Flexible Optimization Problems

1. Flexible Optimization Problems A. Akavia, S. Safra

2. Motivation What do we do with all the NP-hard optimization problems ???

3. Relaxations: 2 Parameters • Optimization function approximation • Input flexibility Example – graph coloring problem: • Optimization function – • find an approximation of the min coloring. • Input flexibility – • find a k-coloring with few monochromatic edges.

4. Talk Plan • Approximation • Input flexibility • Flexible optimization problems • Examples • Definitions • Hardness results

5. Relaxation 1: Approximation • An approximation algorithm is an algorithm that returns an answer C which “g-approximates” the optimal solution C*. • C  g  C*(minimization) • 1/g  C*  C(maximization)

6. example: 2-colorability Relaxation 2: Input FlexibilityExample: Graph Editing problems complexity and approximation results are w/r to the number of modifications Input: • a graph G, and • a desired property Goal: find a small set of edge-modifications (addition/deletion/both) that transforms Ginto G’with the desired property

7. Our Work: Flexible-Approximation-Problems • Combining both relaxations – • approximating the optimization function, • while allowing input flexibility. • Example: Given a graph G, find a coloring with fewcolors, and perhaps few monochromatic edges.

8. Natural Flexible-Approximation-Problems: • Min Non-Deterministic Automaton • Min Synthesis Graph

9. Non-Deterministic Finite Automaton (NFA) • Many applications: • program verification • speech recognition • natural language processing • … Approximating the minimum NFA, which accepts a given language L is hard.

10. Flexible Min NFA Sometimes it suffices to find: • a smallNFA,(Approximation) • accepting a language “similar” to the input one.(Input Flexibility)

11. Example: Automata-Theoretic Approach to Program Verification • ProgramP is correct with respect to a specificationsTif: L(P)  L(T) • In concurrent programming: processesP1,..., Pn are correct w/r to specificationTif: L(P1x…x Pn)  L(T) P . . . P1 . . . P2 . . . P3 . . . in the worst case, |P1x…xPn| is exponential in n

12. Example: Automata-Theoretic Approach to Program Verification • Coping with this state-explosion [K94]: • finding a small automaton P’ (Optimization) • such that L(P1x…x Pn)  L(P’)(Input Flexibility) • and then checking whether L(P')  L(T) L(P1x…x Pn)  L(P’)  L(P1x…x Pn)  L(T) L(P’)  L(T)

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14. a b c a b c Node - grow step Label - appended unit a b c a b c b a b c a a b b c c Produces each string s s.t.s = labels concatenation along a path a b b a b a b a b c a Synthesis Graph a b b b b b b c c a a a b b c c c c Split Synthesis [F91,L91,CS99]

15. Flexible Min Synthesis Graph Output: A smallsynthesis-graphproducing S’, which is similar to S Input: A set of strings S a b c a b c a b c a b • Input flexibitily– • producing S’ and not S • Approximation – • finding a small synthesis graph • not the minimum

16. Definitions, Theorems and Proofs

17. Assume a distance function(x,x’)to be the smallest number of basic modifications (say, bit-changes) necessary in order to transform x to x’. the ball of radius d around a given input x is ball(x,d) = {x’ | (x,x’)d}

18. d (d,g)-flexible approximation problem • Assume: • a distance function , and • an optimizationfunction f. • In a (d,g)-flexible-approximation problem, given an input x, a solution y’ is returned s.t. • y’ is feasible for some x’ball(x,d), and • f(x’,y’) is g-approximateto the optimum (for x). f(x’,y’) ≤ gf(x,y*) x’ x

19. Biclique Edge Cover - Definition Input: Bipartite graph G Goal: Cover all edges by bicliques(i.e. complete bipartite subgraphs)

20. (d,g)-Biclique Edge Cover • Assume: • (G,G’) = symmetric edge difference • f(G,y) = no. of bicliques in the cover y. G G’ f(G’,y)=2 f(G,y)=3 (G,G’)=7

21. (d,g)-Biclique Edge Cover • In a (d,g)-Biclique Edge Cover ((d,g)-BEC) given an input G, a solution y’ is returned s.t. • y’ is the no. of bicliques in a cover ofG’ball(x,d), and • f(G’,y’) is g-approximateto the min-BECfor G. G G’

22. Hardness of (d,g)-Biclique Edge Cover Thm: >0(d,g)-Biclique-Edge-Cover problem is hard for any g=O(|V|1/5-) and d=O(g), unless NP=ZPP. Proof: later.

