IA Scholar Query: The Karger-Stein Algorithm is Optimal for k-cut.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgFri, 30 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Application and Innovation of Biopsychosocial Model
https://scholar.archive.org/work/62r2ukv33zdcxpplbzsuxwufpq
Vulnerability refers to poorly adapting to stressors and showing inappropriate responses that can become persistent states of stress. In contrast, resilience is linked to being able to perceive stressful events in less threatening ways, promoting adaptive coping strategies. In particular, resilience is important in overcoming harmful effects of stress and maintaining health in the pandemic era. Learning Objectives: 1) to learn the importance of roles of cognitive appraisal and coping in determining vulnerability and resilience, 2) to learn that gene-gene and gene-environment interactions are related to individual differences in stress responses and 3) to learn the functional capacity of the brain structures that mediate mood and emotion determines resilience. Description: Differences in individual vulnerability and resilience occur across sex, age, and culture. Resilience is an active process, not just the absence of pathology. The underlying mechanisms of vulnerability and resilience are known to depend on a combination of genetic and nongenetic factors. In particular, the functional capacity of the brain structures involved in the integrated circuits that mediate mood and emotion determines stress resilience. Overall, psychosocial factors, behavioral factors, neuroendocrine stress responses, genetic and epigenetic mechanisms, and neural circuitry are likely to be involved in vulnerability and resilience to stress. Discussion/George Nasrawork_62r2ukv33zdcxpplbzsuxwufpqFri, 30 Sep 2022 00:00:00 GMTFinding the KT Partition of a Weighted Graph in Near-Linear Time
https://scholar.archive.org/work/dweuhcshyzgelafs4yezmsngt4
In a breakthrough work, Kawarabayashi and Thorup (J. ACM'19) gave a near-linear time deterministic algorithm to compute the weight of a minimum cut in a simple graph G = (V,E). A key component of this algorithm is finding the (1+ε)-KT partition of G, the coarsest partition {P_1, ..., P_k} of V such that for every non-trivial (1+ε)-near minimum cut with sides {S, ̄{S}} it holds that P_i is contained in either S or ̄{S}, for i = 1, ..., k. In this work we give a near-linear time randomized algorithm to find the (1+ε)-KT partition of a weighted graph. Our algorithm is quite different from that of Kawarabayashi and Thorup and builds on Karger's framework of tree-respecting cuts (J. ACM'00). We describe a number of applications of the algorithm. (i) The algorithm makes progress towards a more efficient algorithm for constructing the polygon representation of the set of near-minimum cuts in a graph. This is a generalization of the cactus representation, and was initially described by Benczúr (FOCS'95). (ii) We improve the time complexity of a recent quantum algorithm for minimum cut in a simple graph in the adjacency list model from Õ(n^{3/2}) to Õ(√{mn}), when the graph has n vertices and m edges. (iii) We describe a new type of randomized algorithm for minimum cut in simple graphs with complexity 𝒪(m + n log⁶ n). For graphs that are not too sparse, this matches the complexity of the current best 𝒪(m + n log² n) algorithm which uses a different approach based on random contractions. The key technical contribution of our work is the following. Given a weighted graph G with m edges and a spanning tree T of G, consider the graph H whose nodes are the edges of T, and where there is an edge between two nodes of H iff the corresponding 2-respecting cut of T is a non-trivial near-minimum cut of G. We give a 𝒪(m log⁴ n) time deterministic algorithm to compute a spanning forest of H.Simon Apers, Paweł Gawrychowski, Troy Lee, Amit Chakrabarti, Chaitanya Swamywork_dweuhcshyzgelafs4yezmsngt4Thu, 15 Sep 2022 00:00:00 GMTRecent Advances in Fully Dynamic Graph Algorithms – A Quick Reference Guide
https://scholar.archive.org/work/q2thqx2biba2bn6uqrslqpvuem
In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. Here, we present a quick reference guide to recent engineering and theory results in the area of fully dynamic graph algorithms.Kathrin Hanauer, Monika Henzinger, Christian Schulzwork_q2thqx2biba2bn6uqrslqpvuemFri, 12 Aug 2022 00:00:00 GMTRecent Advances in Fully Dynamic Graph Algorithms (Invited Talk)
https://scholar.archive.org/work/mfjefn7ggbc7lkqgevsjo3c2ra
In recent years, significant advances have been made in the design and analysis of fully dynamic algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. Here, we present a quick reference guide to recent engineering and theory results in the area of fully dynamic graph algorithms.Kathrin Hanauer, Monika Henzinger, Christian Schulz, James Aspnes, Othon Michailwork_mfjefn7ggbc7lkqgevsjo3c2raFri, 29 Apr 2022 00:00:00 GMTCounting and enumerating optimum cut sets for hypergraph k-partitioning problems for fixed k
