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

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

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 compaction

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

minimizing total edge length minimizing area minimizing maximum edge length

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)

formulating a decision problem x 2  x 4 x 1  x 2  x 3  x 1  x 2  x 3  x 4 x3x3  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? 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? variable set ={x 1, x 2, x 3, x 4 }

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

sliding rectangles gadget n times r r r r l l l l r r r r n times r r r r l l l l r r r r n times r r r r l l l l r r r r 123n...

transferable path properties r r r l l l l r r l l l l r r r removing inserting

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

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

NP-hardness proof x 3 false x 2 true x 1 false x 4 true x 5 true clause 1 clause 2 clause 3 clause 4

clause gadget x i is falsex i is true x i does not occur in the clause x i occurs in theclause with a positive literal x i occurs in the clause with a negative literal ? ? ? ? one is missing!

clause gadget example variable set ={x 1, x 2, x 3 } x 1  x 2 clause true false true false true false x1 x2x1 x2 x 1  x 2 but we have only five “A”-shaped structures!

an example x 2  x 4 x 1  x 2  x 3  x 1  x 2  x 3  x 4 x3x3  clause 2clause 3clause 4clause 1 x 1 false x 3 false x 2 true x 4 true clause 1 clause 2 clause 3 clause 4

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

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

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

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

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 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? open problems