Detecting wheels

A \emph{wheel} is a graph made of a cycle of length at least~4 together with a vertex that has at least three neighbors in the cycle. We prove that the problem whose instance is a graph $G$ and whose question is "does $G$ contains a wheel as an induced subgraph" is NP-complete. We also settle the complexity of several similar problems

Building Efficient and Compact Data Structures for Simplicial Complexes

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its functionalities. In addition, we propose two new data structures called the Maximal Simplex Tree (MxST) and the Simplex Array List (SAL). We analyze the compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List under various settings.Comment: An extended abstract appeared in the proceedings of SoCG 201

On the size of homogeneous and of depth four formulas with low individual degree

International audienceLet r ≥ 1 be an integer. Let us call a polynomial f (x_1,...,x_N) ∈ F[x] as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. We prove lower bounds for several subclasses of such circuits. Specifically, first define the formal degree of a node α with respect to a variable x_i inductively as follows. For a leaf α it is 1 if α is labelled with x_i and zero otherwise; for an internal node α labelled with × (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x_i. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits

Towards Optimal Depth-Reductions for Algebraic Formulas

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e., d<<s). In particular, for the setting of d=O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this ``low-degree" and ``low-depth" setting. We also show that these results cannot be improved in the monotone setting, even for commutative formulas

Bornes inférieures et supérieures dans les circuits arithmétiques

Arithmetic complexity is the study of the required ressources for computing poynomials using only arithmetic operations. In the last of the 70s, Valiant defined (similarly to the boolean complexity) some classes of polynomials. The polynomials which have polynomial size circuits form the class VP. Exponential sums of these polynomials correspond to the class VNP. Valiant’s hypothesis is the conjecture that VP is different tVNP.Although this conjecture is still open, it seems more accessible than its boolean counterpart. The induced algebraic structure limits the possibilities of the computation. In particular, an important result states that the low de- gree polynomials can be efficiently computed in parallel. Moreover, if we allow a fair increasement of the size, it is possible to compute them with a constant depth. As this last model is very particular, some lower bounds are known.Bürgisser showed that a conjecture (τ-conjecture) which bounds the number of roots of some univariate polynomials, implies lower bounds in arithmetic complexity. But, what happens if we try to reduce as before the depth of the circuits for the polynomials? Bounding the number of real roots of some families of polynomials would imply a separation between VP and VNP. Finally we willstudy these upper bounds on the number of real roots.La complexité arithmétique est l’étude des ressources nécessaires pour calcu- ler des polynômes en n’utilisant que des opérations arithmétiques. À la fin des années 70, Valiant a défini (de manière semblable à la complexité booléenne) des classes de polynômes. Les polynômes, ayant des circuits de taille polyno- miale, considérés faciles forment la classe VP. Les sommes exponentielles de ces derniers correpondent alors à la classe VNP. L’hypothèse de Valiant est la conjecture que VP ̸= VNP.Bien que cette conjecture soit encore grandement ouverture, cette dernière semble toutefois plus accessible que son homologue booléen. La structure algé- brique sous-jacente limite les possibilités de calculs. En particulier, un résultat important du domaine assure que les polynômes faciles peuvent aussi être cal- culés efficacement en paralèlle. De plus, quitte à autoriser une augmentation raisonnable de la taille, il est possible de les calculer avec une profondeur de calcul bornée par une constante. Comme ce dernier modèle est très restreint, de nombreuses bornes inférieures sont connues. Nous nous intéresserons en premier temps à ces résultats sur les circuits de profondeur constante.Bürgisser a montré qu’une conjecture (la τ-conjecture) qui borne supérieu- rement le nombre de racines de certains polynômes univariés, impliquait des bornes inférieures en complexité arithmétique. Mais, que se passe-t-il alors, si on essaye de réduire, comme précédemment, la profondeur du polynôme consi- déré? Borner le nombre de racines réelles de certaines familles de polynômes permetterait de séparer VP et VNP. Nous étudierons finalement ces bornes su- périeures sur le nombre de racines réelles

Upper and lower bounds for arithmetic circuits

La complexité arithmétique est l’étude des ressources nécessaires pour calcu- ler des polynômes en n’utilisant que des opérations arithmétiques. À la fin des années 70, Valiant a défini (de manière semblable à la complexité booléenne) des classes de polynômes. Les polynômes, ayant des circuits de taille polyno- miale, considérés faciles forment la classe VP. Les sommes exponentielles de ces derniers correpondent alors à la classe VNP. L’hypothèse de Valiant est la conjecture que VP ̸= VNP.Bien que cette conjecture soit encore grandement ouverture, cette dernière semble toutefois plus accessible que son homologue booléen. La structure algé- brique sous-jacente limite les possibilités de calculs. En particulier, un résultat important du domaine assure que les polynômes faciles peuvent aussi être cal- culés efficacement en paralèlle. De plus, quitte à autoriser une augmentation raisonnable de la taille, il est possible de les calculer avec une profondeur de calcul bornée par une constante. Comme ce dernier modèle est très restreint, de nombreuses bornes inférieures sont connues. Nous nous intéresserons en premier temps à ces résultats sur les circuits de profondeur constante.Bürgisser a montré qu’une conjecture (la τ-conjecture) qui borne supérieu- rement le nombre de racines de certains polynômes univariés, impliquait des bornes inférieures en complexité arithmétique. Mais, que se passe-t-il alors, si on essaye de réduire, comme précédemment, la profondeur du polynôme consi- déré? Borner le nombre de racines réelles de certaines familles de polynômes permetterait de séparer VP et VNP. Nous étudierons finalement ces bornes su- périeures sur le nombre de racines réelles.Arithmetic complexity is the study of the required ressources for computing poynomials using only arithmetic operations. In the last of the 70s, Valiant defined (similarly to the boolean complexity) some classes of polynomials. The polynomials which have polynomial size circuits form the class VP. Exponential sums of these polynomials correspond to the class VNP. Valiant’s hypothesis is the conjecture that VP is different tVNP.Although this conjecture is still open, it seems more accessible than its boolean counterpart. The induced algebraic structure limits the possibilities of the computation. In particular, an important result states that the low de- gree polynomials can be efficiently computed in parallel. Moreover, if we allow a fair increasement of the size, it is possible to compute them with a constant depth. As this last model is very particular, some lower bounds are known.Bürgisser showed that a conjecture (τ-conjecture) which bounds the number of roots of some univariate polynomials, implies lower bounds in arithmetic complexity. But, what happens if we try to reduce as before the depth of the circuits for the polynomials? Bounding the number of real roots of some families of polynomials would imply a separation between VP and VNP. Finally we willstudy these upper bounds on the number of real roots