https://scholar.archive.org/work/lg3hrdjt7vc6lkhco56shvr2ia
We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems – namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph G=(V, E) with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k non-empty parts (V_1, V_2, ..., V_k) – known as a k-partition – so as to minimize an objective of interest. 1. If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition. 2. If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut. We give the first polynomial bound on the number of minmax-k-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an n^O(k)p-time deterministic algorithm to enumerate all min-k-cut-sets in hypergraphs, thus improving on the previously known n^O(k^2)p-time deterministic algorithm, where n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).Calvin Beideman, Karthekeyan Chandrasekaran, Weihang Wangwork_lg3hrdjt7vc6lkhco56shvr2iaWed, 20 Apr 2022 00:00:00 GMTDEUQUA2022 Conference: Connecting Geoarchives; Abstract Volume
https://scholar.archive.org/work/nbdte77wgnadnhpnplctvfonoi
00 Award ceremony and DEUQUA meeting 12:00 -13:00 Poster Session for Session 6 with oral introduction 13:00 -14:00 Lunch break 14:00 -15:00 Poster Session for Session 6 with oral introduction 15:00 -15:30 Coffee Break 15:30 -16:30 Renate Treffeisen & Klaus Grosfeld (AWI) From science to society -on the importance of knowledge transfer in modern science -examples of products and tools (Impulsvortrag und Diskussion) 16:30 -16:45 Closing remarks 17:00 -18:30 Short course K1: Overcoming challenges in publishing research dataAchim Brauer, Markus Schwabwork_nbdte77wgnadnhpnplctvfonoiBeyond GNNs: An Efficient Architecture for Graph Problems
https://scholar.archive.org/work/f4nu5wh3cnbmlptlfvdmrwpoh4
Despite their popularity for graph structured data, existing Graph Neural Networks (GNNs) have inherent limitations for fundamental graph problems such as shortest paths, kconnectivity, minimum spanning tree and minimum cuts. In these instances, it is known that one needs GNNs of high depth, scaling at a polynomial rate with the number of nodes n, to provably encode the solution space, in turn affecting their statistical efficiency. In this work we propose a new hybrid architecture to overcome this limitation. Our proposed architecture that we call as GNN + networks involve a combination of multiple parallel low depth GNNs along with simple pooling layers involving low depth fully connected networks. We provably demonstrate that for many graph problems, the solution space can be encoded by GNN + networks using depth that scales only poly-logarithmically in the number of nodes. This also has statistical advantages that we demonstrate via generalization bounds for GNN + networks. We empirically show the effectiveness of our proposed architecture for a variety of graph problems and real world classification problems.Pranjal Awasthi, Abhimanyu Das, Sreenivas Gollapudiwork_f4nu5wh3cnbmlptlfvdmrwpoh4Breaking the n^k Barrier for Minimum k-cut on Simple Graphs
https://scholar.archive.org/work/2bjdh2fwpbhvbkjc43gdhqykte
In the minimum k-cut problem, we want to find the minimum number of edges whose deletion breaks the input graph into at least k connected components. The classic algorithm of Karger and Stein runs in Õ(n^2k-2) time, and recent, exciting developments have improved the running time to O(n^k). For general, weighted graphs, this is tight assuming popular hardness conjectures. In this work, we show that perhaps surprisingly, O(n^k) is not the right answer for simple, unweighted graphs. We design an algorithm that runs in time O(n^(1-ϵ)k) where ϵ>0 is an absolute constant, breaking the natural n^k barrier. This establishes a separation of the two problems in the unweighted and weighted cases.Zhiyang He, Jason Liwork_2bjdh2fwpbhvbkjc43gdhqykteWed, 01 Dec 2021 00:00:00 GMTPreconditioning and Locality in Algorithm Design
https://scholar.archive.org/work/bhmxmgqql5autfkhf775x74kia
Algorithms is a broad, rich, and fast-growing field. For the latter half of last century, many branches of algorithms have emerged and grown in popularity, and many different techniques have been invented to solve the central problems in each area. Some of these techniques, such as the push-relabel algorithm for maximum flow, are specially designed to solve a single problem. Other techniques, such as the multiplicative weights update method, are more general and applicable to a wide range of problems. And others, such as dynamic programming, divide and conquer, and linear programming relaxation and rounding, are so fundamental that they have not only pervaded every branch of algorithms, but have ultimately reshaped the way we approach algorithm design. This thesis is devoted to studying two more modern algorithmic techniques,namely preconditioning and locality, which were