The Dominating Set Problem in Intersection Graphs

Transcription

1 The Dominating Set Problem in Intersection Graphs Mark de Berg Sándor Kisfaludi-Bak Gerhard Woeginger IPEC, 6 September / 17

2 Dominating Set in intersection graphs Problem (Dominating Set) Given a graph and an integer k, is there a vertex subset D of size k such that all vertices are in D or have a neighbor in D? 2 / 17

3 Dominating Set in intersection graphs Problem (Dominating Set) Given a graph and an integer k, is there a vertex subset D of size k such that all vertices are in D or have a neighbor in D? Intersection graph: vertices are shapes in R d, edges connect intersecting shape pairs. 2 / 17

4 Dominating Set in intersection graphs Problem (Dominating Set) Given a graph and an integer k, is there a vertex subset D of size k such that all vertices are in D or have a neighbor in D? Intersection graph: vertices are shapes in R d, edges connect intersecting shape pairs. 2 / 17

5 Parameterized complexity classes Two parameterized problems are equivalent if they are FPT-reducible to each other. W -hierarchy: complexity classes based on FPT reductions. W[3] W[2] W[1] FPT P Dominating Set Independent Set Steiner Tree on k terminals Vertex cover 3 / 17

6 What is known so far? What is new? Graph class Complexity Reference Interval P Chang 1998 (unit) 2-interval W[1]-complete Fellows et al Planar NP-complete, FPT Downey and Fellows 1995 Unit disk W[1]-hard Marx 2006 General W[2]-complete Downey and Fellows / 17

7 What is known so far? What is new? Graph class Complexity Reference Interval P Chang 1998 (unit) 2-interval W[1]-complete Fellows et al Planar NP-complete, FPT Downey and Fellows 1995 Unit disk W[1]-hard Marx 2006 General W[2]-complete Downey and Fellows dim. patterns it depends... this talk 4 / 17

8 What is known so far? What is new? Graph class Complexity Reference Interval P Chang 1998 (unit) 2-interval W[1]-complete Fellows et al Planar NP-complete, FPT Downey and Fellows 1995 Unit disk W[1]-hard Marx 2006 General W[2]-complete Downey and Fellows dim. patterns it depends... this talk translates of a polygon W[1]-hard see paper 4 / 17

9 What is known so far? What is new? Graph class Complexity Reference Interval P Chang 1998 (unit) 2-interval W[1]-complete Fellows et al Planar NP-complete, FPT Downey and Fellows 1995 Unit disk W[1]-hard Marx 2006 General W[2]-complete Downey and Fellows dim. patterns it depends... this talk translates of a polygon W[1]-hard see paper const. compl. shapes contained in W[1] this talk 4 / 17

10 One-dimensional patterns 5 / 17

11 Dominating Set in 1-dim. intersection graph classes Take a finite point/interval pattern Q on the line Q 6 / 17

12 Dominating Set in 1-dim. intersection graph classes Take a finite point/interval pattern Q on the line Take some translates of Q on the line. Q 6 / 17

13 Dominating Set in 1-dim. intersection graph classes Take a finite point/interval pattern Q on the line Take some translates of Q on the line. Q This defines an intersection graph (where vertices = translates of Q). 6 / 17

14 Dominating Set in 1-dim. intersection graph classes Take a finite point/interval pattern Q on the line Take some translates of Q on the line. Q This defines an intersection graph (where vertices = translates of Q). Problem (Q-intersection dominating set) Input: x 1, x 2,..., x n R, k Z, parameter: k Output: Is there a Dominating Set of size k in the intersection graph induced by x 1 + Q, x 2 + Q,..., x n + Q? 6 / 17

15 The classification Theorem Q-intersection dominating set has the following complexity: Polynomially solvable if there is an interval in Q. Polynomially solvable if Q is a rational point pattern. It is NP-complete and FPT in all remaining cases. 7 / 17

16 The classification Theorem Q-intersection dominating set has the following complexity: Polynomially solvable if there is an interval in Q. Polynomially solvable if Q is a rational point pattern. It is NP-complete and FPT in all remaining cases. The problem becomes W[2]-complete if Q is part of the input: Theorem Every graph can be obtained as an intersection graph of the translates of a suitable one-dimensional pattern. 7 / 17

