Size Of Minimum Cut Research Articles

Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs. In this problem, we want to build a data structure that can provide (1 ± ε)-approximation of cut values on a graph with n vertices. For arbitrary directed graphs, such a data structure requires Ω(n 2 ) bits even for constant ε. To circumvent this, recent works study β-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times the total weight in the other direction. We consider the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ømega(n √β/ε) lower bound in the for-each model to ~Ω (n √β /ε) and we improve the previous Ω(n β/ε) lower bound in the for-all model to Ω(n β/ε 2 ). This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We prove an ΩL(min m, m/ε 2 k R) lower bound for this problem, which improves the previous ΩL(m/k R) lower bound, where m is the number of edges, k is the minimum cut size, and we seek a (1+ε)-approximation. In addition, we show that existing upper bounds with minor modifications match our lower bound up to logarithmic factors.

The energy complexity of diameter and minimum cut computation in bounded-genus networks

This paper investigates the energy complexity of distributed graph problems in multi-hop radio networks, where the energy cost of an algorithm is measured by the maximum number of awake rounds of a vertex. Recent works revealed that some problems, such as broadcast, breadth-first search, and maximal matching, can be solved with energy-efficient algorithms that consume only polylog⁡n energy. However, there exist some problems, such as computing the diameter of the graph, that require Ω(n) energy to solve. To improve energy efficiency for these problems, we focus on a special graph class: bounded-genus graphs. We present algorithms for computing the exact diameter, the exact global minimum cut size, and a (1±ϵ)-approximate s-t minimum cut size with O˜(n) energy for bounded-genus graphs. Our approach is based on a generic framework that divides the vertex set into high-degree and low-degree parts and leverages the structural properties of bounded-genus graphs to control the number of certain connected components in the subgraph induced by the low-degree part.

Geometrical configuration of hydrocyclone for improving the separation performance

The inherent defect of particle misplacement in traditional hydrocyclones is the main reason for the deterioration of separation accuracy and has obtained wide attentions. This paper presents three novel hydrocyclones based on the traditional cylindrical-conical type and cylindrical type. The effect of the hydrocyclone structure on the flow field characteristics and separation performance is investigated by a validated two fluid model. The numerical results show that a higher turbulence intensity deteriorates the separation performance of CCB type (cylindrical-conical-flat bottom type) and C type (cylindrical type), while a greater tangential velocity and velocity gradient improve the separation accuracy of MS type (multi-stage cylindrical type). Hydrocyclones with a flat-bottom structure reduces the misplaced fine particles due to the effect of the internal swirling flow and the axial circulation flow in the spigot. The MS type has the lowest imperfection value and the cut size of 76.5 μm implying a sharpness separation sharpness and an appropriate separation result. CC type has the minimum cut size of 62.21 μm, while the maximum cut size of 120.68 μm for C type.

Improvements on Spectral Bisection

In this paper, the third eigenvalue of the Laplacian matrix is used to provide a lower bound on the minimum cutsize. This result has algorithmic implications that are exploited in this paper. Besides, combinatorial properties of certain configurations of a graph partition which are related to the minimality of a cut are investigated. It is shown that such configurations are related to the third eigenvector of the Laplacian matrix. It is well known that the second eigenvector encodes structural information, and that can be used to approximate a minimum bisection. In this paper, it is shown that the third eigenvector carries structural information as well. Then a new spectral bisection algorithm using both eigenvectors is provided. The new algorithm is guaranteed to return a cut that is smaller or equal to the one returned by the classic spectral bisection. Also, a spectral algorithm that can refine a given partition and produce a smaller cut is provided.

An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Given an edge-weighted graph G with a set Q of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the size of minimum cut between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph G being either an arbitrary graph or coming from a specific graph class. We show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with k terminals that require 2^{k-2} edges in any mimicking network. This nearly matches an upper bound of mathcal {O}(k 2^{2k}) of Krauthgamer and Rika (in: Khanna (ed) Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, SODA 2013, New Orleans, 2013) and is in sharp contrast with the upper bounds of mathcal {O}(k^2) and mathcal {O}(k^4) under the assumption that all terminals lie on a single face (Goranci et al., in: Pruhs and Sohler (eds) 25th Annual European symposium on algorithms (ESA 2017), 2017, arXiv:1702.01136; Krauthgamer and Rika in Refined vertex sparsifiers of planar graphs, 2017, arXiv:1702.05951). As a side result we show a tight example for double-exponential upper bounds given by Hagerup et al. (J Comput Syst Sci 57(3):366–375, 1998), Khan and Raghavendra (Inf Process Lett 114(7):365–371, 2014), and Chambers and Eppstein (J Gr Algorithms Appl 17(3):201–220, 2013).

