Efficient Algorithms for Graph Bisection of Sparse Planar Graphs. Gerold Jäger University of Halle Germany

Transcription

1 Efficient Algorithms for Graph Bisection of Sparse Planar Graphs Gerold Jäger University of Halle Germany

2 Overview 1 Definition of MINBISECTION 2 Approximation Results 3 Previous Algorithms Notations Simple-Greedy-Algorithm Kernighan-Lin-Algorithm Randomized-Black-Holes-Algorithm

3 Overview 4 New Algorithms Notations Black-Holes-Algorithm Longest-Path-Algorithm 5 Experimental Results First Example Second Example Third Example Fourth Example

4 Definition of MINBISECTION Let G = (V, E) be an undirected and unweighted graph with V = n. A bisection is a partition (A, B) of V with A = n 2. The bisection width is defined as the minimum number of edges between A and B among all possible bisections (A, B). MINBISECTION is the NP-hard problem of finding a bisection with a minimum bisection width.

5 Approximation Results There is a polynomial-time approximation algorithm of MINBISECTION with a factor n/2. This algorithm does not approximate it with a better factor (Saran, Vazirani). The approximation factor can be improved to n log n (Feige, Krautgamer, Nissim). There is a polynomial time algorithm for MINBISECTION of grid graphs (Papadimitriou, Sideri). Another algorithm is able to compute a lower bound for the bisection width and equals it for a random class of graphs (Boppana).

6 Previous Algorithms Notations Let G = (V, E) be a graph and A, B V with A B = and A B = V. For a A denote with I(a) the inner costs, i.e. the number of edges (a, c) E with c A \ {a}. Analogously we define I(b) for b B. For a A denote with O(a) the outer costs, i.e. the number of edges (a, c) E with c B. Analogously we define O(b) for b B. For a A, b B let { 1, if (a, b) E ω(a, b) :=. 0, otherwise For a A, b B let S(a, b) := O(a) I(a) + O(b) I(b) 2ω(a, b).

7 Previous Algorithms Simple-Greedy-Algorithm Input Graph G = (V, E) with V = n. 1 Choose a bisection (A, B), uniformly at random among all possible bisections. 2 Choose a A, b B with S(a, b) > 0. 3 Swap the vertices a and b. 4 Repeat the steps 2 and 3, until there are no a A, b B with S(a, b) > 0. Output Bisection (A, B) with small bisection width. The Simple-Greedy-Algorithm needs O(n 4 ) steps.

8 Previous Algorithms Kernighan-Lin-Algorithm Input Graph G = (V, E) with V = n. 1 Choose a bisection (A, B), uniformly at random among all possible bisections. 2 Copy (A, B) to (A, B ). 3 Choose a A, b B with max. S(a, b) (maybe 0). 4 Swap the vertices a and b. 5 A := A \ {a}, B := B \ {b}. 6 Repeat the steps 3 to 5, until A :=, B :=. 7 Choose the bisection (A, B) as the bisection having min. bisection width among all bisections obtained after step 4. 8 Repeat the steps 2 to 7, until there is no improvement. Output Bisection (A, B) with small bisection width. The Kernighan-Lin-Algorithm needs O(n 5 ) steps.

9 Previous Algorithms Randomized-Black-Holes-Algorithm Input Graph G = (V, E) with V = n. 1 A :=, B :=. 2 Choose uniformly at random an edge between V \ {A B} and A and add the corresponding vertex in V \ {A B} to A. If there is no such edge, choose uniformly at random a vertex among all vertices in V \ {A B} and add it to A. 3 Do step 2 for B. 4 Repeat the steps 2 and 3, until (A, B) is a bisection. Output Bisection (A, B) with small bisection width. The Randomized-Black-Holes-Algorithm needs O(n 3 ) steps.

10 New Algorithms Notations Let G = (V, E) be a graph and A, B V with A B = and A B V. For c V \ {A B} let d 1 (c) be the number of edges (c, a) E with a A. For c V \ {A B} let d 2 (c) be the number of edges (c, b) E with b B.

11 New Algorithms Black-Holes-Algorithm Input Graph G = (V, E) with V = n. 1 A :=, B :=. 2 Choose a vertex c from V \ {A B} with maximum d 1 (c) d 2 (c) and add it to A. If there are two c with the same d 1 (c) d 2 (c), choose the c with the smaller d 1 (c). 3 Choose a vertex c from V \ {A B} with maximum d 2 (c) d 1 (c) and add it to B. If there are two c with the same d 2 (c) d 1 (c), choose the c with the smaller d 1 (c). 4 Repeat the steps 2 and 3, until (A, B) is a bisection. Output Bisection (A, B) with small bisection width. The Black-Holes-Algorithm needs O(n 3 ) steps.

12 New Algorithms Longest-Path-Algorithm Input Graph G = (V, E) with V = n. 1 Choose uniformly at random a vertex z. 2 Z := {z}. 3 List all neighbors of vertices from Z and add it to Z. 4 Repeat step 3, until Z has no neighbors. 5 Choose one of the vertices, added in the last run of step 3, and denote it with x. 6 Repeat steps 2 to 5 with x instead of z. The resulted vertex is denoted with y. 7 X := {x}, Y := {y}. 8 List all neighbors of vertices from X and add each neighbor to X, except for X n 2.

13 New Algorithms Longest-Path-Algorithm 9 Repeat step 8, if X < n 2 and if X has neighbors. 10 List all neighbors of vertices from Y and add each neighbor to Y, except for Y n Repeat step 10, if Y < n 2 and if Y has neighbors. 12 Add the remaining vertices in an arbitrary way, so that X = n 2 and Y = n 2. Output Bisection (A, B) with small bisection width. The Longest-Path-Algorithm needs O(n 3 ) steps.

14 Experimental Results First Example SG-Alg.: 98 (0 s.) KL-Alg.: 88 (153 s.) RBH-Alg.: 69 (37 s.) BH-Alg.: 54 (0 s.) LP-Alg.: 35 (0 s.) LP+SG-Alg. / OPT: 33 (0 s.)

15 Experimental Results Second Example SG-Alg.: 33 (0 s.) KL-Alg.: 31 (21 s.) RBH-Alg.: 39 (3 s.) BH-Alg. / OPT: 13 (0 s.) LP-Alg.: 17 (0 s.) LP+SG-Alg. / OPT: 13 (0 s.)

16 Experimental Results Third Example SG-Alg.: 134 (1 s.) KL-Alg.: 103 (958 s.) RBH-Alg.: 73 (118 s.) BH-Alg.: 41 (0 s.) LP-Alg.: 64 (0 s.) LP+SG-Alg. / OPT: 25 (1 s.)

17 Experimental Results Fourth Example SG-Alg.: 362 (3 s.) KL-Alg.: 291 (5154 s.) RBH-Alg.: 22 (1030 s.) BH-Alg.: 151 (3 s.) LP-Alg.: 12 (1 s.) LP+SG-Alg. / OPT: 11 (4 s.)