17 Patterns that contain an interval Suppose the longest interval in Q has length 1. Left endpoints of a minimum dominating set D: D left endpoints x 2 + Q x 1 + Q 1 w : length of the pattern 8 / 17

18 Patterns that contain an interval Suppose the longest interval in Q has length 1. Left endpoints of a minimum dominating set D: D left endpoints x 2 + Q x 1 + Q 1 w : length of the pattern Lemma A minimum dominating set D has at most 3w left endpoints in any window of length w. Dynamic programming by shifting a window gives O(n 6w+4 ) running time. 8 / 17

19 Non-rational point patterns 1 5 Example: Q Consider the translates a(1 + 5) + b 5 + Q (a, b Z). Observe: (a, b) intersects (a, b ) (a a, b b ) { (1, 0), (0, 1), ( 1, 0), (0, 1), (1, 1), ( 1, 1), (0, 0) } 9 / 17

20 The translates a(1 + 5) + b 5 + Q (a, b Z) define a graph isomorphic to the triangular grid. b (-1,1) (0,1) (1,1) (-1,0) (0,0) (1,0) a (-1,-1) (0,-1) (1,-1) Any induced triangular grid graph is a Q-intersection graph.... and DS is NP-complete on induced triangular grid graphs. 10 / 17

21 Constant-complexity shapes and W[1] 11 / 17

22 Containment in W[1] special RAMs Theorem (Chen, Flum, Grohe 2003) A parameterized language is in W[1] iff it has nondeterministic RAM that runs in f (k)poly(n) time, uses f (k)poly(n) registers, storing integers f (k)poly(n), it has nondeterministic steps only among the last g(k) instructions. 12 / 17

23 Containment in W[1] special RAMs Theorem (Chen, Flum, Grohe 2003) A parameterized language is in W[1] iff it has nondeterministic RAM that runs in f (k)poly(n) time, uses f (k)poly(n) registers, storing integers f (k)poly(n), it has nondeterministic steps only among the last g(k) instructions. In practice: deterministic preprocessing in FPT time 12 / 17

24 Containment in W[1] special RAMs Theorem (Chen, Flum, Grohe 2003) A parameterized language is in W[1] iff it has nondeterministic RAM that runs in f (k)poly(n) time, uses f (k)poly(n) registers, storing integers f (k)poly(n), it has nondeterministic steps only among the last g(k) instructions. In practice: deterministic preprocessing in FPT time guess the solution (typically O(k) time) 12 / 17

25 Containment in W[1] special RAMs Theorem (Chen, Flum, Grohe 2003) A parameterized language is in W[1] iff it has nondeterministic RAM that runs in f (k)poly(n) time, uses f (k)poly(n) registers, storing integers f (k)poly(n), it has nondeterministic steps only among the last g(k) instructions. In practice: deterministic preprocessing in FPT time guess the solution (typically O(k) time) check the solution in g(k) time 12 / 17

26 Example: unit disks and the vertical decomposition Claim: Dominating set on unit disk graphs is in W[1]. S: input set of unit disks For D S define: D + : radius 2 disks with same centers A(D + ) : arrangement induced by D + D is a dominating set D + contains all disk centers from S. Take a vertical decomposition of the arrangement A(D + ). D is dominating Sum of # of centers in cells covered by D + is n. 13 / 17

27 Example: unit disks and the vertical decomposition, cont. associate cells with 4-tuples of disks O(n 4 ) possible cells 14 / 17

28 Example: unit disks and the vertical decomposition, cont. The NRAM program: Preprocessing: count the # of points in each possible cell, put it in a lookup table. (O(n 4 n) time) associate cells with 4-tuples of disks O(n 4 ) possible cells 14 / 17

29 Example: unit disks and the vertical decomposition, cont. The NRAM program: Preprocessing: count the # of points in each possible cell, put it in a lookup table. (O(n 4 n) time) Using nondeterminism, guess D S, the k solution disks. associate cells with 4-tuples of disks O(n 4 ) possible cells 14 / 17

