918 research outputs found
Fine-grained dichotomies for the Tutte plane and Boolean #CSP
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of
evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard
almost everywhere, and the remaining points admit polynomial-time algorithms.
Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination
with Curticapean [7], extended the #P-hardness results to tight lower bounds
under the counting exponential time hypothesis #ETH, with the exception of the
line , which was left open. We complete the dichotomy theorem for the
Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs
of a given -vertex graph cannot be determined in time unless
#ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann
[6] for counting the number of satisfying assignments to a constraint
satisfaction problem instance over the Boolean domain. We prove that all
#P-hard cases are also hard under #ETH. The main ingredient is to prove that
the number of independent sets in bipartite graphs with vertices cannot be
computed in time unless #ETH fails. In order to prove our results,
we use the block interpolation idea by Curticapean [7] and transfer it to
systems of linear equations that might not directly correspond to
interpolation.Comment: 16 pages, 1 figur
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) 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/polylog 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
The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid
We show that computing the Tutte polynomial of a linear matroid of dimension
on points over a field of elements requires
time unless the \#ETH---a counting extension of the Exponential
Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell {\em et al.}
[ACM TALG 2014]---is false. This holds also for linear matroids that admit a
representation where every point is associated to a vector with at most two
nonzero coordinates. We also show that the same is true for computing the Tutte
polynomial of a binary matroid of dimension on points with at
most three nonzero coordinates in each point's vector. This is in sharp
contrast to computing the Tutte polynomial of a -vertex graph (that is, the
Tutte polynomial of a {\em graphic} matroid of dimension ---which is
representable in dimension over the binary field so that every vector has
two nonzero coordinates), which is known to be computable in
time [Bj\"orklund {\em et al.}, FOCS 2008]. Our lower-bound proofs proceed via
(i) a connection due to Crapo and Rota [1970] between the number of tuples of
codewords of full support and the Tutte polynomial of the matroid associated
with the code; (ii) an earlier-established \#ETH-hardness of counting the
solutions to a bipartite -CSP on vertices in time; and
(iii) new embeddings of such CSP instances as questions about codewords of full
support in a linear code. We complement these lower bounds with two algorithm
designs. The first design computes the Tutte polynomial of a linear matroid of
dimension~ on points in operations. The second design
generalizes the Bj\"orklund~{\em et al.} algorithm and runs in
time for linear matroids of dimension defined over the
-element field by points with at most two nonzero coordinates
each.Comment: This version adds Theorem
Exponential Time Complexity of Weighted Counting of Independent Sets
We consider weighted counting of independent sets using a rational weight x:
Given a graph with n vertices, count its independent sets such that each set of
size k contributes x^k. This is equivalent to computation of the partition
function of the lattice gas with hard-core self-repulsion and hard-core pair
interaction. We show the following conditional lower bounds: If counting the
satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time
2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in
time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph
needs time 2^{\Omega(n)} and weighted counting of independent sets needs time
2^{\Omega(n/log^3 n)} for all rational weights x\neq 0.
We have two technical ingredients: The first is a reduction from 3SAT to
independent sets that preserves the number of solutions and increases the
instance size only by a constant factor. Second, we devise a combination of
vertex cloning and path addition. This graph transformation allows us to adapt
a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by
a family of reductions, each of which increases the instance size only
polylogarithmically.Comment: Introduction revised, differences between versions of counting
independent sets stated more precisely, minor improvements. 14 page
Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth
We consider the multivariate interlace polynomial introduced by Courcelle
(2008), which generalizes several interlace polynomials defined by Arratia,
Bollobas, and Sorkin (2004) 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.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor
improvements. 44 pages, 14 figure
How proofs are prepared at Camelot
We study a design framework for robust, independently verifiable, and
workload-balanced distributed algorithms working on a common input. An
algorithm based on the framework is essentially a distributed encoding
procedure for a Reed--Solomon code, which enables (a) robustness against
byzantine failures with intrinsic error-correction and identification of failed
nodes, and (b) independent randomized verification to check the entire
computation for correctness, which takes essentially no more resources than
each node individually contributes to the computation. The framework builds on
recent Merlin--Arthur proofs of batch evaluation of Williams~[{\em Electron.\
Colloq.\ Comput.\ Complexity}, Report TR16-002, January 2016] with the
observation that {\em Merlin's magic is not needed} for batch evaluation---mere
Knights can prepare the proof, in parallel, and with intrinsic
error-correction.
The contribution of this paper is to show that in many cases the verifiable
batch evaluation framework admits algorithms that match in total resource
consumption the best known sequential algorithm for solving the problem. As our
main result, we show that the -cliques in an -vertex graph can be counted
{\em and} verified in per-node time and space on
compute nodes, for any constant and
positive integer divisible by , where is the
exponent of matrix multiplication. This matches in total running time the best
known sequential algorithm, due to Ne{\v{s}}et{\v{r}}il and Poljak [{\em
Comment.~Math.~Univ.~Carolin.}~26 (1985) 415--419], and considerably improves
its space usage and parallelizability. Further results include novel algorithms
for counting triangles in sparse graphs, computing the chromatic polynomial of
a graph, and computing the Tutte polynomial of a graph.Comment: 42 p
Exact Covers via Determinants
Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such
that every hyperedge includes one vertex from each part, the k-dimensional
matching problem asks whether there is a disjoint collection of the hyperedges
which covers all vertices. We show it can be solved by a randomized polynomial
space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors
polynomial in n and k.
When we drop the partition constraint and permit arbitrary hyperedges of
cardinality k, we obtain the exact cover by k-sets problem. We show it can be
solved by a randomized polynomial space algorithm in time O*(c_k^n), where
c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k.
Both results substantially improve on the previous best algorithms for these
problems, especially for small k, and follow from the new observation that
Lovasz' perfect matching detection via determinants (1979) admits an embedding
in the recently proposed inclusion-exclusion counting scheme for set covers,
despite its inability to count the perfect matchings
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