Dominique Rossin

On the Sandpile Group of Dual Graphs

The group of recurrent configurations in the sandpile model, introduced by Dhar , may be considered as a finite abelian group associated with any graph G; we call it the sandpile group of G. The aim of this paper is to prove that the sandpile group of planar graph is isomorphic to that of its dual. A combinatorial point of view on the subject is also developed.

Polynomial ideals for sandpiles and their Gröbner bases

A polynomial ideal encoding topplings in the abelian sandpile model on a graph is introduced. A Grbner basis of this ideal is interpreted combinatorially in terms of well-connected subgraphs. This gives rise to algorithms to determine the identity and the operation in the group of recurrent configurations.

On the identity of the sandpile group

The sandpile automaton was introduced by physicists in 1987. This cellular automaton presents the same critical exponents as some natural systems. Moreover, this is the simplest model known. Dhar et al. show that the recurrent configurations of this automaton form an finite abelian group. We study here the identity of this group if the automaton is defined on rectangular grids. We prove that the identity on such grids of size p × q with p > √2q can be decomposed into three parts. This proof is based on the study of the toppling of a single pile of chips on an infinite grid filled uniformly.

A variant of the tandem duplication — random loss model of genome rearrangement

In [K. Chaudhuri, K. Chen, R. Mihaescu, S. Rao, On the tandem duplication-random loss model of genome rearrangement, in: SODA, 2006, pp. 564-570], Chaudhuri, Chen, Mihaescu and Rao study algorithmic properties of the tandem duplication - random loss model of genome rearrangement, well-known in evolutionary biology. In their model, the cost of one step of duplication-loss of width k is αk for α=1 or α≥2. In this paper, we study a variant of this model, where the cost of one step of width k is 1 if k≤K and ∞ if k>K, for any value of the parameter Keℕ∪{∞}. We first show that permutations obtained after p steps of width K define classes of pattern-avoiding permutations. We also compute the numbers of duplication-loss steps of width K necessary and sufficient to obtain any permutation of Sn, in the worst case and on average. In this second part, we may also consider the case K=K(n), a function of the size n of the permutation on which the duplication-loss operations are performed.

AVERAGE-CASE ANALYSIS OF PERFECT SORTING BY REVERSALS

A sequence of reversals that takes a signed permutation to the identity is perfect if at no step a common interval is broken. Determining a parsimonious perfect sequence of reversals that sorts a signed permutation is NP-hard. Here we show that, despite this worst-case analysis, with probability one, sorting can be done in polynomial time. Further, we find asymptotic expressions for the average length and number of reversals in commuting permutations, an interesting sub-class of signed permutations.

Longest Common Separable Pattern Among Permutations

In this paper, we study the problem of finding the longest common separable pattern among several permutations. We first give a polynomial-time algorithm when the number of input permutations is fixed and next show that the problem is NP-hard for an arbitrary number of input permutations even if these permutations are separable. On the other hand, we show that the NP-hard problem of finding the longest common pattern between two permutations cannot be approximated better than within a ratio of √opt (where opt is the size of an optimal solution) when taking common patterns belonging to patternavoiding permutation classes.

Average-Case Analysis of Perfect Sorting by Reversals

A sequence of reversals that takes a signed permutation to the identity is perfect if it preserves all common intervals between the permutation and the identity. The problem of computing a parsimonious perfect sequence of reversals is believed to be NP-hard, as the more general problem of sorting a signed permutation by reversals while preserving a given subset of common intervals is NP-hard. The only published algorithms that compute a parsimonious perfect reversals sequence have an exponential time complexity. Here we show that, despite this worst-case analysis, with probability one, sorting can be done in polynomial time. Further, we find asymptotic expressions for the average length and number of reversals in commuting permutations, an interesting sub-class of signed permutations.

An algorithm computing combinatorial specifications of permutation classes

This article presents a methodology that automatically derives a combinatorial specification for a permutation class C, given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations. The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in C. The method presented is fully algorithmic.

2-Stack Sorting is polynomial.

This article deals with deciding whether a permutation is sortable with two stacks in series. Whether this decision problem lies in P or is NP-complete is a longstanding open problem since the introduction of serial compositions of stacks by Knuth in The Art of Computer Programming in 1973. We hereby prove that this decision problem lies in P by giving a polynomial algorithm to solve it. This algorithm uses the concept of pushall sorting, which was previously defined and studied by the authors.