30 Example: unit disks and the vertical decomposition, cont. associate cells with 4-tuples of disks O(n 4 ) possible cells The NRAM program: Preprocessing: count the # of points in each possible cell, put it in a lookup table. (O(n 4 n) time) Using nondeterminism, guess D S, the k solution disks. Compute vertical decomposition of A(D + ). 14 / 17

31 Example: unit disks and the vertical decomposition, cont. associate cells with 4-tuples of disks O(n 4 ) possible cells The NRAM program: Preprocessing: count the # of points in each possible cell, put it in a lookup table. (O(n 4 n) time) Using nondeterminism, guess D S, the k solution disks. Compute vertical decomposition of A(D + ). Accept if the sum of points covered by the cells inside D + is n. 14 / 17

32 Example: unit disks and the vertical decomposition, cont. associate cells with 4-tuples of disks O(n 4 ) possible cells The NRAM program: Preprocessing: count the # of points in each possible cell, put it in a lookup table. (O(n 4 n) time) Using nondeterminism, guess D S, the k solution disks. Compute vertical decomposition of A(D + ). Accept if the sum of points covered by the cells inside D + is n. Steps 2,3 and 4 run in poly(k) time. 14 / 17

33 Constant complexity shapes Semi-algebraic set: a subset of R d characterized by a boolean expression of polynomial inequalites over R. Semi-algebraic family: semi-algebraic sets characterized by a single expression that contains a constant number of parameters. 15 / 17

34 Constant complexity shapes Semi-algebraic set: a subset of R d characterized by a boolean expression of polynomial inequalites over R. Semi-algebraic family: semi-algebraic sets characterized by a single expression that contains a constant number of parameters. Examples: (hollow) spheres in R 3 : (a, b, c, r) { (x, y, z) R 3 (a x) 2 + (b y) 2 + (c z) 2 = r 2} diagrams of polynomials of degree 5 triangles in R 4 15 / 17

35 Constant complexity shapes Semi-algebraic set: a subset of R d characterized by a boolean expression of polynomial inequalites over R. Semi-algebraic family: semi-algebraic sets characterized by a single expression that contains a constant number of parameters. Examples: (hollow) spheres in R 3 : (a, b, c, r) { (x, y, z) R 3 (a x) 2 + (b y) 2 + (c z) 2 = r 2} diagrams of polynomials of degree 5 triangles in R 4 Theorem Let F be a semi-algebraic family, and let C be the class of intersection graphs gained from finite subsets of F. Then Dominating Set on C is contained in W[1]. 15 / 17

36 Conclusion In R 1 : Characterized the complexity of Dominating Set for patterns 16 / 17

37 Conclusion In R 1 : Characterized the complexity of Dominating Set for patterns If the pattern is part of the input: W[2]-complete 16 / 17

38 Conclusion In R 1 : Characterized the complexity of Dominating Set for patterns If the pattern is part of the input: W[2]-complete In R 2 and higher dimensions: W[1]-hard for translates of a simple polygon 16 / 17

39 Conclusion In R 1 : Characterized the complexity of Dominating Set for patterns If the pattern is part of the input: W[2]-complete In R 2 and higher dimensions: W[1]-hard for translates of a simple polygon In W[1] if there is a single parameterized first order formula 16 / 17

40 Conclusion In R 1 : Characterized the complexity of Dominating Set for patterns If the pattern is part of the input: W[2]-complete In R 2 and higher dimensions: W[1]-hard for translates of a simple polygon In W[1] if there is a single parameterized first order formula W[2]-complete for separately defined convex polygons in R 2 16 / 17

41 Q&A Q is a pattern with two unit intervals at distance l (part of input). Is Dominating Set in FPT with parameter k or k + l? l 1 1 Q Take n translates of a regular n-gon in R 2. Is Dominating Set in W[1] here? 17 / 17

42 Q&A Q is a pattern with two unit intervals at distance l (part of input). Is Dominating Set in FPT with parameter k or k + l? l 1 1 Q Take n translates of a regular n-gon in R 2. Is Dominating Set in W[1] here? Thank you. Any questions? 17 / 17