Recognizing Strings in NP Marcus Schaefer, Eric Sedgwick, Daniel tefankovic

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Recognizing Strings in NPMarcus Schaefer, Eric
Sedgwick, Daniel tefankovic

• Presentation by
• Robert Salazar

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Definitions

• Given a collection of curves Ci where i I,
  the intersection graph is defined as (I, (i, j)
  Ci and Cj intersect)
• The size of a collection of curves is the number
  of intersection points
• String graph A graph isomorphic to the
  intersection graph of a collection of curves
• String graph problem asks how string graphs can
  be recognized. This problem was previously shown
  to be NP-hard.

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Definitions

• Graph Drawing problem
• Given graph G (V, E) and a set
• A drawing D is a weak realization of (G, R) if
  only pairs of edges which are in R are allowed to
  intersect. These edges do not necessarily
  intersect.
• (G, R) is weakly realizible

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String graph problem reduction

• Given G (V, E), let G
• Let
• G is a string graph if and only if (G, R) is
  weakly realizible

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(No Transcript)
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Lemmas

• Let M be a compact, orientable surface with a
  boundary. A properly embedded arc ? has both
  endpoints on the boundary dM and all internal
  points on the interior of M.

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Lemmas
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Theorem

• Let G (V, E) be a graph with m edges,
  such that (G, R) is weakly realizable, and let D
  be a weak realization of (G, R) with the minimal
  number of intersections. Then for any edge e
  G there are less than 2m intersections for the
  curve realizing e in D.
• Let M be a compact, orientable surface with a
  boundary. A properly embedded arc ? has both
  endpoints on the boundary dM and all internal
  points on the interior of M.

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Proof

• Let G (V, E) be a graph. Let M be the surface
  obtained from the plane by drilling V holes.
  Each hole is labeled by a vertex of G. Let .
  A set S of properly embedded arcs on M is called
  a weak realization with holes of (G, R) if for
  each e u, v E there is a properly
  embedded arc in S connecting hole u to hole v,
  and if then the properly embedded arcs e, f
  are isotopically disjoint.

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Proof (II)

• Given a weak realization D, drill small holes in
  place of the vertices to obtain a weak
  realization with holes.
• By Lemma 3.2, there is a weak realization with
  holes in which for the properly embedded arcs e,
  f are disjoint.
• Contracting the holes yields a weak realization
  of (G, R)

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Proposition / Lemma
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Proof

• Construct a triangulation T with 3n vertices,
  using 3 vertices for each boundary component
  (i.e. hole)
• T has 9n 6 edged by Eulers formula

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Proof (II)

• Consider Weak realization problem
• Graph H such that
• and
• There are edges to all vertices of T which lie
  on hole v.
• Pairs P of edges that may intersect are
• All pairs in R
• For every edge select an edge
  ev may intersect with edges in EG going to v
• Any edge in T which is not on the boundary dM can
  intersect any edge in EG

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Proof (III)

• (G, R) is weakly realizible if and only if (H, P)
  is weakly realizible.
• Considering the realization of H with the fewest
  intersections By theorem 4.1, there is a
  realization such that there are at most
  intersections.

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Theorem

• The weak realizability problem is in NP.

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Proof

• Assume that (G, R) is weakly realizable.
• By proposition 4.2, it has a weak realization
  with holes.
• By lemma 4.3, there is a weak realization with
  holes in whiche each edge of triangulation T is
  intersected at most 212nm times.
• For any arc ? and edge , then the binary
  encoding of the number of intersections between ?
  and e is polynomial in n.

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Proof (II)

• To verify weak realizibiltity with holes, guess
  for each edge of G a labeling of the edges of
  T.
• By Lemma 3.5, for any it is possible to
  check in polynomial time that e and f are
  isotopically disjoint for the guessed set of
  lablings.
• By Lemma 3.2, this will guarantee a weak
  realization of (G, R)

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Conclusions

• The string graph problem is NP-Complete
• The weak realizability problem is
  NP-Complete
• The pairwise crossing number problem, the
  existential theory of diagrams, and the
  existential fragment of topological inference are
  NP-Complete.