195 research outputs found
Weighted interlace polynomials
The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend
to invariants of graphs with vertex weights, and these weighted interlace
polynomials have several novel properties. One novel property is a version of
the fundamental three-term formula
q(G)=q(G-a)+q(G^{ab}-b)+((x-1)^{2}-1)q(G^{ab}-a-b) that lacks the last term. It
follows that interlace polynomial computations can be represented by binary
trees rather than mixed binary-ternary trees. Binary computation trees provide
a description of that is analogous to the activities description of the
Tutte polynomial. If is a tree or forest then these "algorithmic
activities" are associated with a certain kind of independent set in . Three
other novel properties are weighted pendant-twin reductions, which involve
removing certain kinds of vertices from a graph and adjusting the weights of
the remaining vertices in such a way that the interlace polynomials are
unchanged. These reductions allow for smaller computation trees as they
eliminate some branches. If a graph can be completely analyzed using
pendant-twin reductions then its interlace polynomial can be calculated in
polynomial time. An intuitively pleasing property is that graphs which can be
constructed through graph substitutions have vertex-weighted interlace
polynomials which can be obtained through algebraic substitutions.Comment: 11 pages (v1); 20 pages (v2); 27 pages (v3); 26 pages (v4). Further
changes may be made before publication in Combinatorics, Probability and
Computin
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
Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan
The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the
generalized fan have been calculated exactly for arbitrary size as well as
arbitrary individual edge and node reliabilities, using transfer matrices of
dimension four at most. While the all-terminal reliabilities of these graphs
are identical, the special case of identical edge () and node ()
reliabilities shows that their two-terminal reliabilities are quite distinct,
as demonstrated by their generating functions and the locations of the zeros of
the reliability polynomials, which undergo structural transitions at
On the chromatic roots of generalized theta graphs
The generalized theta graph \Theta_{s_1,...,s_k} consists of a pair of
endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1.
We prove that the roots of the chromatic polynomial $pi(\Theta_{s_1,...,s_k},z)
of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)]
k/\log k, uniformly in the path lengths s_i. Moreover, we prove that
\Theta_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 +
o(1)] k/\log k. Finally, for k \le 8 we prove that the generalized theta graph
with a chromatic root that maximizes |z-1| is the one with all path lengths
equal to 2; we conjecture that this holds for all k.Comment: LaTex2e, 25 pages including 2 figure
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