28,012 research outputs found
Robust Assignments via Ear Decompositions and Randomized Rounding
Many real-life planning problems require making a priori decisions before all
parameters of the problem have been revealed. An important special case of such
problem arises in scheduling problems, where a set of tasks needs to be
assigned to the available set of machines or personnel (resources), in a way
that all tasks have assigned resources, and no two tasks share the same
resource. In its nominal form, the resulting computational problem becomes the
\emph{assignment problem} on general bipartite graphs.
This paper deals with a robust variant of the assignment problem modeling
situations where certain edges in the corresponding graph are \emph{vulnerable}
and may become unavailable after a solution has been chosen. The goal is to
choose a minimum-cost collection of edges such that if any vulnerable edge
becomes unavailable, the remaining part of the solution contains an assignment
of all tasks.
We present approximation results and hardness proofs for this type of
problems, and establish several connections to well-known concepts from
matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape
The matching relaxation for a class of generalized set partitioning problems
This paper introduces a discrete relaxation for the class of combinatorial
optimization problems which can be described by a set partitioning formulation
under packing constraints. We present two combinatorial relaxations based on
computing maximum weighted matchings in suitable graphs. Besides providing dual
bounds, the relaxations are also used on a variable reduction technique and a
matheuristic. We show how that general method can be tailored to sample
applications, and also perform a successful computational evaluation with
benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented
at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial
Optimization), with proceedings on Electronic Notes in Discrete Mathematic
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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