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46
Checking Computations in Polylogarithmic Time
, 1991
"... . Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every no ..."
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Cited by 275 (11 self)
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. Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every nondeterministic computational task S(x; y), defined as a polynomial time relation between the instance x, representing the input and output combined, and the witness y can be modified to a task S 0 such that: (i) the same instances remain accepted; (ii) each instance/witness pair becomes checkable in polylogarithmic Monte Carlo time; and (iii) a witness satisfying S 0 can be computed in polynomial time from a witness satisfying S. Here the instance and the description of S have to be provided in errorcorrecting code (since the checker will not notice slight changes). A modification of the MIP proof was required to achieve polynomial time in (iii); the earlier technique yields N O(log log N)...
Threecoloring trianglefree graphs on surfaces I. Extending a coloring to a disk with one triangle
, 2010
"... Let G be a plane graph with with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3coloring of C does not extend to a 3coloring of G if and only if C has length exactly six and there is a color x such that eit ..."
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Cited by 23 (8 self)
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Let G be a plane graph with with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3coloring of C does not extend to a 3coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series.
Automatic Visual to Tactile Translation, Part II: Evaluation of the TACTile Image Creation System
"... This is the second part of a twopart paper that develops a method for the automatic conversion of images from visual to tactile form. In Part I, a variety of topics were reviewed including issues in human factors, access technology for tactile graphics production, and image processing. In this part ..."
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Cited by 20 (2 self)
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This is the second part of a twopart paper that develops a method for the automatic conversion of images from visual to tactile form. In Part I, a variety of topics were reviewed including issues in human factors, access technology for tactile graphics production, and image processing. In this part, the material presented in the first part is used to motivate, develop and support the methods used in the development of a prototype visualtotactile translator called the TACTile Image Creation System (TACTICS). The specific choices made in the design of the system are discussed and justified, including selection of software platform, tactile output format, tactile image creation procedure, aggregate image processing sequences used, and principles from the discipline of psychophysics. The results of four experiments on tactile image discrimination, identification and comprehension are reported and discussed, and future directions in this area are proposed. Keywords blindness, image p...
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 12 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
The Number of Knight's Tours Equals 33,439,123,484,294  Counting with Binary Decision Diagrams
 Electronic Journal of Combinatorics
, 1996
"... The number of knight's tours, i. e. Hamiltonian circuits, on an 8x8 chessboard is computed with decision diagrams which turn out to be a useful tool for counting problems. 1 Introduction Binary decision diagrams are representations of Boolean functions with many applications in hardware verific ..."
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Cited by 10 (0 self)
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The number of knight's tours, i. e. Hamiltonian circuits, on an 8x8 chessboard is computed with decision diagrams which turn out to be a useful tool for counting problems. 1 Introduction Binary decision diagrams are representations of Boolean functions with many applications in hardware verification and computeraided design (Bryant (1992)). We believe that binary decision diagrams also have many applications in combinatorics and graph theory. To support this claim we determine the number of cycle coverings of the knight's graph on an 8x8 chessboard as well as the number of knight's tours with binary and slightly more general multi decision diagrams. We have chosen the knight's tour problem because of its long history (famous mathematicians like Euler, Legendre, and Vandermonde (see Rouse Ball and Coxeter (1987)) have worked on this problem) and since it is a combinatorial chess problem known to everybody. Our results are the following ones. Theorem 1 The number of cycl...
On the Complexity of Approximating ColoredGraph Problems (Extended Abstract)
 In COCOON
, 1999
"... ) Andrea E.F. Clementi Universit`a degli Studi di Roma "Tor Vergata" and Pierluigi Crescenzi Universit`a degli Studi di Firenze and Gianluca Rossi Universit`a degli Studi di Roma "La Sapienza" In this paper we prove explicit lower bounds on the approximability of some gr ..."
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Cited by 6 (0 self)
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) Andrea E.F. Clementi Universit`a degli Studi di Roma "Tor Vergata" and Pierluigi Crescenzi Universit`a degli Studi di Firenze and Gianluca Rossi Universit`a degli Studi di Roma "La Sapienza" In this paper we prove explicit lower bounds on the approximability of some graph problems restricted to instances which are already colored with a constant number of colors. As far as we know, this is the first time these problems are explicitily defined and analyzed. This allows us to drastically improve the previously known inapproximability results which were mainly a consequence of the analysis of boundeddegree graph problems. Moreover, we apply one of these results to obtain new lower bounds on the approximabiluty of the minimum delay schedule problem on storeandforward networks of bounded diameter. Finally, we propose a generalization of our analysis of the complexity of approximating coloredgraph problems to the complexity of approximating approximated optimization problems. ...