HEPart: A balanced hypergraph partitioning algorithm for big data applications

Minimizing the query cost among multi-hosts is important to data processing for big data applications. Hypergraph is good at modelingdata and data relationships of complex networks, the typical big data applications, by representing multi-way relationships or interactions as hyperedges. Hypergraph partitioning (HP) helps to partition the query loads on several hosts, enabling the horizontal scaling of large-scale networks. Existing heuristic HP algorithms are generally vertex hypergraph partitioning, designed to minimize the number of cut hyperedges while satisfying the balance requirements of part weights regarding vertices. However, since workloads are mainly produced by group operations, minimizing query costs landing on hyperedges and balancing the workloads should be the objectives in horizontal scaling. We thus propose a heuristic hyperedge partitioning algorithm, HEPart. Specifically, HEPart directly partitions the hypergraph into K sub-hypergraphs with a minimum cutsize for vertices, while satisfying the balance constraint on hyperedge weights, based on the effective move of hyperedges. The performance of HEPart is evaluated using several complex network datasets modeled by undirected hypergraphs, under different cutsize metrics. The partitioning quality of HEPart is then compared with alternative hyperedge partitioners and vertex hypergraph partitioning algorithms. The experimental findings demonstrate the utility of HEPart (e.g. low cut cost while keeping load balancing as required, especially over scale-free networks).

The peculiar phase structure of random graph bisection

The mincut graph bisection problem involves partitioning the n vertices of a graph into disjoint subsets, each containing exactly n/2 vertices, while minimizing the number of "cut" edges with an endpoint in each subset. When considered over sparse random graphs, the phase structure of the graph bisection problem displays certain familiar properties, but also some surprises. It is known that when the mean degree is below the critical value of 2 log 2, the cutsize is zero with high probability. We study how the minimum cutsize increases with mean degree above this critical threshold, finding a new analytical upper bound that improves considerably upon previous bounds. Combined with recent results on expander graphs, our bound suggests the unusual scenario that random graph bisection is replica symmetric up to and beyond the critical threshold, with a replica symmetry breaking transition possibly taking place above the threshold. An intriguing algorithmic consequence is that although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio to the optimal value approaches 1 asymptotically) in polynomial time for typical instances near the phase transition.

Intratissue surgery with 80 MHz nanojoule femtosecond laser pulses in the near infrared

The use of 1 nanojoule near infrared 80 MHz femtosecond laser pulses for highly precise intratissue processing, in particular for intraocular refractive surgery, was evaluated. Destructive optical breakdown at TW/cm2 light intensities in a subfemtoliter intrastromal volume was obtained by diffraction-limited focussing with an 40x objective (N.A. 1.3) and beam scanning 50 to 140 microm below the epithelial surface. Using the same system at GW/cm2 intensities two-photon excited autofluorescence imaging was used to determine the target of interest and to visualize intraocular laser effects. Histological examination of laser-exposed porcine eyes reveal a minimum cut size below 1 microm without destructive effects to surrounding tissues.

Minimum replication min-cut partitioning

Logic replication has been shown to be very effective in reducing the number of cut nets in partitioned circuits. Liu et al. (see IEEE Trans. Computer-Aided Design, vol. 14, p. 623-30, May 1995) considered the circuit partitioning problem with logic replication for separating two given nodes and presented an algorithm to determine a partitioning of the minimum possible cut size. In general, there are many possible partitioning solutions with the minimum cut size and the difference in the required amount of replication by these solutions can be significant. Since there is a size constraint on each component of the partitioning in practice, it is desirable to also minimize the amount of replication. In this paper, we present a network-flow based algorithm to determine an optimum replication min-cut partitioning that requires minimum replication. We show that the algorithm can be generalized to separate two given subsets of nodes giving an optimum partitioning of the minimum possible cut size using the least possible amount of replication. We also show that our algorithm can be used to improve the solutions produced by any existing size-constrained replication min-cut partitioning algorithm by reducing the cut size and shrinking the replication set.

Laser microdissection of metaphase chromosomes and characterization by atomic force microscopy

A new experimental setup has been constructed in which a UV laser microbeam and atomic force microscopy (AFM) have been combined on an inverted microscope in order to manipulate and visualize chromosomes with high resolution. The laser beam has been used to dissect Muntjak metaphase chromosomes and was aimed to optimize the physical size of the cuts. The capability of the AFM to visualize biological material with relative ease has been used to characterize the microdissected chromosomes. This work demonstrates that chromosome fiber material can be removed completely at the cut site using appropriate laser power. The minimum cut size achieved with a 337-nm nitrogen UV laser was between 600 and 800 nm. The smallest distance between the cuts was around 500 nm, corresponding to the finest probe for further biochemical use after physical translocation such as the polymerase chain reaction. Limitations on minimizing the cut size due to diffraction-limited focusing and the effects of laser ablation of biomaterial are discussed. © 1997 Society of Photo-Optical Instrumentation Engineers.

Removal of dust by impingement of gas on a water surface

The impingement of gas on a water surface was investigated as a means for removing dust. The dust-removal efficiency was found to be greater than 95% for our experimental conditions. The efficiency and the size distribution of the particulates in the cleaned gas were affected by the nozzel diameter, nozzle height and gas velocity. Nearly 100% efficiency was achieved in removing dust particles with size greater than 7.8 μm. The minimum cut size D 50 of the dust obtained in the cleaned gas was around 2.5 μm.