1,295 research outputs found
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
Optimality of Treating Interference as Noise: A Combinatorial Perspective
For single-antenna Gaussian interference channels, we re-formulate the
problem of determining the Generalized Degrees of Freedom (GDoF) region
achievable by treating interference as Gaussian noise (TIN) derived in [3] from
a combinatorial perspective. We show that the TIN power control problem can be
cast into an assignment problem, such that the globally optimal power
allocation variables can be obtained by well-known polynomial time algorithms.
Furthermore, the expression of the TIN-Achievable GDoF region (TINA region) can
be substantially simplified with the aid of maximum weighted matchings. We also
provide conditions under which the TINA region is a convex polytope that relax
those in [3]. For these new conditions, together with a channel connectivity
(i.e., interference topology) condition, we show TIN optimality for a new class
of interference networks that is not included, nor includes, the class found in
[3].
Building on the above insights, we consider the problem of joint link
scheduling and power control in wireless networks, which has been widely
studied as a basic physical layer mechanism for device-to-device (D2D)
communications. Inspired by the relaxed TIN channel strength condition as well
as the assignment-based power allocation, we propose a low-complexity
GDoF-based distributed link scheduling and power control mechanism (ITLinQ+)
that improves upon the ITLinQ scheme proposed in [4] and further improves over
the heuristic approach known as FlashLinQ. It is demonstrated by simulation
that ITLinQ+ provides significant average network throughput gains over both
ITLinQ and FlashLinQ, and yet still maintains the same level of implementation
complexity. More notably, the energy efficiency of the newly proposed ITLinQ+
is substantially larger than that of ITLinQ and FlashLinQ, which is desirable
for D2D networks formed by battery-powered devices.Comment: A short version has been presented at IEEE International Symposium on
Information Theory (ISIT 2015), Hong Kon
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