Júlia Pap

Recognizing conic TDI systems is hard

In this note we prove that the problem of deciding whether or not a set of integer vectors forms a Hilbert basis is co-NP-complete. Equivalently, deciding whether a conic linear system is totally dual integral or not, is co-NP-complete. These statements are true even if the vectors in the set or respectively the coefficient vectors of the inequalities are 0–1 vectors having at most three ones.

Stable Multicommodity Flows

We extend the stable flow model of Fleiner to multicommodity flows. In addition to the preference lists of agents on trading partners for each commodity, every trading pair has a preference list on the commodities that the seller can sell to the buyer. A blocking walk (with respect to a certain commodity) may include saturated arcs, provided that a positive amount of less preferred commodity is traded along the arc. We prove that a stable multicommodity flow always exists, although it is PPAD-hard to find one.

A note on kernels and Sperner's Lemma

The kernel-solvability of perfect graphs was first proved by Boros and Gurvich, and later Aharoni and Holzman gave a shorter proof. Both proofs were based on Scarfs Lemma. In this note we show that a very simple proof can be given using a polyhedral version of Sperners Lemma. In addition, we extend the Boros-Gurvich theorem to h-perfect graphs and to a more general setting.

Total Dual Integrality of Rothblum's Description of the Stable-Marriage Polyhedron

Rothblum showed that the convex hull of the stable matchings of a bipartite preference system can be described by an elegant system of linear inequalities. In this paper we prove that the description given by Rothblum is totally dual integral. We give a constructive proof based on the results of Gusfield and Irving on rotations, which gives rise to a strongly polynomial algorithm for finding an integer optimal dual solution.

Characterizing and recognizing generalized polymatroids

Generalized polymatroids are a family of polyhedra with several nice properties and applications. One property of generalized polymatroids used widely in existing literature is “total dual laminarity;” we make this notion explicit and show that only generalized polymatroids have this property. Using this we give a polynomial-time algorithm to check whether a given linear program defines a generalized polymatroid, and whether it is integral if so. Additionally, whereas it is known that the intersection of two integral generalized polymatroids is integral, we show that no larger class of polyhedra satisfies this property.

PPAD-completeness of polyhedral versions of Sperner’s Lemma

Abstract We show that certain polyhedral versions of Sperner’s Lemma, where the colouring is given explicitly as part of the input, are PPAD -complete. The proofs are based on two recent results on the complexity of computational problems in game theory: the PPAD -completeness of 2-player Nash, proved by Chen and Deng, and of Scarf’s Lemma, proved by Kintali. We give a strengthening of the latter result, show how colourings of polyhedra provide a link between the two, and discuss a special case related to vertex covers.

Algorithms for multiplayer multicommodity flow problems

We investigate the multiplayer multicommodity flow problem: several players have different networks and commodities over a common node set. Pairs of players have contracts where one of them agrees to route the flow of the other player (up to a given capacity) between two specified nodes. In return, the second player pays an amount proportional to the flow value. We show that the social optimum can be computed by linear programming, and we propose algorithms based on column generation and Lagrangian relaxation. In contrast, we prove that it is hard to decide if an equilibrium solution exists, although some natural conditions guarantee its existence.