434 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Improved Approximation Algorithms for Steiner Connectivity Augmentation Problems
The Weighted Connectivity Augmentation Problem is the problem of augmenting
the edge-connectivity of a given graph by adding links of minimum total cost.
This work focuses on connectivity augmentation problems in the Steiner setting,
where we are not interested in the connectivity between all nodes of the graph,
but only the connectivity between a specified subset of terminals.
We consider two related settings. In the Steiner Augmentation of a Graph
problem (-SAG), we are given a -edge-connected subgraph of a graph
. The goal is to augment by including links and nodes from of
minimum cost so that the edge-connectivity between nodes of increases by 1.
In the Steiner Connectivity Augmentation Problem (-SCAP), we are given a
Steiner -edge-connected graph connecting terminals , and we seek to add
links of minimum cost to create a Steiner -edge-connected graph for .
Note that -SAG is a special case of -SCAP.
All of the above problems can be approximated to within a factor of 2 using
e.g. Jain's iterative rounding algorithm for Survivable Network Design. In this
work, we leverage the framework of Traub and Zenklusen to give a -approximation for the Steiner Ring Augmentation Problem (SRAP):
given a cycle embedded in a larger graph and
a subset of terminals , choose a subset of links of minimum cost so that has 3 pairwise edge-disjoint paths
between every pair of terminals.
We show this yields a polynomial time algorithm with approximation ratio for -SCAP. We obtain an improved approximation
guarantee of for SRAP in the case that , which
yields a -approximation for -SAG for any
LP-based approximation algorithms for partial-ordered scheduling and matroid augmentation
In this thesis, we study two NP-hard problems from Combinatorial Optimization, from the perspective of approximation algorithms. The first problem we study is called Partial-Order Scheduling on Parallel Machines, which we abbreviate to PO Scheduling. Here, we are given a partially ordered set of jobs which we want to schedule to a set of machines. Each job has some weight and some processing time associated to it. On each machine, the order of the jobs scheduled to it must agree with the given partial order, i.e., a job can only be started once all its predecessors scheduled to the same machine have been completed. However, two jobs scheduled to different machines are not constrained in any way. Thus, PO Scheduling deviates from the well-studied problem of precedence-constrained scheduling in this regard. The goal of PO Scheduling is to find a feasible schedule which minimizes the sum of weighted completion times of the jobs. PO Scheduling generalizes an already NP-hard version of scheduling introduced by Bosman, Frascaria, Olver, Sitters and Stougie [3], where they ask the same question as in PO Scheduling for the case where the jobs are totally ordered. The authors above present a constant-factor approximation algorithm for their problem. We conjecture that there is a constant-factor approximation algorithm for PO Scheduling as well. While we do not solve the problem, we give approximation algorithms for the special case that the partial order consists of disjoint totally ordered chains of linearly bounded length. Additionally, we give a structural result for optimal schedules in the case that the partial order consists of disjoint, backwardly ordered (with regard to the Smith-ratio) chains. We point towards some potential research directions. For the Weighted Tree Augmentation Problem, we are given a graph with a distinguished spanning tree. Each non tree-edge has a cost associated to it. The goal is to find a cost-minimal set of edges such that when we add them to the tree-edges, the resulting graph is 2-edge-connected. Weighted tree augmentation is NP-hard. There has been recent progress in decreasing the best-known approximation factor for the problem by Traub and Zenklusen to (1.5 + ε) [51, 52]. We study a generalization of weighted tree augmentation, called the Weighted Matroid Augmentation Problem, which we abbreviate to WMAP. In WMAP, we consider a matroid with a distinguished basis and a cost function on the non-basis elements. The goal is to find a cost-minimal set such that the union of the fundamental circuits of the elements in the set with regard to the distinguished basis cover that basis. We conjecture that there is a 2-approximation algorithm for the problem in the case that the matroid is regular. While we do not solve the problem, we give an approximation algorithm for the special case of the cographic matroid and show that there is no constant-factor approximation algorithm for WMAP for representable matroids unless P = NP
Augmentation varieties and disk potentials
We elaborate on a suggestion of Aganagic-Ekholm-Ng-Vafa, that in order for
Lagrangian fillings such as the Harvey-Lawson filling to define augmentations
of Chekanov-Eliashberg differential graded algebras, one should count
configurations of holomorphic disks connected by gradient trajectories. We
propose a definition of the Chekanov-Eliashberg dga in higher dimensions which