23. (d,g)-Non-Deterministic Finite Automaton (NFA) • Assume: • (L,L’) = symmetric difference between L and L’ • f(L,A) = no. of states in an NFA A accepting L. • In a (d,g)-NFA, given a language L, a solution y’ is returned s.t. • y’ is the no. of states in an NFA accepting L’ball(L,d), and • f(L’,y’) is g-approximateto the optimum for L.

24. Hardness of (d,g)-NFA Thm: >0 (d,g)-NFA problem is hard for any g=O(|L|1/10-) and d=O(g), unless NP=ZPP

25. qF q0 {uv| (u,v)E} Bipartite graph G Strings set S NFA Reduction Outlines • Reduction from Biclique-Edge-Cover:

26. v5 v4 v3 v2 v1 Define L Define L = {vu| (v,u)E} u5 u4 u3 u2 u1 qF q0 A - an automaton acceptin L Proof Reduction from flexible approximation Biclique Edge Cover: k-biclique edge cover of G’in d-distance from G  NFA with k+2 states,accepting L’ in d-distance from L Reduction from Biclique Edge Cover: k-biclique edge cover of G  NFA with k+2 states,accepting L (d,g)-BECis hard for g=O(|V|1/5-) and d=O(g)  (d,g)-NFAis hard for g=O(|L|1/10-) and d=O(g) G=(V,U,E) a bipartite graph.

27. Reminder • A synthesis-graph Hproduces a string s if:s is a label concatenation along a path from first to last layers in H.

28. (d,g)-Synthesis Graph Given a set of strings S, output a small synthesis-graph producing S’, which is similar to S • Assume: • (S,S’) = symmetric difference between S and S’ • f(S,H) = no. of internal nodes in a synthesis graph H that produces the strings S. • In a (d,g)-Synthesis Graph, given an input S, a solution y’ is returned s.t. • y’ is the no. of internal nodes in a synthesis graph H producing S’ball(S,d), and • f(S’,y’) g-approximatethe optimum for S.

29. Hardness of (d,g)-Synthesis Graph Thm: >0 (d,g)-Synthesis Graph problem is hard for any g=O(|S|1/10-) and d=O(g), unless NP=ZPP Proof:

30. A A {uAv| (u,v)E} A Bipartite graph G Synthesis graph H Strings set S Reduction Outlines • Reduction from Biclique-Edge-Cover:

31. Define S = {vAu| (v,u)E} Define S Proof Reduction from flexible approximation Biclique Edge Cover: k-biclique edge coverof G’in d-distance from G synthesis graph H with k internal nodes producing S’ in d-distance from S Reduction from Biclique Edge Cover [CS99]: k-biclique edge coverof G synthesis graph H with k internal nodes producing S A A (d,g)-BECis hard for g=O(|V|1/5-) and d=O(g)  (d,g)-SGis hard for g=O(|S|1/10-) and d=O(g) A G=(V,U,E) a bipartite graph. H - a graph constructing S

32. Hardness Proof of(d,g)-Biclique-Edge-Cover((d,g)-BEC)

33. (d,g)-Biclique Edge Cover ((d,g)-BEC) • In a (d,g)-Biclique Edge Cover ((d,g)-BEC) given an input G, a solution y’ is returned s.t. • y’ is the no. of bicliques in the cover of G’ball(G,d), and • f(G’,y’) is g-approximateto the min-BECfor G. G f(G,y)=3

34. Hardness of (d,g)-BEC Thm: (d,g)-BEC is hard for any g=O(|V|1/5-) and d=O(g), unless NP=ZPP. Proof:

35. flexiblevs. non-flexiblesolutions Lemma: from any solution y’ to G’ball(G,d),we may construct a solution y to G,s.t. f(G,y) ≤ f(G’,y’) + d G G’

