732,786 research outputs found
Maharam's problem
We construct an exhaustive submeasure that is not equivalent to a measure.
This solves problems of J. von Neumann (1937) and D. Maharam (1947)
Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT
In directed graphs, we investigate the problems of finding: 1) a minimum
feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2)
a feedback vertex set inducing an acyclic graph (also called the Vertex
2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a
minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS).
We show that these problems are strongly related to (variants of) Monotone
3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in
positive form. As a consequence, we deduce several NP-completeness results on
restricted versions of these problems. In particular, we define the 2-Choice
version of an optimization problem to be its restriction where the optimum
value is known to be either D or D+1 for some integer D, and the problem is
reduced to decide which of D or D+1 is the optimum value. We show that the
2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones
Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of
Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth
assignment minimize the number of variables set to true.
Finally, we propose two classes of directed graphs for which Acyclic FVS is
polynomially solvable, namely flow reducible graphs (for which MFVS is already
known to be polynomially solvable) and C1P-digraphs (defined by an adjacency
matrix with the Consecutive Ones Property)
A Decentralized Method for Joint Admission Control and Beamforming in Coordinated Multicell Downlink
In cellular networks, admission control and beamforming optimization are
intertwined problems. While beamforming optimization aims at satisfying users'
quality-of-service (QoS) requirements or improving the QoS levels, admission
control looks at how a subset of users should be selected so that the
beamforming optimization problem can yield a reasonable solution in terms of
the QoS levels provided. However, in order to simplify the design, the two
problems are usually seen as separate problems. This paper considers joint
admission control and beamforming (JACoB) under a coordinated multicell MISO
downlink scenario. We formulate JACoB as a user number maximization problem,
where selected users are guaranteed to receive the QoS levels they requested.
The formulated problem is combinatorial and hard, and we derive a convex
approximation to the problem. A merit of our convex approximation formulation
is that it can be easily decomposed for per-base-station decentralized
optimization, namely, via block coordinate decent. The efficacy of the proposed
decentralized method is demonstrated by simulation results.Comment: 2012 IEEE Asilomar Conference on Signals, Systems, and Computer
Tropical polyhedra are equivalent to mean payoff games
We show that several decision problems originating from max-plus or tropical
convexity are equivalent to zero-sum two player game problems. In particular,
we set up an equivalence between the external representation of tropical convex
sets and zero-sum stochastic games, in which tropical polyhedra correspond to
deterministic games with finite action spaces. Then, we show that the winning
initial positions can be determined from the associated tropical polyhedron. We
obtain as a corollary a game theoretical proof of the fact that the tropical
rank of a matrix, defined as the maximal size of a submatrix for which the
optimal assignment problem has a unique solution, coincides with the maximal
number of rows (or columns) of the matrix which are linearly independent in the
tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius
theory.Comment: 28 pages, 5 figures; v2: updated references, added background
materials and illustrations; v3: minor improvements, references update
- …