arXiv subjects

Gregory B. Sorkin

Publications and source records attributed to Gregory B. Sorkin.

22 records · Page 2Linked to original sources

arXiv2004

The Interlace Polynomial of a Graph

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable ``interlace polynomial'' for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial. It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers.

math.CO↗

arXiv2004

A Two-Variable Interlace Polynomial

We introduce a new graph polynomial in two variables. This ``interlace'' polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix. The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs. We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the ``vertex-nullity interlace polynomial'', is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte-Martin polynomial on isotropic systems previously considered by Bouchet. Another, the ``vertex-rank interlace polynomial'', is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial.

math.CO↗

arXiv2003

The Satisfiability Threshold of Random 3-SAT Is at Least 3.52

We prove that a random 3-SAT instance with clause-to-variable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement.

math.CO↗

arXiv2003

Random MAX SAT, Random MAX CUT, and Their Phase Transitions

Given a 2-SAT formula $F$ consisting of $n$ variables and $\cn$ random clauses, what is the largest number of clauses $\max F$ satisfiable by a single assignment of the variables? We bound the answer away from the trivial bounds of $(3/4)cn$ and $cn$. We prove that for $c<1$, the expected number of clauses satisfiable is $\cn-Θ(1/n)$; for large $c$, it is $((3/4)c + Θ(\sqrt{c}))n$; for $c = 1+\eps$, it is at least $(1+\eps-O(\eps^3))n$ and at most $(1+\eps-Ω(\eps^3/\ln \eps))n$; and in the ``scaling window'' $c= 1+Θ(n^{-1/3})$, it is $cn-Θ(1)$. In particular, just as the decision problem undergoes a phase transition, our optimization problem also undergoes a phase transition at the same critical value $c=1$. Nearly all of our results are established without reference to the analogous propositions for decision 2-SAT, and as a byproduct we reproduce many of those results, including much of what is known about the 2-SAT scaling window. We consider ``online'' versions of MAX-2-SAT, and show that for one version, the obvious greedy algorithm is optimal. We can extend only our simplest MAX-2-SAT results to MAX-k-SAT, but we conjecture a ``MAX-k-SAT limiting function conjecture'' analogous to the folklore satisfiability threshold conjecture, but open even for $k=2$. Neither conjecture immediately implies the other, but it is natural to further conjecture a connection between them. Finally, for random MAXCUT (the size of a maximum cut in a sparse random graph) we prove analogous results.

math.CO↗