In this article, we present a new algorithm for computing generating sets and Gröbner bases of lattice ideals. In contrast to other existing methods, our algorithm starts computing in projected subspaces and then iteratively lifts the results back into higher dimensions, by using a completion procedure, until the original dimension is reached. We give a completely geometric presentation of our Projectand-Lift algorithm and describe also the two other existing main algorithms in this geometric framework. We then give more details on an efficient implementation of this algorithm, in particular on critical-pair criteria specific to lattice ideal computations. Finally, we conclude the paper with a computational comparison of our implementation of the Project-and-Lift algorithm in 4ti2 with algorithms for lattice ideal computations implemented in CoCoA and Singular. Our algorithm outperforms the other algorithms in every single instance we have tried.Recently, there has been renewed interest in toric ideal computations because of applications in algebraic statistics. Here, we are interested in Markov bases, which are used in a Monte-Carlo Markov-Chain (MCMC) process to test validity of statistical models via sampling. Diaconis and Sturmfels (Diaconis and Sturmfels,where A σ and Aσ are the sub-matrices of A whose columns are indexed by σ andσ respectively, and A σ Z := {A σ z : z ∈ Z |σ| }. In the special case where σ = ∅, we set A σ Z := {0}, and the problem IP ∅ A,c,b simplifies to IP A,c,b := min{cz : Az = b, z ∈ N n }. Note that group relaxations and extended group relaxations of IP A,c,b are also of the form IP σ A,c,b for some cost vectorc ∈ Q |σ| (Hosten and Thomas, 2002). Without loss of generality, we assume that c is generic meaning that IP σ A,c,b has a unique optimal solution for every feasible b ∈ Z d . We can always easily perturb a given c so that it is generic.if T contains an improving direction t for every non-optimal feasible solution z ∈ N |σ| of IP σ A,c,b ; that is, z − t is also feasible and c(z − t) < cz. Clearly, z − t being feasible implies that t is an element of the latticeMoreover, a set T ⊆ L σ A is called a test set for IP σ A,c the class of integer programs IP σ A,c,b for all b ∈ Z d , if T is a test set for every integer program in IP σ A,c . Graver showed that there exist finite sets T that are test sets for IP A,c (σ = ∅). In fact, his sets also constitute finite test sets for IP σ A,c for arbitrary

Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.

We investigate the Sherali-Adams lift & project hierarchy applied to a graph isomorphism polytope whose integer points encode the isomorphisms between two graphs. In particular, the Sherali-Adams relaxations characterize a new vertex classification algorithm for graph isomorphism, which we call the generalized vertex classification algorithm. This algorithm generalizes the classic vertex classification algorithm and generalizes the work of Tinhofer on polyhedral methods for graph automorphism testing. We establish that the Sherali-Adams lift & project hierarchy when applied to a graph isomorphism polytope needs Ω(n) iterations in the worst case before converging to the convex hull of integer points. We also show that this generalized vertex classification algorithm is also strongly related to the well-known Weisfeiler-Lehman algorithm, which we show can also be characterized in terms of the Sherali-Adams relaxations of a semi-algebraic set whose integer points encode graph isomorphisms.Note that Q G,H ∩ {0, 1} n×n = Ψ G,H because the equations X u1v1 X u2v2 = 0 enforce that edges must map onto edges and non-edges (2-vertex independent sets) must map onto non-edges. We define Q G = Q G,G , and Q k G,H (respectively Q k G ) as the kth Sherali-Adams relaxations of Q G,H (respectively Q G ). The following theorem summarizes the relationship between the W-L algorithm and the Sherali-Adams relaxations of Q G,H . Theorem 1.2. Let k > 1. The polytope Q k+1 G = {I n } if and only if the k-dim W-L algorithm determines that G is asymmetric. The polytope Q k+1 G,H = ∅ if and only if the k-dim W-L algorithm determines that G and H are not isomorphic.Again, we will actually prove a more general and stronger result than the above theorem that illustrates that the the polytopes Q k G and Q G,H are geometric analogues of the k-dim W-L automorphism and isomorphism algorithms respectively (see Section 4).

Abstract. The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.