On the Approximability of the Maximum Interval Constrained Coloring Problem

Transcription

1 S. Canzar 1 K. Elbassioni 2 A. Elmasry 3 R. Raman 4 On the Approximability of the Maximum Interval Constrained Coloring Problem 1 Centrum Wiskunde & Informatica, Amsterdam, The Netherlands 2 Max-Planck-Institut für Informatik, Saarbrücken, Germany 3 Datalogisk Institut, University of Copenhagen, Denmark 4 DIMAP and Department of Computer Science, University of Warwick, UK ISAAC'10

2 Motivation S. Canzar On the Approximability of the ICC Problem 2

3 Motivation can be determined by X-ray crystal diraction, NMR,... but limitations! (concentration, conformational changes... ) S. Canzar On the Approximability of the ICC Problem 2

4 Motivation can be determined by X-ray crystal diraction, NMR,... but limitations! (concentration, conformational changes... ) Exchange of labile hydrogens for deuteriums (HDX) S. Canzar On the Approximability of the ICC Problem 2

5 Motivation can be determined by X-ray crystal diraction, NMR,... but limitations! (concentration, conformational changes... ) Exchange of labile hydrogens for deuteriums (HDX) exchange rates as measure of solvent accessibility structure! S. Canzar On the Approximability of the ICC Problem 2

6 Motivation can be determined by X-ray crystal diraction, NMR,... but limitations! (concentration, conformational changes... ) Exchange of labile hydrogens for deuteriums (HDX) exchange rates as measure of solvent accessibility structure! monitored by mass spectrometry (MS) (NMR limited) S. Canzar On the Approximability of the ICC Problem 2

7 Motivation can be determined by X-ray crystal diraction, NMR,... but limitations! (concentration, conformational changes... ) Exchange of labile hydrogens for deuteriums (HDX) exchange rates as measure of solvent accessibility structure! monitored by mass spectrometry (MS) (NMR limited) output: aggregate exchange data for peptic fragments S. Canzar On the Approximability of the ICC Problem 2

8 Motivation can be determined by X-ray crystal diraction, NMR,... but limitations! (concentration, conformational changes... ) Exchange of labile hydrogens for deuteriums (HDX) exchange rates as measure of solvent accessibility structure! monitored by mass spectrometry (MS) (NMR limited) output: aggregate exchange data for peptic fragments Increase resolution from fragments to single residues! S. Canzar On the Approximability of the ICC Problem 2

9 Problem Illustration G R Y R G Y R R Y S. Canzar On the Approximability of the ICC Problem 3

10 Problem Illustration G R Y R G Y R R Y Experiments [1, 5] (2, 2, 1) [3, 7] (2, 1, 2) [6, 8] (2, 0, 1) [5, 9] (2, 1, 2) S. Canzar On the Approximability of the ICC Problem 3

11 Problem Illustration G R Y R G Y R R Y Experiments [1, 5] (2, 2, 1) [3, 7] (2, 1, 2) [6, 8] (2, 0, 1) [5, 9] (2, 1, 2) feasible assignment S. Canzar On the Approximability of the ICC Problem 3

12 Mathematical Abstraction Protein of n residues Set V = [n] S. Canzar On the Approximability of the ICC Problem 4

13 Mathematical Abstraction Protein of n residues Set V = [n] Peptic fragments Set I of intervals dened on V S. Canzar On the Approximability of the ICC Problem 4

14 Mathematical Abstraction Protein of n residues Set V = [n] Peptic fragments Set I of intervals dened on V H/D exchange rates Set of color classes [k] S. Canzar On the Approximability of the ICC Problem 4

15 Mathematical Abstraction Protein of n residues Set V = [n] Peptic fragments Set I of intervals dened on V H/D exchange rates Set of color classes [k] Requirement function r : I [k] N S. Canzar On the Approximability of the ICC Problem 4

16 Mathematical Abstraction Protein of n residues Set V = [n] Peptic fragments Set I of intervals dened on V H/D exchange rates Set of color classes [k] Requirement function r : I [k] N Feasibility I is feasible or colorable if there exists a coloring χ : V [k] such that for every I I we have {j I χ(j) = c} = r(i, c). S. Canzar On the Approximability of the ICC Problem 4

