on the complexity of orthogonal compaction

1. on the complexity of orthogonal compaction maurizio patrignani univ. rome III

2. circuit schematics entity relationship diagrams industrial plants integrated circuits network topologies data flow diagrams orthogonal drawings

3. 6 1 5 2 3 6 1 5 2 3 6 4 5 2 1 3 4 topology-shape-metrics approach V={1,2,3,4,5,6} E={(1,4),(1,5),(1,6), (2,4),(2,5),(2,6), (3,4),(3,5),(3,6)} planarization orthogonalization 4 compaction

4. /2 /2  2 3/2  3/2  /2 /2 /2 /2  3/2  /2 /2 3/2  /2 /2 3/2  3/2  the compaction step without loss of generality we will consider only graphs without bends input: an orthogonal representation or shape = 2 1) 2) a(f) · - 2 a(f) · + 2 a(f) = number of vertices of face f output: an orthogonal grid drawing

5. minimizing total edge length minimizing area minimizing maximum edge length

6. state of the art orthogonal compaction wrt area was mentioned as open problem (G. Vijayan and A. Widgerson) linear time compaction heuristic based on rectangularization (R. Tamassia) optimal compaction wrt total edge length by means of ILP + branch & bound or branch & cut techniques (G. W. Klau and P. Mutzel) polynomial time compaction heuristic based on turn-regularization (S. Bridgeman, G. Di Battista, W. Didimo, G. Liotta, R. Tamassia, and L. Vismara) 1985 1987 1998 1998

7. x2 x4  x1 x2 x3 x4  x3  x1 x2 x3 formulating a decision problem problem: orthogonal area compaction instance: an orthogonal representation H and a value k question: can an orthogonal drawing of H be found such that its area is less or equal to k? problem: satisfiability (SAT) instance: a set of clauses, each containing literals from a set of boolean variables question: can truth values be assigned to the variables so that each clause contains at least one true literal? variable set ={x1 , x2 , x3 , x4}

8. not compacted as much as possible compacted as much as possible reduction SAT instance SAT solution compacted drawing local and global properties

9. r r r r l l l l r r r r r r r r l l l l r r r r n times n times n times r r r r l l l l r r r r sliding rectangles gadget 1 2 3 ... n

10. transferable path properties removing r r r l l l l r inserting r l l l l r r r

11. a global property made local an exponential number of orthogonal drawings with the minimum area a variant of the sliding rectangles gadget

12. ( parenthesis different “shapes”... ... all sharing the same orthogonal shape parenthesis )

13. x2 true x4 true x5 true x1 false x3 false NP-hardness proof clause 1 clause 2 clause 3 clause 4

14. ? ? ? ? xi is false xi is true clause gadget one is missing! xi does not occur in the clause xi occurs in theclause with a positive literal xi occurs in the clause with a negative literal

15. clause x1 x2 x1 x2 true true false x1 x2 true true false true false false clause gadget example variable set ={x1 , x2 , x3} but we have only five “A”-shaped structures!

16. x2 x4  x1 x2 x3 x4  x3  x1 x2 x3 clause 1 clause 2 clause 3 clause 4 x1 false x2 true x3 false x4 true an example clause 1 clause 2 clause 3 clause 4

17. NP-completeness property: the compaction problem with respect to area is NP-hard property: the compaction problem with respect to area is in NP theorem: the compaction problem with respect to area is NP-complete

18. compaction with respect to total edge length corollary: the compaction problem with respect to total edge length is NP-complete

19. compaction with respect to maximum edge length corollary: the compaction problem with respect to maximum edge length is NP-complete

20. approximability considerations 3 3 does not admit a polinomial-time approximation scheme (not in PTAS)

21. conclusions • we have shown that the compaction problem with respect to area, total edge length, or maximum edge length is NP-complete • we have shown that the three problems are not in PTAS • it is possible to modify the constructions so to have biconnected orthogonal representations open problems • does an orthogonal representation consisting in a single cycle retain the complexity of the three general problems? • how many classes (rectangular, turn-regular, ...) of orthogonal representations admit a polynomial solution?