47 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Irreducibility of Generalized Permutohedra, Supermodular Functions, and Balanced Multisets
We study generalized permutohedra and supermodular functions. Specifically we
analyze decomposability and irreducibility for these objects and establish some
asymptotic behavior. We also study a related problem on irreducibility for
multisets
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Computing the Nucleolus of Matching and b-Matching Games
In the classical weighted matching problem the optimizer is given a graph with edge weights and their goal is to find a matching which maximizes the sum of the weights of edges in the matching. It is typically assumed in this process that the optimizer has unilateral control over the decision to take each edge. Where cooperative game theory intersects combinatorial optimization this assumption is subverted. In a cooperative matching game each vertex of the graph is controlled by a distinct player, and an edge can only be taken into a matching with the cooperation of the players at each of its vertices. One can think of the weight of an edge as representing the value the players of that edge generate by collaborating in partnership. In this setting the question is more than simply can we find an optimal matching, as in the classic matching problem, but also how should the players share the total value of the matching amongst themselves.
The players should share the value they generate in a way that fairly respects the contributions of each player, and which encourages as well as possible the stable participation of every player in the network. Cooperative game theory formulates such fair distributions of wealth as solution concepts. One classical and beautiful solution concept is the nucleolus. Intuitively the nucleolus distributes value so that the worst off groups of players are as satisfied as possible, and subject to that the second worst off groups, and so on. Here we think of satisfaction as the difference between how much value the players were distributed versus how much they could have generated on their own had they seceded from the grand coalition.
This thesis studies the nucleolus of matching games, and their generalization to b-matching games where each player can take on multiple partnerships simultaneously, from a computational perspective. We study when the nucleolus of a b-matching game can be computed efficiently and when it is intractable to do so.
Chapter 2 describes an algorithm for computing the nucleolus of any weighted cooperative matching game in polynomial time. Chapter 3 studies the computational complexity of b-matching games. We show that computing the nucleolus of such games is NP-hard even when every vertex has b-value 3, the graph is unweighted, bipartite, and of maximum degree 7. Finally, in Chapter 4 we show that when the problem of determining the worst off coalition under a given allocation in a cooperative game can be formulated as a dynamic program then the nucleolus of the game can be computed in time which is only a polynomial factor larger than the time it takes to solve said dynamic program. We apply this result to show that nucleolus of b-matching games can be computed in polynomial time on graphs of bounded treewidth
Coxeter submodular functions and deformations of Coxeter permutahedra
We describe the cone of deformations of a Coxeter permutahedron, or
equivalently, the nef cone of the toric variety associated to a Coxeter
complex. This family of polytopes contains polyhedral models for the
Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and
associahedra. Our description extends the known correspondence between
generalized permutahedra, polymatroids, and submodular functions to any finite
reflection group.Comment: Minor edits. To appear in Advances of Mathematic
Shard polytopes
For any lattice congruence of the weak order on permutations, N. Reading
proved that gluing together the cones of the braid fan that belong to the same
congruence class defines a complete fan, called a quotient fan, and V. Pilaud
and F. Santos showed that it is the normal fan of a polytope, called a
quotientope. In this paper, we provide a simpler approach to realize quotient
fans based on Minkowski sums of elementary polytopes, called shard polytopes,
which have remarkable combinatorial and geometric properties. In contrast to
the original construction of quotientopes, this Minkowski sum approach extends
to type .Comment: 73 pages, 35 figures; Version 2: minor corrections for final versio
Contribution of František Matúš to the research on conditional independence
summary:An overview is given of results achieved by F. Matúš on probabilistic conditional independence (CI). First, his axiomatic characterizations of stochastic functional dependence and unconditional independence are recalled. Then his elegant proof of discrete probabilistic representability of a matroid based on its linear representability over a finite field is recalled. It is explained that this result was a basis of his methodology for constructing a probabilistic representation of a given abstract CI structure. His embedding of matroids into (augmented) abstract CI structures is recalled and his contribution to the theory of semigraphoids is mentioned as well. Finally, his results on the characterization of probabilistic CI structures induced by four discrete random variables and by four regular Gaussian random variables are recalled. Partial probabilistic representability by binary random variables is also mentioned