17 Mathematical Abstraction Protein of n residues Set V = [n] Peptic fragments Set I of intervals dened on V H/D exchange rates Set of color classes [k] Requirement function r : I [k] N Feasibility I is feasible or colorable if there exists a coloring χ : V [k] such that for every I I we have {j I χ(j) = c} = r(i, c). MaxFeasibleColoring (MFC) Given non-negative weights w : I R +, nd a maximum weight colorable subset I I. S. Canzar On the Approximability of the ICC Problem 4

18 Previous Results & Contribution [Althaus et al. SAC'08]: ILP formulation, polynomial for k = 2 [Althaus et al. SWAT'08]: N P-hard in general ±1 violation by LP rounding (1 + ɛ) violation in quasi-polynomial time [Byrka et al. LATIN'10]: N P-hard/APX -hard for k = 3 [Komusiewicz et al. CPM'09] Fixed-parameter tractable w.r.t. max. interval length, max. no. of intervals containing a vertex. S. Canzar On the Approximability of the ICC Problem 5

19 Previous Results & Contribution [Althaus et al. SAC'08]: ILP formulation, polynomial for k = 2 [Althaus et al. SWAT'08]: N P-hard in general ±1 violation by LP rounding (1 + ɛ) violation in quasi-polynomial time [Byrka et al. LATIN'10]: N P-hard/APX -hard for k = 3 [Komusiewicz et al. CPM'09] Fixed-parameter tractable w.r.t. max. interval length, max. no. of intervals containing a vertex. Our Results O( OPT )-approximation algorithm (for constant k) Mfc is APX -hard for k = 2 (reduction from Max2SAT) S. Canzar On the Approximability of the ICC Problem 5

20 Approximation Algorithm Idea Trim solution space and show it still contains a good solution. S. Canzar On the Approximability of the ICC Problem 6

21 Approximation Algorithm Idea Trim solution space and show it still contains a good solution. (i) Solve easier problem variants optimally: a) Tower b) Staircase S. Canzar On the Approximability of the ICC Problem 6

22 Approximation Algorithm Idea Trim solution space and show it still contains a good solution. (i) Solve easier problem variants optimally: a) Tower b) Staircase (ii) Pick best (combined) solution S. Canzar On the Approximability of the ICC Problem 6

23 Optimal Tower Consider partially ordered set (P, ): I 2 I 1 (2, 3) a 1 b 1 (3, 5) a 2 b 2 I 1 I 2 ( a 1, 2, 3, b 1 ) ( a 2, 3, 5, b 2 ) S. Canzar On the Approximability of the ICC Problem 7

24 Optimal Tower Consider partially ordered set (P, ): I 2 I 1 (2, 3) a 1 b 1 (3, 5) a 2 b 2 I 1 I 2 ( a 1, 2, 3, b 1 ) ( a 2, 3, 5, b 2 ) S. Canzar On the Approximability of the ICC Problem 7

25 Optimal Tower Consider partially ordered set (P, ): I 2 I 1 (2, 3) a 1 b 1 (3, 5) a 2 b 2 I 1 I 2 ( a 1, 2, 3, b 1 ) ( a 2, 3, 5, b 2 ) S. Canzar On the Approximability of the ICC Problem 7

26 Optimal Tower Consider partially ordered set (P, ): I 2 I 1 (2, 3) a 1 b 1 (3, 5) a 2 b 2 I 1 I 2 ( a 1, 2, 3, b 1 ) ( a 2, 3, 5, b 2 ) Observation: I I is a feasible tower i it is a chain in P S. Canzar On the Approximability of the ICC Problem 7

27 Optimal Tower Consider partially ordered set (P, ): I 2 I 1 (2, 3) a 1 b 1 (3, 5) a 2 b 2 I 1 I 2 ( a 1, 2, 3, b 1 ) ( a 2, 3, 5, b 2 ) Observation: I I is a feasible tower i it is a chain in P Find a maximum-weight chain in P (in polynomial time)! S. Canzar On the Approximability of the ICC Problem 7

28 Optimal Staircase I 1 I t } I l χ S. Canzar On the Approximability of the ICC Problem 8

29 Optimal Staircase I t I 1 I t } I l χ S. Canzar On the Approximability of the ICC Problem 8

30 Optimal Staircase I 1 I t χ { I t } I l χ S. Canzar On the Approximability of the ICC Problem 8

31 Optimal Staircase I 1 I t χ { I t } I l χ } {v I α χ(v) = c} = {v I α χ (v) = c}, for all colors c. I α S. Canzar On the Approximability of the ICC Problem 8

