We study the expressive power of inflationary fixed-point logic IFP and inflationary fixed-point logic with counting IFP+C on planar graphs. We prove the following results: (1) IFP captures polynomial time on 3-connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2) Planar graphs can be characterized up to isomorphism in a logic with finitely many variables a...

We design a fully polynomial time approximation scheme (FPTAS) for counting the number of matchings (packings) in arbitrary 3-uniform hypergraphs of maximum degree three, referred to as (3, 3)hypergraphs. It is the first polynomial time approximation scheme for that problem, which includes also, as a special case, the 3D Matching counting problem for 3-partite (3, 3)-hypergraphs. The proof tech...

We prove that integer programming with three alternating quantifiers is NPcomplete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two alternating quantifiers can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes P,Q ⊂ R, c...

Two important algebraic proof systems are the Nullstellensatz system [1] and the polynomial calculus [2] (also called the Gröbner system). The Nullstellensatz system is a propositional proof system based on Hilbert’s Nullstellensatz, and the polynomial calculus (PC) is a proof system which allows derivations of polynomials, over some £eld. The complexity of a proof in these systems is measured ...

The complexity of computing the Tutte polynomial T(~/c,x,y) is determined for transversal matroid ,4s and algebraic numbers x and y. It is shown that for fixed x and y the problem of computing T(~,x,y) for JA a transversal matroid is ~pP-complete unless the numbers x and y satisfy (x 1)(y 1) = 1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a t...

We devise a framework for proving tight lower bounds under the counting exponential-time hypothesis #ETH introduced by Dell et al. (ACM Transactions on Algorithms, 2014). Our framework allows us to convert classical #P-hardness results for counting problems into tight lower bounds under #ETH, thus ruling out algorithms with running time 2o(n) on graphs with n vertices and O(n) edges. As exempla...

We introduce Fibonacci gates as a polynomial time computable primitive, and develop a theory of holographic algorithms based on these gates. The Fibonacci gates play the role of matchgates in Valiant’s theory [16]. We develop a signature theory and characterize all realizable signatures for Fibonacci gates. For bases of arbitrary dimensions we prove a universal bases collapse theorem. We apply ...

We show how to generate labeled and unlabeled outerplanar graphs with n vertices uniformly at random in polynomial time in n. To generate labeled outerplanar graphs, we present a counting technique using the decomposition of a graph according to its block structure, and compute the exact number of labeled outerplanar graphs. This allows us to make the correct probabilistic choices in a recursiv...

A graph is called (generically) rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here we deal with the problem of determining the maximum number of planar Euclidean embeddings as a function of the number of the vertices. We obtain polynomial systems which totally capture the structure...

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting ...