Complexity

  38. Correctness Complexity

  39. Transformation T:V[q] for any v1,v2,v3,v4,v5,v6, T(v1)+T(v2)+T(v3) T(v4)+T(v5)+T(v6) (mod q) {v1,v2,v3}={v4,v5,v6} T is unique for triplets Complexity

  40. Observations • Such T is unique for pairs and for single vertices as well: • If T(x)+T(u)=T(v)+T(w), then {x,u}={v,w} • If T(x)=T(y) (mod q), then x=y Complexity

  41. v5 forbidden values v1 v2 v3 v4 feasible values vertices we determined Greedy Construction v6 Complexity

  42. Greedy Construction - Analysis At most values are ruled out totally, so for q=n5the greedy construction works. Corollary: There exists a polynomial time algorithm which constructs a triplet unique transformation with q=n5 Complexity

  43. Using the Transformation vi vj CLIQUE T(vj)=4 T(vi)=1 CCN 0 1 2 3 4 … (q-1) Complexity

  44. Completing the CCN Graph Construction T(s) (s,t)ECLIQUE  (T(s),T(t))ECCN T(t) Complexity

  45. Completing the CCN Graph Construction Close the set of edges under shift: For every (x,y)E, if x’-y’=x-y (mod q), then (x’,y’)E T(s) T(t) Complexity

  46. Max Clique of G-clique and G-ccn • Lemma:Max-Clique(G-clique) = Max-Clique(G-CCN) • Corollary: • MAX-clique(G-clique) = m CCN(G-ccn)=q • MAX-clqiue(G-clique) < m CCN(G-ccn)> q Complexity

  47. Edge Origin Unique T(s) First Observation: This edge comes only from (s,t) T(t) Complexity

  48. Triangle Consistency Second Observation: A triangle only come from a triangle Complexity

  49. Clique Preservation Corollary: {c1,…,ck} is a clique in the CCN graph iff {T(c1),…,T(ck)} is a clique in the CLIQUE graph. Complexity

  50. Summary • We’ve seen how to show hardness of approximation results in general, • and even proven several such using the PCP theorem: • CLIQUE • CHROMATIC NUMBER Complexity

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