36. Calculating Approximation Factor Claim: Let G be a graph, if v’=f(G’,y’)’g-approximate(d,g)-BEC, where d=O(g),then y, f(G,y)v’+d, s.t. v=v’+dO(g)-approximateBEC Proof: By the lemma y, f(G,y)a’+d, and v  O(g)optG v = v’ + d  goptG + d  2goptG(since d = O(g)) Assume BEC is hard for g=O(|V|1/5-) then: (d,g)-BEC is hard for g=O(|V|1/5-) and d=O(g)

37. Hardness of Approximation of BEC Proof Outlines: • Construction [Simon90] • Graph coloring is hard to approximate by g=O(|V|1-) [FK98] • Simple calculation, improves Simon’s bound from a constant factor to g=O(|V|1/5-).

38. 1 1 1 1 2 2 2 5 5 5 3 3 3 3 4 4 4 2 5 4 Clique Cover Problem Input: A graph G Goal: Cover all vertices by cliques Clique Cover is equivalent to Graph Coloring

39. 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ 1 1’ 1 2 2’ 2 5 3 3’ 3 4 4 4’ 5 5’ 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ H Construction [Simon ‘90] GI G

40. 1 1’ 2 2’ 3 3’ 4 4’ 5 5’ G GI Propositions 1 2 5 3 4 • a clique-cover in G translates into abiclique-cover of the horizontaledges in GI, and vice versa • a clique-cover in G translates into t2biclique-covers of the horizontaledges in H. No. of GI’s copies in H.

41. H 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ 1 2 5 3 4 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ Propositions G=(V,E) • 2|E|t bicliques are necessary and sufficient in order to cover the diagonal edges of H.

42. 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ H H Propositions • horizontaledges in different copies of GIcannotbe members of the samebiclique. Ascending, hence cannot connect copies within the same row.

43. H 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ 1 2 5 3 4 1 1’ 1 1’ 1 1’ 2 2’ 2 2’ 2 2’ 3 3’ 3 3’ 3 3’ 4 4’ 4 4’ 4 4’ 5 5’ 5 5’ 5 5’ Lemma G=(V,E) • sclique-cover in G(t2s + 2|E|t)biclique-cover in H • rbiclique-cover in H(r – 2|E|t) / t2clique-cover in G

44. Calculating Approximation Factor • Given a solution with rbicliques for H, we may construct a solution with scliques for G, and • it is easy to verify that • if ris O(|VH|x)-approximate to ropt • then sisO(|V|5x)-approximate to sopt Clique-Cover is hard to approximate by O(|V|1-), >0, unless NP=ZPP (from [FK98])  BEC is hard to approx within g=O(|VH|1/5-), >0, , unless NP=ZPP.

45. Related Work • Property testing [GGR] • Sample size • Typically seeking an exact solution and not an approximated one • Bi-criteria optimization[MRSR] • optimization (vs. relaxation). • Might be easy when only one criterion is considered. • Bi-criteria: hard with one criterionhard in the bi-criterionversion. • Typically hostile objectives.

46. relevant case for (d,g)-SG hard easy d=0 d=O(|V|1/5-) d=O(|V|) d=O(|E|) d=|V|2-|E| Graph coloring is easy Discussion • Similar proof is valid for any problem, which is • hard to approximate, and • “v  vd + d” • The gap is still too large:

47. Future Work • Improving our results • Extending our results to other problems • Achieving positive results • Parameterized polynomial solution • Approximation algorithm

49. Property Testing[GGR] • Distinguish, by a small no. of probes, between instances x that • satisfy a given property accept • no set of |x| modifications causes x to satisfy the property reject back

50. Bi-criteria Optimization [MRSR] • Bi-criteria network design problem is a tuple (A,B,S) • A,B are two minimization objectives • S specify a membership requirement in a class of sub-graphs • The problem specifies a budget value on objective A • And seeks minimum over B (within the budget) • A (k,l)-approximation algorithm for an (A,B,S)-bi-criteria optimization problem is a poly.-time algorithm, that produces a solution that belongs to the sub-graph class S, in which • the objective A is at most k times the budget, and • the objective B is at most l times the minimum back