Some important problems and their complexity status

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Some important problems and their complexity
status

• Mohammad KAYKOBAD
• Visiting Professor
• Computer Engineering Department
• Kyung Hee University
• CSE Department, North South University

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Resolved problems

• There were 12 important problems whose complexity
  status was unresolved. Of them 8 cases have been
  resolved
• Most important of them are as follows
• Linear Programming solved in 1979 by LG Khachiyan
  a Russian mathematician by discovering Ellipsoid
  Algorithm later on Karmarkar also devised a
  polynomial time algorithm. This discovery was in
  full compliance of prediction of theory of
  computational complexity

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Solved Problems(contd)

• Subgraph Homeomorphism problem solved by
  Robertson and Seymour(1986)
• Spanning Tree Parity Problem solved by Lovasz in
  1980 to be polynomial
• Total Unimodularity solved by Seymour in 1980 to
  be polynomial
• Graph Genus problem has been shown to be
  NP-Complete by Thomassen

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Solved Problems(contd)

• Chordal Graph Completion proved to be NP-Complete
  by Yannakakis in 1981
• Chromatic Index proved to be NP-Complete by
  Holyer in 1981.
• Partial order Dimension proved to be NP-Complete
  by Yannakakis in 1982.
• Crossing number proved to be NP-Complete by Garey
  and Johnson in 1981.

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Solved Problems(contd)

• Graph Thickness proved to be NP-Complete by
  Mansfield 1983.
• Linear Complementarity proved to be NP-Complete
  by Chung 1979.
• Primality Testing has been shown to be polynomial
  by M Aggrawal in 2002.

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Important Unresolved Problems

• Graph Isomorphism- The Complexity of Some
  Isomorphism Problems 
• Many of the mathematical problems are used in a
  wide variety of applications, e.g., storing
  images in JPEG format, storing songs in MP3
  format, error-free and secure transmission of
  data etc.In this writeup, three problems from
  mathematics are discussed Graph Isomorphism,
  F-Algebra Isomorphism, and Cubic Form Equivalence.

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• The Graph Isomorphism problem is as follows. We
  are given two undirected simple graphs over n
  vertices, say G and H. The vertices in both the
  graphs are numbered from 1 to n. The problem is
  to decide if the two graphs are isomorphic. In
  other words, to check if renumbering the vertices
  of G make it equal to H. This problem has had a
  long history.

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Unresolved problems (contd)

• It is useful in several places, for example in
  classifying the structure of large molecules (the
  atoms are "vertices" and bonds between them are
  "edges"). It is known that this problem is
  unlikely to be NP-hard and so the problem might
  be easy to solve. At the same time, no efficient
  algorithm is known for solving the problem.

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Unresolved problems (contd)

• The F-Algebra Isomorphism problem is as follows.
  Given two algebras over field F (algebras are
  commutative rings with identity), test if they
  are isomorphic. This is one of the basic problems
  in mathematics. When the two algebras are fields,
  it is easy to test the isomorphism. However, in
  general, the problem is not easy to solve.

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Unresolved problems (contd)

• When F is an algebraically closed field (for
  example, the field of complex numbers), then --
  using the famous Hilbert's Nullstellensatz -- one
  can show that the problem can be solved within
  polynomial space. When F is the field of reals,
  then -- using another famous result by Tartski on
  decidability of first-order theory of reals --
  one can show that the problem can be solved in
  doubly exponential time.

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Unresolved problems (contd)

• When F is the field of rational numbers, the
  status of the problem is open -- it is not even
  known to be decidable! When F is a finite field,
  its complexity is very similar to that of Graph
• Isomorphism.The Cubic Form Equivalence problem is
  as follows. Given two homogeneous, degree three
  polynomials over field F, test if the first one
  becomes equal to the second one under a linear
  transformation. This problem is also very well
  studied in mathematics. In case of degree two
  polynomials (instead of degree three), the
  problem has a very elegant and efficient solution
  for any F. However, degree three case does not
  appear so easy.

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Unresolved problems (contd)

• The complexity of the problem also depends on the
  field F and behaves in the same way as the
  complexity of the F-Algebra Isomorphism problem.
  Interestingly, although these three problems do
  not seem related to each other, it can be shown
  that, in fact, they are! The Graph Isomorphism
  problem can be transformed to F-Algebra
  Isomorphism problem which, in turn, can be
  transformed to Cubic Form Equivalence problem.
  (Manindra Agrawal)

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Unresolved problems (contd)

• Still unresolved are problems like graceful
  labelling of trees- Given an arbitrary tree on n
  vertices is it possible to label them with
  distinct integers 0,1,,n-1 so that edges get
  labels 1,2,, n-1, where edge label is absolute
  difference of vertex labels.

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Unresolved problems (contd)

• Graph Reconstruction Problem asks to reconstruct
  the graph given n subgraphs induced by the
  vertices of the original graph removing everytime
  a different vertex one at a time. The problem has
  been shown to be polynomially solvable if the
  graph is a tree.

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