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Let K be an infinite field and let I = (f_1,...,f_r) be an ideal in the polynomial ring R = K[x_1,...,x_n] generated by generic forms of degrees d_1,...,d_r. In the case r=n, following an effective method by Gao, Guan and Volny, we give a description of the initial ideal of I with respect to the degree reverse lexicographic order. Thanks to a theorem due to Pardue (2010), we apply our result to give a partial solution to a longstanding conjecture stated by Froeberg (1985) on the Hilbert series of R/I., Comment: 18 pages

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain non-trivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being non-trivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that $\omega$-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking a different approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a non-trivial system of linear identities, and obtain non-trivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every $\omega$-categorical structure is homomorphically equivalent to a model-complete core., Comment: 24 pages

The paper deals with recursive constructions for simple 3-designs based on other 3-designs having $(1, \sigma)$-resolution. The concept of $(1, \sigma)$-resolution may be viewed as a generalization of the parallelism for designs. We show the constructions and their applications to produce many previously unknown infinite families of simple 3-designs. We also include a discussion of $(1,\sigma)$-resolvability of the constructed designs., Comment: 18 pages

In this paper we determine the complexity of a broad class of problems that extends the temporal constraint satisfaction problems. To be more precise we study the problems Poset-SAT($\Phi$), where $\Phi$ is a given set of quantifier-free $\leq$-formulas. An instance of Poset-SAT($\Phi$) consists of finitely many variables $x_1,\ldots,x_n$ and formulas $\phi_i(x_{i_1},\ldots,x_{i_k})$ with $\phi_i \in \Phi$; the question is whether this input is satisfied by any partial order on $x_1,\ldots,x_n$ or not. We show that every such problem is NP-complete or can be solved in polynomial time, depending on $\Phi$. All Poset-SAT problems can be formalized as constraint satisfaction problems on reducts of the random partial order. We use model-theoretic concepts and techniques from universal algebra to study these reducts. In the course of this analysis we establish a dichotomy that we believe is of independent interest in universal algebra and model theory., Comment: 29 pages

We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains for example the rooted triple consistency problem, forbidden subtree problems, the quartet consistency problem, and many other problems studied in the bioinformatics literature. The studied problems can be described as \emph{constraint satisfaction problems} where the constraints have a first-order definition over the rooted triple relation. We show that every such phylogeny problem can be solved in polynomial time or is NP-complete. On the algorithmic side, we generalize a well-known polynomial-time algorithm of Aho, Sagiv, Szymanski, and Ullman for the rooted triple consistency problem. Our algorithm repeatedly solves linear equation systems to construct a solution in polynomial time. We then show that every phylogeny problem that cannot be solved by our algorithm is NP-complete. Our classification establishes a dichotomy for a large class of infinite structures that we believe is of independent interest in universal algebra, model theory, and topology. The proof of our main result combines results and techniques from various research areas: a recent classification of the model-complete cores of the reducts of the homogeneous binary branching C-relation, Leeb's Ramsey theorem for rooted trees, and universal algebra., Comment: 48 pages, 2 figures. In this version we fix several bugs in the proofs of the previous versions

Let (L;C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of (L;C), i.e., the structures with domain L that are first-order definable in (L;C). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of (L;C). We also study the endomorphism monoids of such reducts and show that they fall into four categories., Comment: 39 pages, 4 figures

The rotor-router model is a popular deterministic analogue of random walk. In this paper we prove that all orbits of the rotor-router operation have the same size on a strongly connected directed graph (digraph) and give a formula for the size. By using this formula we address the following open question about orbits of the rotor-router operation: Is there an infinite family of non-Eulerian strongly connected digraphs such that the rotor-router operation on each digraph has a single orbit? It turns out that on a strongly connected digraph the stationary distribution of the simple random walk coincides with the frequency of vertices in a rotor walk. In this common aspect a rotor walk simulates a random walk. This gives one similarity between two models on (finite) digraphs., Comment: 9 pages, 4 figures. To appear in Advances in Applied Mathematics

The Chip-firing game is a discrete dynamical system played on a graph, in which chips move along edges according to a simple local rule. Properties of the underlying graph are of course useful to the understanding of the game, but since a conjecture of Biggs that was proved by Merino L\'opez, we also know that the study of the Chip-firing game can give insights on the graph. In particular, a strong relation between the partial Tutte polynomial $T_G(1,y)$ and the set of recurrent configurations of a Chip-firing game (with a distinguished sink vertex) has been established for undirected graphs. A direct consequence is that the generating function of the set of recurrent configurations is independent of the choice of the sink for the game, as it characterizes the underlying graph itself. In this paper we prove that this property also holds for Eulerian directed graphs (digraphs), a class on the way from undirected graphs to general digraphs. It turns out from this property that the generating function of the set of recurrent configurations of an Eulerian digraph is a natural and convincing candidate for generalizing the partial Tutte polynomial $T_G(1,y)$ to this class. Our work also gives some promising directions of looking for a generalization of the Tutte polynomial to general digraphs., Comment: 28 pages, 12 figures

In this paper we present further studies of recurrent configurations of Chip-firing games on Eulerian directed graphs (simple digraphs), a class on the way from undirected graphs to general directed graphs. A computational problem that arises naturally from this model is to find the minimum number of chips of a recurrent configuration, which we call the minimum recurrent configuration (MINREC) problem. We point out a close relationship between MINREC and the minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and prove that both problems are NP-hard., Comment: 18 pages, 6 figures

Sand Pile Models are discrete dynamical systems emphasizing the phenomenon of Self-Organized Criticality. From a configuration composed of a finite number of stacked grains, we apply on every possible positions (in parallel) two grain moving transition rules. The transition rules permit one grain to fall to its right or left (symmetric) neighboring column if the difference of height between those columns is larger than 2. The model is nondeterministic and grains always fall downward. We propose a study of the set of fixed points reachable in the Parallel Symmetric Sand Pile Model (PSSPM). Using a comparison with the Symmetric Sand Pile Model (SSPM) on which rules are applied once at each iteration, we get a continuity property. This property states that within PSSPM we can't reach every fixed points of SSPM, but a continuous subset according to the lexicographic order. Moreover we define a successor relation to browse exhaustively the sets of fixed points of those models., Comment: 13 pages

It is well-known that the class of lattices generated by Chip Firing games (CFGs) is strictly included in the class of upper locally distributive lattices (ULD). However a necessary and sufficient criterion for this class is still an open question. In this paper we settle this problem by giving such a criterion. This criterion provides a polynomial-time algorithm for constructing a CFG which generates a given lattice if such a CFG exists. Going further we solve the same problem on two other classes of lattices which are generated by CFGs on the classes of undirected graphs and directed acyclic graphs., Comment: Some informations about the term "Abelian Sandpile model" have been added to this version