On Approximately Counting Colorings of Small Degree Graphs

Bubley, Russ; Dyer, Martin; Greenhill, Catherine; Jerrum, Mark

doi:10.1137/s0097539798338175

Cited by 35 publications

(54 citation statements)

References 16 publications

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“…The problem ⊕mCol: for an undirected planar graph G of maximum degree m determine the parity of the number of equivalence classes of 3-colorings of G. The problem ⊕mFCol: for an undirected planar graph G of maximum degree m and an integer k determine the parity of the number of 3-colorings that are invariant under permutations of the remaining two colors when exactly k nodes are given the first color. We note that the corresponding counting problems for 3-colorability of degree three graphs are #P-complete [BDGJ99].…”

Section: Holographic Algorithm For the Parity Of The Number Of Connecmentioning

confidence: 97%

Some Observations on Holographic Algorithms

Valiant

2010

LATIN 2010: Theoretical Informatics

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Abstract. We define the notion of diversity for families of finite functions, and express the limitations of a simple class of holographic algorithms in terms of limitations on diversity. We go on to describe polynomial time holographic algorithms for computing the parity of the following quantities for degree three planar undirected graphs: the number of 3-colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case the parity can be computed for any slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes. These holographic algorithms use bases of three components, rather than two.

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Section: Holographic Algorithm For the Parity Of The Number Of Connecmentioning

confidence: 97%

Some Observations on Holographic Algorithms

Valiant

2010

LATIN 2010: Theoretical Informatics

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“…In other words, efficient ranking and unranking of X = {X G } is at least as hard counting the elements of each X X . But a result of Bubley, Dyer, Greenhill, and Jerrum [7,Section 6] says that it is #P -complete to count the number of proper κ-colorings of a maximum-degree-∆ graph, even for a fixed κ, ∆ ≥ 3. As a consequence, one can't FPE via RtE on our specification X = {X G } assuming P = #P. Note that P = NP, or the existence of a one-way function, already implies that P = #P.…”

Section: Fpe Without Rankingmentioning

confidence: 99%

Format-Preserving Encryption

Bellare

Ristenpart

Rogaway

et al. 2009

Lecture Notes in Computer Science

210

129

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Abstract. Format-preserving encryption (FPE) encrypts a plaintext of some specified format into a ciphertext of identical format-for example, encrypting a valid credit-card number into a valid creditcard number. The problem has been known for some time, but it has lacked a fully general and rigorous treatment. We provide one, starting off by formally defining FPE and security goals for it. We investigate the natural approach for achieving FPE on complex domains, the "rank-then-encipher" approach, and explore what it can and cannot do. We describe two flavors of unbalanced Feistel networks that can be used for achieving FPE, and we prove new security results for each. We revisit the cycle-walking approach for enciphering on a non-sparse subset of an encipherable domain, showing that the timing information that may be divulged by cycle walking is not a damaging thing to leak.

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“…Similar to its decision counterpart, the Counting Constraint Satisfaction Problem (#CSP) can be used to provide a generic framework for numerous counting combinatorial problems that arise frequently in a wide range of areas from logic, graph theory, and artificial intelligence [4,13,21,26,33,41,45,51,52,55,56], to statistical physics [3,11,39].…”

Section: Introductionmentioning

confidence: 99%

“…#CSP terms include problems from propositional logic [13,52], classical combinatorial problems such as #Clique, Graph Reliability, Antichain, Permanent [41,51,55,56], counting graph homomorphisms, and many others [4,21,26,33].…”

Section: Introductionmentioning

confidence: 99%

Towards a dichotomy theorem for the counting constraint satisfaction problem

Bulatov¹,

Dalmau²

44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.

136

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The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number of assignments of values to variables that satisfy all the constraints. The #CSP provides a general framework for numerous counting combinatorial problems including counting satisfying assignments to a propositional formula, counting graph homomorphisms, graph reliability and many others. This problem can be parametrized by the set of relations that may appear in a constraint. In this paper we start a systematic study of subclasses of the #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of the #CSP which are solvable in polynomial time from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision constraint satisfaction problem. Then we prove that if a subclass of the #CSP is solvable in polynomial time, then constraints allowed by the class satisfy some very restrictive condition: they need to have a Mal'tsev polymorphism, that is a ternary operation m(x, y, z) such that m(x, y, y) = m(y, y, x) = x. This condition uniformly explains many existing complexity results for particular cases of the #CSP, including the dichotomy results for the problem of counting graph homomorphisms, and it allows us to obtain new results.

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On Approximately Counting Colorings of Small Degree Graphs

Cited by 35 publications

References 16 publications

Some Observations on Holographic Algorithms

Some Observations on Holographic Algorithms

Format-Preserving Encryption

Towards a dichotomy theorem for the counting constraint satisfaction problem

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