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Publication# On some problems related to 2-level polytopes

Résumé

In this thesis we investigate a number of problems related to 2-level polytopes, in particular from the point of view of the combinatorial structure and the extension complexity. 2-level polytopes were introduced as a generalization of stable set polytopes of perfect graphs, and despite their apparently simple structure, are at the center of many open problems ranging from information theory to semidefinite programming. The extension complexity of a polytope P is a measure of the complexity of representing P: it is the smallest size of an extended formulation of P, which in turn is a linear description of a polyhedron that projects down to P.

In the first chapter, we examine several classes of 2-level polytopes arising in combinatorial settings and we prove a relation between the number of vertices and facets of such polytopes, which is conjectured to hold for all 2-level polytopes. The proofs are obtained through an improved understanding of the combinatorial structure of such polytopes, which in some cases leads to results of independent interest.

In the second chapter, we study the extension complexity of a restricted class of 2-level polytopes, the stable set polytopes of bipartite graphs, for which we obtain non-trivial lower and upper bounds.

In the third chapter we study slack matrices of 2-level polytopes, important combinatorial objects related to extension complexity, defining operations on them and giving algorithms for the following recognition problem: given a matrix, determine whether it is a slack matrix of some special class of 2-level polytopes.

In the fourth chapter we address the problem of explicitly obtaining small size extended formulations whose existence is guaranteed by communication protocols. In particular we give an algorithm to write down extended formulations for the stable set polytope of perfect graphs, making a well known result by Yannakakis constructive, and we extend this to all deterministic protocols.

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Concepts associés (12)

Combinatoire

En mathématiques, la combinatoire, appelée aussi analyse combinatoire, étudie les configurations de collections finies d'objets ou les combinaisons d'ensembles finis, et les dénombrements.
Géné

Graphe biparti

En théorie des graphes, un graphe est dit biparti si son ensemble de sommets peut être divisé en deux sous-ensembles disjoints U et V tels que chaque arête ait une extrémi

Complexité

La complexité caractérise le comportement d'un système dont les composants interagissent
localement et de façon non linéaire, ce qui se traduit par un comportement difficilement prédictible. La compl

Publications associées (18)

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Aurore Amaudruz, Christina Fragouli

A long-standing open question in information theory is to characterize the unicast capacity of a wireless relay network. The difficulty arises due to the complex signal interactions induced in the network, since the wireless channel inherently broadcasts the signals and there is interference among transmissions. Recently, Avestimehr, Diggavi and Tse proposed a linear binary deterministic model that takes into account the shared nature of wireless channels, focusing on the signal interactions rather than the background noise. They generalized the min-cut max-flow theorem for graphs to networks of deterministic channels and proved that the capacity can be achieved using information theoretical tools. They showed that the value of the minimum cut is in this case the minimum rank of all the binary adjacency matrices describing source-destination cuts. However, since there exists an exponential number of cuts, identifying the capacity through exhaustive search becomes infeasible. In this work, we develop a polynomial time algorithm that discovers the relay encoding strategy to achieve the min-cut value in binary linear deterministic (wireless) networks, for the case of a unicast connection. Our algorithm crucially uses a notion of linear independence between edges to calculate the capacity in polynomial time. Moreover, we can achieve the capacity by using very simple one-bit processing at the intermediate nodes, thereby constructively yielding finite length strategies that achieve the unicast capacity of the linear deterministic (wireless) relay network.

2009Haxell's condition [14] is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph perfect matching that is guaranteed to exist when Haxell's condition is satisfied. We prove the existence of an efficient algorithm to find perfect matchings in bipartite hypergraphs whenever a stronger version of Haxell's condition holds. Our algorithm can be seen as a generalization of the classical Hungarian algorithm for finding perfect matchings in bipartite graphs. The techniques we use to achieve this result could be of use more generally in other combinatorial problems on hypergraphs where disjointness structure is crucial, e.g., Set Packing

In this thesis we investigate a number of problems related to 2-level polytopes, in particular regarding their combinatorial structure and extension complexity. 2-level polytopes have been introduced as a generalization of stable set polytopes of perfect graphs, and despite their apparently simple structure, are at the center of many open problems: these include connection with communication complexity and the separation between linear and semidefinite programming. The extension complexity of a polytope P is a measure of the complexity of representing P: it is the smallest size of an extended formulation of P, which in turn is a linear description of a polyhedron that projects down to P. In the first chapter we introduce themain concepts that will be used through the thesis and we motivate our interest in 2-level polytopes. In the second chapter we examine several classes of 2-level polytopes arising in combinatorial settings and we prove a relation between the number of vertices and facets of such polytopes, which is conjectured to hold for all 2-level polytopes. The proofs are obtained through an improved understanding of the combinatorial structure of such polytopes, which in some cases leads to results of independent interest. In the third chapter we study the extension complexity of a restricted class of 2-level polytopes, the stable set polytopes of bipartite graphs, for which we obtain improved lower and upper bounds. In the fourth chapter we study slack matrices of 2-level polytopes, important combinatorial objects related to extension complexity, defining operations on them and giving algorithms for the following recognition problem: given a matrix, determine whether it is a slack matrix of some special class of 2-level polytopes. In the fifth chapter we address the problem of explicitly obtaining small size extended formulations whose existence is guaranteed by communication protocols. In particular we give an output-efficient algorithmto write down extended formulations for the stable set polytope of perfect graphs, making a well known result by Yannakakis constructive, and we extend this to all deterministic protocols. We then conclude the thesis outlining themain open questions that stem from our work.