32 Optimal Staircase χ = I 1 χ I t χ { I t } I l χ } {v I α χ(v) = c} = {v I α χ (v) = c}, for all colors c. I α S. Canzar On the Approximability of the ICC Problem 8

33 Optimal Staircase χ = I 1 χ χ I t χ { I t } I l χ } } I β I α {v I α χ(v) = c} = {v I α χ (v) = c}, for all colors c. S. Canzar On the Approximability of the ICC Problem 8

34 Optimal Staircase χ = I 1 χ χ χ χ I t χ { I t } I l χ } } I β I α {v I α χ(v) = c} = {v I α χ (v) = c}, for all colors c. S. Canzar On the Approximability of the ICC Problem 8

35 Optimal Staircase χ = I 1 χ χ χ χ I t χ { I t } I l χ } } I β I α {v I α χ(v) = c} = {v I α χ (v) = c}, for all colors c. D[t, r(i α )] := weight of optimal solution from I 1 to I t that can be satised by coloring satisfying r on I α S. Canzar On the Approximability of the ICC Problem 8

36 Optimal Staircase χ = I 1 χ χ χ χ I t χ { I t } I l χ } } I β I α {v I α χ(v) = c} = {v I α χ (v) = c}, for all colors c. D[t, r(i α )] := weight of optimal solution from I 1 to I t that can be satised by coloring satisfying r on I α DP: guess predecessor I t and coloring on I α S. Canzar On the Approximability of the ICC Problem 8

37 Putting it all together Dene Poset P = (OPT, ). Dilworth Theorem There is either a chain or an anti-chain in P of weight at least w(opt ) 2 OPT. S. Canzar On the Approximability of the ICC Problem 9

38 Putting it all together Dene Poset P = (OPT, ). Dilworth Theorem There is either a chain or an anti-chain in P of weight at least w(opt ) 2 OPT. chain (= tower): S. Canzar On the Approximability of the ICC Problem 9

39 Putting it all together Dene Poset P = (OPT, ). Dilworth Theorem There is either a chain or an anti-chain in P of weight at least w(opt ) 2 OPT. chain (= tower): anti-chain: S. Canzar On the Approximability of the ICC Problem 9

40 Putting it all together Dene Poset P = (OPT, ). Dilworth Theorem There is either a chain or an anti-chain in P of weight at least w(opt ) 2 OPT. chain (= tower): anti-chain: can be partitioned into 2 sets of independent staircases S. Canzar On the Approximability of the ICC Problem 9

41 Putting it all together Dene Poset P = (OPT, ). Dilworth Theorem There is either a chain or an anti-chain in P of weight at least w(opt ) 2 OPT. chain (= tower): anti-chain: can be partitioned into 2 sets of independent staircases feasible set of weight at least w(opt ) 4 OPT S. Canzar On the Approximability of the ICC Problem 9

42 Independent Set of Staircases u v S. Canzar On the Approximability of the ICC Problem 10

43 Independent Set of Staircases w u,v S. Canzar On the Approximability of the ICC Problem 10

44 Independent Set of Staircases w u,v S. Canzar On the Approximability of the ICC Problem 10

45 Independent Set of Staircases w u,v S. Canzar On the Approximability of the ICC Problem 10

46 Independent Set of Staircases w u,v S. Canzar On the Approximability of the ICC Problem 10

47 Independent Set of Staircases w u,v S. Canzar On the Approximability of the ICC Problem 10

48 Algorithm Algorithm 1 MfcApprox(V, I, r, w) 1: W 1 = max Tower I w(i ) 2: for every u, v V s.t. u < v do 3: w u,v = max Staircase(u,v) I w(i ) 4: end for 5: Let W 2 = max Indep.set I {[u,v]:u,v V, u<v} w (I ) 6: return max{w 1, W 2 } S. Canzar On the Approximability of the ICC Problem 11

49 Open Questions Improve O( OPT )-approximation or hardness? Approximation if k part of the input? S. Canzar On the Approximability of the ICC Problem 12