201 research outputs found
Strong Parameterized Deletion: Bipartite Graphs
The purpose of this article is two fold: (a) to formally introduce a stronger version of graph deletion problems; and (b) to study this version in the context of bipartite graphs. Given a family of graphs F, a typical instance of parameterized graph deletion problem consists of an undirected graph G and a positive integer k and the objective is to check whether we can delete at most k vertices (or k edges) such that the resulting graph belongs to F. Another version that has been recently studied is the one where the input contains two integers k and l and the objective is to check whether we can delete at most k vertices and l edges such that the resulting graph belongs to F. In this paper, we propose and initiate the study of a more general version which we call strong deletion. In this problem, given an undirected graph G and positive integers k and l, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G-S can be transformed into a graph in F by deleting at most l edges. In this paper we study this stronger version of deletion problems for the class of bipartite graphs. In particular, we study Strong Bipartite Deletion, where given an undirected graph G and positive integers k and l, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G-S can be made bipartite by deleting at most l edges. While fixed-parameter tractability when parameterizing by k or l alone is unlikely, we show that this problem is fixed-parameter tractable (FPT) when parameterized by both k and l
Belief-Invariant and Quantum Equilibria in Games of Incomplete Information
Drawing on ideas from game theory and quantum physics, we investigate
nonlocal correlations from the point of view of equilibria in games of
incomplete information. These equilibria can be classified in decreasing power
as general communication equilibria, belief-invariant equilibria and correlated
equilibria, all of which contain the familiar Nash equilibria. The notion of
belief-invariant equilibrium has appeared in game theory before, in the 1990s.
However, the class of non-signalling correlations associated to
belief-invariance arose naturally already in the 1980s in the foundations of
quantum mechanics.
Here, we explain and unify these two origins of the idea and study the above
classes of equilibria, and furthermore quantum correlated equilibria, using
tools from quantum information but the language of game theory. We present a
general framework of belief-invariant communication equilibria, which contains
(quantum) correlated equilibria as special cases. It also contains the theory
of Bell inequalities, a question of intense interest in quantum mechanics, and
quantum games where players have conflicting interests, a recent topic in
physics.
We then use our framework to show new results related to social welfare.
Namely, we exhibit a game where belief-invariance is socially better than
correlated equilibria, and one where all non-belief-invariant equilibria are
socially suboptimal. Then, we show that in some cases optimal social welfare is
achieved by quantum correlations, which do not need an informed mediator to be
implemented. Furthermore, we illustrate potential practical applications: for
instance, situations where competing companies can correlate without exposing
their trade secrets, or where privacy-preserving advice reduces congestion in a
network. Along the way, we highlight open questions on the interplay between
quantum information, cryptography, and game theory
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