includes as generators both Reeb chords and the space of chains on the
Legendrian, similar to the definition of immersed Lagrangian Floer theory whose
generators are chains on the Lagrangian as well as self-intersection points. We
prove that for connected Legendrian covers of monotone Lagrangian tori, the
augmentation variety in this model is equal to the image of the zero level set
of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In
particular, we show that Legendrian lifts of Vianna's exotic tori are not
Legendrian isotopic, as conjectured in Dimitroglou-Rizell-Golovko. Using
related ideas, we show that the Legendrian lift of the Clifford torus admits no
exact fillings, extending the results of Dimitroglou-Rizell and Treumann-Zaslow
in dimension two. We consider certain disconnected Legendrians, and show,
similar to another suggestion of Aganagic-Ekholm-Ng-Vafa, that the components
of the augmentation variety correspond to certain partitions and each component
is defined by a (not necessarily exact) Lagrangian filling. An adaptation of
the theory of holomorphic quilts shows that the cobordism maps associated to
bounding chains are independent of all choices up to chain homotopy.Comment: 157 page
Scaling Package Queries to a Billion Tuples via Hierarchical Partitioning and Customized Optimization
A package query returns a package -- a multiset of tuples -- that maximizes
or minimizes a linear objective function subject to linear constraints, thereby
enabling in-database decision support. Prior work has established the
equivalence of package queries to Integer Linear Programs (ILPs) and developed
the SketchRefine algorithm for package query processing. While this algorithm
was an important first step toward supporting prescriptive analytics scalably
inside a relational database, it struggles when the data size grows beyond a
few hundred million tuples or when the constraints become very tight. In this
paper, we present Progressive Shading, a novel algorithm for processing package
queries that can scale efficiently to billions of tuples and gracefully handle
tight constraints. Progressive Shading solves a sequence of optimization
problems over a hierarchy of relations, each resulting from an ever-finer
partitioning of the original tuples into homogeneous groups until the original
relation is obtained. This strategy avoids the premature discarding of
high-quality tuples that can occur with SketchRefine. Our novel partitioning
scheme, Dynamic Low Variance, can handle very large relations with multiple
attributes and can dynamically adapt to both concentrated and spread-out sets
of attribute values, provably outperforming traditional partitioning schemes
such as KD-Tree. We further optimize our system by replacing our off-the-shelf
optimization software with customized ILP and LP solvers, called Dual Reducer
and Parallel Dual Simplex respectively, that are highly accurate and orders of
magnitude faster
Parallel and Flow-Based High Quality Hypergraph Partitioning
Balanced hypergraph partitioning is a classic NP-hard optimization problem that is a fundamental tool in such diverse disciplines as VLSI circuit design, route planning, sharding distributed databases, optimizing communication volume in parallel computing, and accelerating the simulation of quantum circuits.
Given a hypergraph and an integer , the task is to divide the vertices into disjoint blocks with bounded size, while minimizing an objective function on the hyperedges that span multiple blocks.
In this dissertation we consider the most commonly used objective, the connectivity metric, where we aim to minimize the number of different blocks connected by each hyperedge.
The most successful heuristic for balanced partitioning is the multilevel approach, which consists of three phases.
In the coarsening phase, vertex clusters are contracted to obtain a sequence of structurally similar but successively smaller hypergraphs.
Once sufficiently small, an initial partition is computed.
Lastly, the contractions are successively undone in reverse order, and an iterative improvement algorithm is employed to refine the projected partition on each level.
An important aspect in designing practical heuristics for optimization problems is the trade-off between solution quality and running time.
The appropriate trade-off depends on the specific application, the size of the data sets, and the computational resources available to solve the problem.
Existing algorithms are either slow, sequential and offer high solution quality, or are simple, fast, easy to parallelize, and offer low quality.
While this trade-off cannot be avoided entirely, our goal is to close the gaps as much as possible.
We achieve this by improving the state of the art in all non-trivial areas of the trade-off landscape with only a few techniques, but employed in two different ways.
Furthermore, most research on parallelization has focused on distributed memory, which neglects the greater flexibility of shared-memory algorithms and the wide availability of commodity multi-core machines.
In this thesis, we therefore design and revisit fundamental techniques for each phase of the multilevel approach, and develop highly efficient shared-memory parallel implementations thereof.
