59,095 research outputs found
Local Guarantees in Graph Cuts and Clustering
Correlation Clustering is an elegant model that captures fundamental graph
cut problems such as Min Cut, Multiway Cut, and Multicut, extensively
studied in combinatorial optimization. Here, we are given a graph with edges
labeled or and the goal is to produce a clustering that agrees with the
labels as much as possible: edges within clusters and edges across
clusters. The classical approach towards Correlation Clustering (and other
graph cut problems) is to optimize a global objective. We depart from this and
study local objectives: minimizing the maximum number of disagreements for
edges incident on a single node, and the analogous max min agreements
objective. This naturally gives rise to a family of basic min-max graph cut
problems. A prototypical representative is Min Max Cut: find an cut
minimizing the largest number of cut edges incident on any node. We present the
following results: an -approximation for the problem of
minimizing the maximum total weight of disagreement edges incident on any node
(thus providing the first known approximation for the above family of min-max
graph cut problems), a remarkably simple -approximation for minimizing
local disagreements in complete graphs (improving upon the previous best known
approximation of ), and a -approximation for
maximizing the minimum total weight of agreement edges incident on any node,
hence improving upon the -approximation that follows from
the study of approximate pure Nash equilibria in cut and party affiliation
games
Curriculum learning for multilevel budgeted combinatorial problems
Learning heuristics for combinatorial optimization problems through graph
neural networks have recently shown promising results on some classic NP-hard
problems. These are single-level optimization problems with only one player.
Multilevel combinatorial optimization problems are their generalization,
encompassing situations with multiple players taking decisions sequentially. By
framing them in a multi-agent reinforcement learning setting, we devise a
value-based method to learn to solve multilevel budgeted combinatorial problems
involving two players in a zero-sum game over a graph. Our framework is based
on a simple curriculum: if an agent knows how to estimate the value of
instances with budgets up to , then solving instances with budget can
be done in polynomial time regardless of the direction of the optimization by
checking the value of every possible afterstate. Thus, in a bottom-up approach,
we generate datasets of heuristically solved instances with increasingly larger
budgets to train our agent. We report results close to optimality on graphs up
to nodes and a speedup on average compared to the quickest
exact solver known for the Multilevel Critical Node problem, a max-min-max
trilevel problem that has been shown to be at least -hard
Joint Pilot Design and Uplink Power Allocation in Multi-Cell Massive MIMO Systems
This paper considers pilot design to mitigate pilot contamination and provide
good service for everyone in multi-cell Massive multiple input multiple output
(MIMO) systems. Instead of modeling the pilot design as a combinatorial
assignment problem, as in prior works, we express the pilot signals using a
pilot basis and treat the associated power coefficients as continuous
optimization variables. We compute a lower bound on the uplink capacity for
Rayleigh fading channels with maximum ratio detection that applies with
arbitrary pilot signals. We further formulate the max-min fairness problem
under power budget constraints, with the pilot signals and data powers as
optimization variables. Because this optimization problem is non-deterministic
polynomial-time hard due to signomial constraints, we then propose an algorithm
to obtain a local optimum with polynomial complexity. Our framework serves as a
benchmark for pilot design in scenarios with either ideal or non-ideal
hardware. Numerical results manifest that the proposed optimization algorithms
are close to the optimal solution obtained by exhaustive search for different
pilot assignments and the new pilot structure and optimization bring large
gains over the state-of-the-art suboptimal pilot design.Comment: 16 pages, 8 figures. Accepted to publish at IEEE Transactions on
Wireless Communication
Solving Min-Max Capacitated Vehicle Routing Problem by Local Search
Vehicle routing is a class of combinatorial optimization problems in transportation and logistics. Min-max capacitated vehicle routing is a problem of this class in which the length of the longest route must be minimized. This paper investigates local search approach for solving the min-max capacitated vehicle routing problem with different neighborhood structures. We also propose a combined function instead of the objective function itself for controlling the local search. Experimental results on different datasets show the efficiency of our proposed algorithms compared to previous techniques
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