pioneered by Spielman and Teng [106] in their ground-breaking work on Laplacian system solvers and have seen countless new applications in the past decade. In this thesis, I successfullyapply preconditioning and locality to resolve fundamental open problems from a wide array of algorithmic subfields, from fast, sequential algorithms to deterministicalgorithms to parallel algorithms, thereby demonstrating the power and versatility of the two techniques. Taking one step further, I make my case that preconditioning and locality are more than just powerful tools with countless applications: they are new, fundamental ways of thinking about algorithms that have the potential to revolutionize algorithm design just like dynamic programming and divide and conquer had done in the past.Jason Liwork_bhmxmgqql5autfkhf775x74kiaTue, 09 Nov 2021 00:00:00 GMTFinding the KT partition of a weighted graph in near-linear time
https://scholar.archive.org/work/ynaqzxgclrfazg2qcjypzd65ba
In a breakthrough work, Kawarabayashi and Thorup (J. ACM'19) gave a near-linear time deterministic algorithm for minimum cut in a simple graph G = (V,E). A key component is finding the (1+ε)-KT partition of G, the coarsest partition {P_1, ..., P_k} of V such that for every non-trivial (1+ε)-near minimum cut with sides {S, S̅} it holds that P_i is contained in either S or S̅, for i=1, ..., k. Here we give a near-linear time randomized algorithm to find the (1+ε)-KT partition of a weighted graph. Our algorithm is quite different from that of Kawarabayashi and Thorup and builds on Karger's framework of tree-respecting cuts (J. ACM'00). We describe applications of the algorithm. (i) The algorithm makes progress towards a more efficient algorithm for constructing the polygon representation of the set of near-minimum cuts in a graph. This is a generalization of the cactus representation initially described by Benczúr (FOCS'95). (ii) We improve the time complexity of a recent quantum algorithm for minimum cut in a simple graph in the adjacency list model from O(n^3/2) to O(√(mn)). (iii) We describe a new type of randomized algorithm for minimum cut in simple graphs with complexity O(m + n log^6 n). For slightly dense graphs this matches the complexity of the current best O(m + n log^2 n) algorithm which uses a different approach based on random contractions. The key technical contribution of our work is the following. Given a weighted graph G with m edges and a spanning tree T, consider the graph H whose nodes are the edges of T, and where there is an edge between two nodes of H iff the corresponding 2-respecting cut of T is a non-trivial near-minimum cut of G. We give a O(m log^4 n) time deterministic algorithm to compute a spanning forest of H.Simon Apers, Paweł Gawrychowski, Troy Leework_ynaqzxgclrfazg2qcjypzd65baTue, 02 Nov 2021 00:00:00 GMTDeterministic enumeration of all minimum cut-sets and k-cut-sets in hypergraphs for fixed k
https://scholar.archive.org/work/gas5q5mc2zb2lgghlclcijan4q
We consider the problem of deterministically enumerating all minimum k-cut-sets in a given hypergraph for any fixed k. The input here is a hypergraph G = (V, E) with non-negative hyperedge costs. A subset F of hyperedges is a k-cut-set if the number of connected components in G - F is at least k and it is a minimum k-cut-set if it has the least cost among all k-cut-sets. For fixed k, we call the problem of finding a minimum k-cut-set as Hypergraph-k-Cut and the problem of enumerating all minimum k-cut-sets as Enum-Hypergraph-k-Cut. The special cases of Hypergraph-k-Cut and Enum-Hypergraph-k-Cut restricted to graph inputs are well-known to be solvable in (randomized as well as deterministic) polynomial time. In contrast, it is only recently that polynomial-time algorithms for Hypergraph-k-Cut were developed. The randomized polynomial-time algorithm for Hypergraph-k-Cut that was designed in 2018 (Chandrasekaran, Xu, and Yu, SODA 2018) showed that the number of minimum k-cut-sets in a hypergraph is O(n^2k-2), where n is the number of vertices in the input hypergraph, and that they can all be enumerated in randomized polynomial time, thus resolving Enum-Hypergraph-k-Cut in randomized polynomial time. A deterministic polynomial-time algorithm for Hypergraph-k-Cut was subsequently designed in 2020 (Chandrasekaran and Chekuri, FOCS 2020), but it is not guaranteed to enumerate all minimum k-cut-sets. In this work, we give the first deterministic polynomial-time algorithm to solve Enum-Hypergraph-k-Cut (this is non-trivial even for k = 2). Our algorithms are based on new structural results that allow for efficient recovery of all minimum k-cut-sets by solving minimum (S,T)-terminal cuts. Our techniques give new structural insights even for enumerating all minimum cut-sets (i.e., minimum 2-cut-sets) in a given hypergraph.Calvin Beideman, Karthekeyan Chandrasekaran, Weihang Wangwork_gas5q5mc2zb2lgghlclcijan4qFri, 29 Oct 2021 00:00:00 GMT