scholarly journals The complexity of computing Kronecker coefficients

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Peter Bürgisser ◽  
Christian Ikenmeyer
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Irreducible Representations ◽  
Schur Polynomials ◽  
Product Decomposition ◽  
Nous Montrons ◽  
Produit Tensoriel ◽  
International Audience ◽  
Kronecker Coefficients

International audience Kronecker coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the symmetric group $S_n$. They can also be interpreted as the coefficients of the expansion of the internal product of two Schur polynomials in the basis of Schur polynomials. We show that the problem $\mathrm{KRONCOEFF}$ of computing Kronecker coefficients is very difficult. More specifically, we prove that $\mathrm{KRONCOEFF}$ is #$\mathrm{P}$-hard and contained in the complexity class $\mathrm{GapP}$. Formally, this means that the existence of a polynomial time algorithm for $\mathrm{KRONCOEFF}$ is equivalent to the existence of a polynomial time algorithm for evaluating permanents. Les coefficients de Kronecker sont les multiplicités dans la décomposition du produit tensoriel de deux représentations irréductibles du groupe symétrique. On peut aussi les voir comme les coefficients du développement du produit interne des polynômes de Schur. Nous montrons que le problème $\mathrm{KRONCOEFF}$ de calculer les coefficients de Kronecker est très difficile. Plus précisément, nous prouvons que $\mathrm{KRONCOEFF}$ est #$\mathrm{P}$-dur et que $\mathrm{KRONCOEFF}$ est dans la classe de complexité $\mathrm{GapP}$. Cela veut dire qu'il existe un algorithme pour $\mathrm{KRONCOEFF}$ s'exécutant en temps polynomial si et seulement s'il existe un algorithme pour l'évaluation du permanent s'exécutant en temps polynomial.

scholarly journals A max-flow algorithm for positivity of Littlewood-Richardson coefficients

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Peter Bürgisser ◽  
Christian Ikenmeyer
Keyword(s):  
Polynomial Time ◽  
Linear Optimization ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Product Decomposition ◽  
Flow Algorithm ◽  
Produit Tensoriel ◽  
Flows In Networks ◽  
International Audience ◽  
Programmation Linéaire

International audience Littlewood-Richardson coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group $\mathrm{GL}(n,\mathbb{C})$. They have a wide variety of interpretations in combinatorics, representation theory and geometry. Mulmuley and Sohoni pointed out that it is possible to decide the positivity of Littlewood-Richardson coefficients in polynomial time. This follows by combining the saturation property of Littlewood-Richardson coefficients (shown by Knutson and Tao 1999) with the well-known fact that linear optimization is solvable in polynomial time. We design an explicit $\textit{combinatorial}$ polynomial time algorithm for deciding the positivity of Littlewood-Richardson coefficients. This algorithm is highly adapted to the problem and it is based on ideas from the theory of optimizing flows in networks. Les coefficients de Littlewood-Richardson sont les multiplicités dans la décomposition du produit tensoriel de deux représentations irréductibles du groupe général linéaire $\mathrm{GL}(n,\mathbb{C})$. Ces coefficients ont plusieurs interprétations en combinatoire, en théorie des représentations et en géométrie. Mulmuley et Sohoni ont observé qu'on peut décider si un coefficient de Littlewood-Richardson est positif en temps polynomial. C'est une conséquence de la propriété de saturation des coefficients de Littlewood-Richardson (démontrée par Knutson et Tao en 1999) et le fait bien connu que la programmation linéaire est possible en temps polynomial. Nous décrivons un algorithme $\textit{combinatoire}$ pour décider si un coefficient de Littlewood-Richardson est positif. Cet algorithme est bien adapté au problème et il utilise des idées de la théorie des flots maximaux sur des réseaux.

scholarly journals Some results on stable sets for k-colorable P₆-free graphs and generalizations

2012 ◽  
Vol Vol. 14 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Stable Set ◽  
Stable Sets ◽  
Polynomial Time Algorithms ◽  
Complexity Status ◽  
International Audience ◽  
Hard Problems ◽  
Structure Properties

Graphs and Algorithms International audience This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P-6-free graphs. The complexity status of MWS is open for P-6-free graphs and is open even for P-5-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P-5-free: in particular, MWS can be solved for k-colorable P-5-free graphs in polynomial time for every k (depending on k) and more generally for (P-5, K-p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P-6-free graphs and more generally for (P-6, K-p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P-6-free graphs with respect to stable sets, (ii) two reductions for MWS on (P-6; K-p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P-6-free, for 4-colorable P-6-free, and for (P-6, K-4)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P-6, F)-free graphs in polynomial time where F is any four vertex graph).

scholarly journals An expected polynomial time algorithm for coloring 2-colorable 3-graphs

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Yury Person ◽  
Mathias Schacht
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Running Time ◽  
Uniform Hypergraphs ◽  
Average Running Time ◽  
International Audience

Graphs and Algorithms International audience We present an algorithm that for 2-colorable 3-uniform hypergraphs, finds a 2-coloring in average running time O (n(5) log(2) n).

scholarly journals An upper bound for the chromatic number of line graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Andrew D. King ◽  
Bruce A. Reed ◽  
Adrian R. Vetta
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Upper Bound ◽  
Maximum Degree ◽  
Line Graph ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Line Graphs ◽  
International Audience

International audience It was conjectured by Reed [reed98conjecture] that for any graph $G$, the graph's chromatic number $χ (G)$ is bounded above by $\lceil Δ (G) +1 + ω (G) / 2\rceil$ , where $Δ (G)$ and $ω (G)$ are the maximum degree and clique number of $G$, respectively. In this paper we prove that this bound holds if $G$ is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph $G$ and produces a colouring that achieves our bound.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

A Polynomial-Time Algorithm for the Knapsack Problem with Two Variables

1976 ◽  
Vol 23 (1) ◽  
pp. 147-154 ◽  
Author(s):  
D. S. Hirschberg ◽  
C. K. Wong
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm

scholarly journals Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures

Algorithmica ◽  
2013 ◽  
Vol 71 (1) ◽  
pp. 152-180 ◽  
Author(s):  
Son Hoang Dau ◽  
Yeow Meng Chee
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Tree Structures ◽  
Simple Tree
Sign in / Sign up

Export Citation Format

Share Document