Complexity of the Cover Polynomial

Bläser, Markus; Dell, Holger

doi:10.1007/978-3-540-73420-8_69

Cited by 27 publications

(37 citation statements)

References 19 publications

(15 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…a tuple of numbers ((x a ) a∈V , (y a ) a∈V , u, v), evaluates the multivariate interlace polynomial C(G) at ((x a ) a∈V , (y a ) a∈V , u, v). Whereas this is a #P-hard problem in general [BH08], it is fixed parameter tractable with cliquewidth as parameter [Cou08,Theorem 23,Corollary 33]. This is a consequence of the fact that the interlace polynomial is a monadic second order logic definable polynomial.…”

Section: Results and Related Workmentioning

confidence: 99%

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

Bläser¹,

Hoffmann²

2009

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f (k) · n, where f (k) is doubly exponential in k. Analyzing the GF (2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2 3k 2 +O(k) · n arithmetic operations and can be efficiently implemented in parallel.

show abstract

Section: Results and Related Workmentioning

confidence: 99%

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

Bläser¹,

Hoffmann²

2009

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…The correctness of the reduction follows along the lines of [Pap94] and [BD07]. The satisfying assignments stand in bijection to cycle covers of weight (−1) i 2 j where i (resp.…”

Section: The Permanentmentioning

confidence: 99%

Exponential Time Complexity of the Permanent and the Tutte Polynomial

Dell

Husfeldt

Marx

et al. 2014

ACM Trans. Algorithms

Self Cite

View full text Add to dashboard Cite

We show conditional lower bounds for well-studied #P-hard problems:• The number of satisfying assignments of a 2-CNF formula with n variables cannot be computed in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph.• The permanent of an n × n matrix with entries 0 and 1 cannot be computed in time exp(o(n)).• The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x, y) in the case of multigraphs, and it cannot be computed in time exp(o(n/ poly log n)) in the case of simple graphs.Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting.

show abstract

“…Let us turn to matrices with negative entries. Consider (2) has an even number of edges and −1 otherwise. Note that H 2 is the simplest nontrivial Hadamard matrix.…”

Section: Examplesmentioning

confidence: 99%

“…Like the complexity of graph polynomials [2,16,18,20] and constraint satisfaction problems [1,3,4,5,12,15,17], which are both closely related to our partition functions, the complexity of partition functions has already received quite a bit of a attention. Dyer and Greenhill [9] studied the complexity of counting homomorphisms from a given graph G to a fixed graph H without parallel edges.…”

Section: Complexitymentioning

confidence: 99%

See 1 more Smart Citation

A Complexity Dichotomy for Partition Functions with Mixed Signs

Goldberg¹,

Grohe²,

Jerrum³

et al. 2010

SIAM J. Comput.

View full text Add to dashboard Cite

Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of kcolourings or the number of independent sets of a graph and also the partition functions of certain "spin glass" models of statistical physics such as the Ising model.Building on earlier work by Dyer and Greenhill [9] and Bulatov and Grohe [6], we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete.While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices -these turn out to be central in our proofs -we obtain a simple algebraic tractability criterion, which says that the tractable cases are those "representable" by a quadratic polynomial over the field F 2 .

show abstract

Complexity of the Cover Polynomial

Cited by 27 publications

References 19 publications

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

Exponential Time Complexity of the Permanent and the Tutte Polynomial

A Complexity Dichotomy for Partition Functions with Mixed Signs

Contact Info

Product

Resources

About