195 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game
In the past 20 years, increasing complexity in real world data has lead to
the study of higher-order data models based on partitioning hypergraphs.
However, hypergraph partitioning admits multiple formulations as hyperedges can
be cut in multiple ways. Building upon a class of hypergraph partitioning
problems introduced by Li & Milenkovic, we study the problem of minimizing
ratio-cut objectives over hypergraphs given by a new class of cut functions,
monotone submodular cut functions (mscf's), which captures hypergraph expansion
and conductance as special cases.
We first define the ratio-cut improvement problem, a family of local
relaxations of the minimum ratio-cut problem. This problem is a natural
extension of the Andersen & Lang cut improvement problem to the hypergraph
setting. We demonstrate the existence of efficient algorithms for approximately
solving this problem. These algorithms run in almost-linear time for the case
of hypergraph expansion, and when the hypergraph rank is at most .
Next, we provide an efficient -approximation algorithm for finding
the minimum ratio-cut of . We generalize the cut-matching game framework of
Khandekar et. al. to allow for the cut player to play unbalanced cuts, and
matching player to route approximate single-commodity flows. Using this
framework, we bootstrap our algorithms for the ratio-cut improvement problem to
obtain approximation algorithms for minimum ratio-cut problem for all mscf's.
This also yields the first almost-linear time -approximation
algorithms for hypergraph expansion, and constant hypergraph rank.
Finally, we extend a result of Louis & Makarychev to a broader set of
objective functions by giving a polynomial time -approximation algorithm for the minimum ratio-cut problem based on
rounding -metric embeddings.Comment: Comments and feedback welcom
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
バンディット問題における環境適応的リグレット解析
京都大学新制・課程博士博士(情報学)甲第24939号情博第850号京都大学大学院情報学研究科システム科学専攻(主査)准教授 本多 淳也, 教授 田中 利幸, 教授 鹿島 久嗣学位規則第4条第1項該当Doctor of InformaticsKyoto UniversityDFA
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Partitioning Hypergraphs is Hard: Models, Inapproximability, and Applications
We study the balanced -way hypergraph partitioning problem, with a special
focus on its practical applications to manycore scheduling. Given a hypergraph
on nodes, our goal is to partition the node set into parts of size at
most each, while minimizing the cost of the
partitioning, defined as the number of cut hyperedges, possibly also weighted
by the number of partitions they intersect. We show that this problem cannot be
approximated to within a factor of the optimal
solution in polynomial time if the Exponential Time Hypothesis holds, even for
hypergraphs of maximal degree 2. We also study the hardness of the partitioning
problem from a parameterized complexity perspective, and in the more general
case when we have multiple balance constraints.
Furthermore, we consider two extensions of the partitioning problem that are
motivated from practical considerations. Firstly, we introduce the concept of
hyperDAGs to model precedence-constrained computations as hypergraphs, and we
analyze the adaptation of the balanced partitioning problem to this case.
Secondly, we study the hierarchical partitioning problem to model hierarchical
NUMA (non-uniform memory access) effects in modern computer architectures, and
we show that ignoring this hierarchical aspect of the communication cost can
yield significantly weaker solutions.Comment: Published in the 35th ACM Symposium on Parallelism in Algorithms and
Architectures (SPAA 2023
Quantum Thermal State Preparation
Preparing ground states and thermal states is of key importance to simulating
quantum systems on a quantum computer. Despite the hope for practical quantum
advantage in quantum simulation, popular approaches like variational circuits
or adiabatic algorithms appear to face serious difficulties. Monte-Carlo style
quantum Gibbs samplers have emerged as an alternative, but prior proposals have
been unsatisfactory due to technical obstacles related to energy-time
uncertainty. We introduce simple continuous-time quantum Gibbs samplers that
overcome these obstacles by efficiently simulating Nature-inspired quantum
Master Equations (Liouvillians) utilizing the operator Fourier transform. In
addition, we construct the first provably accurate and efficient algorithm for
preparing certain purified Gibbs states (called thermal field double states in
high-energy physics) of rapidly thermalizing systems; this algorithm also
benefits from a Szegedy-type quadratic improvement with respect to the mixing
time. Our algorithms' cost has a favorable dependence on temperature, accuracy,
and the mixing time (or spectral gap) of the relevant Liouvillians. We
contribute to the theory of thermalization by developing a general analytic
framework that handles energy uncertainty through non-asymptotic secular
approximation and approximate detailed balance, establishing our approximation
guarantees and, as a byproduct yielding the first rigorous proof of finite-time
thermalization for physically derived Liouvillians. Given the success of the
classical Metropolis algorithm and the ubiquity of thermodynamics, we
anticipate that quantum Gibbs sampling will become an indispensable tool in
quantum computing.Comment: 68 pages, 11 figure
A Subquadratic Bound for Online Bisection
In the online bisection problem one has to maintain a partition of
elements into two clusters of cardinality . During runtime, an online
algorithm is given a sequence of requests, each being a pair of elements: an
inter-cluster request costs one unit while an intra-cluster one is free. The
algorithm may change the partition, paying a unit cost for each element that
changes its cluster.
This natural problem admits a simple deterministic -competitive
algorithm [Avin et al., DISC 2016]. While several significant improvements over
this result have been obtained since the original work, all of them either
limit the generality of the input or assume some form of resource augmentation
(e.g., larger clusters). Moreover, the algorithm of Avin et al. achieves the
best known competitive ratio even if randomization is allowed.
In this paper, we present a first randomized online algorithm that breaks
this natural barrier and achieves a competitive ratio of
without resource augmentation and for an arbitrary sequence of requests
Tight Approximations for Graphical House Allocation
The Graphical House Allocation (GHA) problem asks: how can houses (each
with a fixed non-negative value) be assigned to the vertices of an undirected
graph , so as to minimize the sum of absolute differences along the edges of
? This problem generalizes the classical Minimum Linear Arrangement problem,
as well as the well-known House Allocation Problem from Economics. Recent work
has studied the computational aspects of GHA and observed that the problem is
NP-hard and inapproximable even on particularly simple classes of graphs, such
as vertex disjoint unions of paths. However, the dependence of any
approximations on the structural properties of the underlying graph had not
been studied.
In this work, we give a nearly complete characterization of the
approximability of GHA. We present algorithms to approximate the optimal envy
on general graphs, trees, planar graphs, bounded-degree graphs, and
bounded-degree planar graphs. For each of these graph classes, we then prove
matching lower bounds, showing that in each case, no significant improvement
can be attained unless P = NP. We also present general approximation ratios as
a function of structural parameters of the underlying graph, such as treewidth;
these match the tight upper bounds in general, and are significantly better
approximations for many natural subclasses of graphs. Finally, we investigate
the special case of bounded-degree trees in some detail. We first refute a
conjecture by Hosseini et al. [2023] about the structural properties of exact
optimal allocations on binary trees by means of a counterexample on a depth-
complete binary tree. This refutation, together with our hardness results on
trees, might suggest that approximating the optimal envy even on complete
binary trees is infeasible. Nevertheless, we present a linear-time algorithm
that attains a -approximation on complete binary trees
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