We consider two iterative improvement algorithms, one based on the Fiduccia-Mattheyses (FM) heuristic, and one based on label propagation.
For these, we propose a variety of techniques to improve the accuracy of gains when moving vertices in parallel, as well as low-level algorithmic improvements.
For coarsening, we present a parallel variant of greedy agglomerative clustering with a novel method to resolve cluster join conflicts on-the-fly.
Combined with a preprocessing phase for coarsening based on community detection, a portfolio of from-scratch partitioning algorithms, as well as recursive partitioning with work-stealing, we obtain our first parallel multilevel framework.
It is the fastest partitioner known, and achieves medium-high quality, beating all parallel partitioners, and is close to the highest quality sequential partitioner.
Our second contribution is a parallelization of an n-level approach, where only one vertex is contracted and uncontracted on each level.
This extreme approach aims at high solution quality via very fine-grained, localized refinement, but seems inherently sequential.
We devise an asynchronous n-level coarsening scheme based on a hierarchical decomposition of the contractions, as well as a batch-synchronous uncoarsening, and later fully asynchronous uncoarsening.
In addition, we adapt our refinement algorithms, and also use the preprocessing and portfolio.
This scheme is highly scalable, and achieves the same quality as the highest quality sequential partitioner (which is based on the same components), but is of course slower than our first framework due to fine-grained uncoarsening.
The last ingredient for high quality is an iterative improvement algorithm based on maximum flows.
In the sequential setting, we first improve an existing idea by solving incremental maximum flow problems, which leads to smaller cuts and is faster due to engineering efforts.
Subsequently, we parallelize the maximum flow algorithm and schedule refinements in parallel.
Beyond the strive for highest quality, we present a deterministically parallel partitioning framework.
We develop deterministic versions of the preprocessing, coarsening, and label propagation refinement.
Experimentally, we demonstrate that the penalties for determinism in terms of partition quality and running time are very small.
All of our claims are validated through extensive experiments, comparing our algorithms with state-of-the-art solvers on large and diverse benchmark sets.
To foster further research, we make our contributions available in our open-source framework Mt-KaHyPar.
While it seems inevitable, that with ever increasing problem sizes, we must transition to distributed memory algorithms, the study of shared-memory techniques is not in vain.
With the multilevel approach, even the inherently slow techniques have a role to play in fast systems, as they can be employed to boost quality on coarse levels at little expense.
Similarly, techniques for shared-memory parallelism are important, both as soon as a coarse graph fits into memory, and as local building blocks in the distributed algorithm
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Reconstruction of machine-made shapes from bitmap sketches
We propose a method of reconstructing 3D machine-made shapes from
bitmap sketches by separating an input image into individual patches and
jointly optimizing their geometry. We rely on two main observations: (1)
human observers interpret sketches of man-made shapes as a collection of
simple geometric primitives, and (2) sketch strokes often indicate occlusion
contours or sharp ridges between those primitives. Using these main observations we design a system that takes a single bitmap image of a shape, estimates image depth and segmentation into primitives with neural networks,
then fits primitives to the predicted depth while determining occlusion contours and aligning intersections with the input drawing via optimization.
Unlike previous work, our approach does not require additional input, annotation, or templates, and does not require retraining for a new category
of man-made shapes. Our method produces triangular meshes that display
sharp geometric features and are suitable for downstream applications, such
as editing, rendering, and shading
Unifying Token and Span Level Supervisions for Few-Shot Sequence Labeling
Few-shot sequence labeling aims to identify novel classes based on only a few
labeled samples. Existing methods solve the data scarcity problem mainly by
designing token-level or span-level labeling models based on metric learning.
However, these methods are only trained at a single granularity (i.e., either
token level or span level) and have some weaknesses of the corresponding
granularity. In this paper, we first unify token and span level supervisions
and propose a Consistent Dual Adaptive Prototypical (CDAP) network for few-shot
sequence labeling. CDAP contains the token-level and span-level networks,
jointly trained at different granularities. To align the outputs of two
networks, we further propose a consistent loss to enable them to learn from
each other. During the inference phase, we propose a consistent greedy
inference algorithm that first adjusts the predicted probability and then
greedily selects non-overlapping spans with maximum probability. Extensive
experiments show that our model achieves new state-of-the-art results on three
benchmark datasets.Comment: Accepted by ACM Transactions on Information System
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