Algorithm For Bipartite Graphs Research Articles

Theoretical Aspects of Generating Instances with Unique Solutions: Pre-assignment Models for Unique Vertex Cover

The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and the existence of only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is ΣP2-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an O(2.1996^n)-time algorithm for general graphs and an O(1.9181^n)-time algorithm for bipartite graphs, where n is the number of vertices. The latter is based on an FPT algorithm with O*(3.6791^τ) time for vertex cover number τ. Furthermore, we show that the problem for trees can be solved in O(1.4143^n) time.

Product Redesign and Innovation Based on Online Reviews: A Multistage Combined Search Method

Online reviews published on the e-commerce platform provide a new source of information for designers to develop new products. Past research on new product development (NPD) using user-generated textual data commonly focused solely on extracting and identifying product features to be improved. However, the competitive analysis of product features and more specific improvement strategies have not been explored deeply. This study fully uses the rich semantic attributes of online review texts and proposes a novel online review–driven modeling framework. This new approach can extract fine-grained product features; calculate their importance, performance, and competitiveness; and build a competitiveness network for each feature. As a result, decision making is assisted, and specific product improvement strategies are developed for NPD beyond existing modeling approaches in this domain. Specifically, online reviews are first classified into redesign- and innovation-related themes using a multiple embedding model, and the redesign and innovation product features can be extracted accordingly using a mutual information multilevel feature extraction method. Moreover, the importance and performance of features are calculated, and the competitiveness and competitiveness network of features are obtained through a personalized unidirectional bipartite graph algorithm. Finally, the importance performance competitiveness analysis plot is constructed, and the product improvement strategy is developed via a multistage combined search algorithm. Case studies and comparative experiments show the effectiveness of the proposed method and provide novel business insights for stakeholders, such as product providers, managers, and designers. History: Accepted by Ram Ramesh, Area Editor for Data Science and Machine Learning. Funding: This work was supported by the National Natural Science Foundation of China [Project 72071151] and the Natural Science Foundation of Hubei Province, China [Grant 2023CFB712]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0333 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0333 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Excluding a planar matching minor in bipartite graphs

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

Reducing Graph Parameters by Contractions and Deletions

We consider the following problem: for a given graph G and two integers k and d, can we apply a fixed graph operation at most k times in order to reduce a given graph parameter π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} by at least d? We show that this problem is NP-hard when the parameter is the independence number and the graph operation is vertex deletion or edge contraction, even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document} and when restricted to chordal graphs. We give a polynomial time algorithm for bipartite graphs when the operation is edge contraction, the parameter is the independence number and d is fixed. Further, we complete the complexity dichotomy for H-free graphs when the parameter is the clique number and the operation is edge contraction by showing that this problem is NP-hard in (C3+P1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_3+P_1)$$\\end{document}-free graphs even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}. When the operation is edge deletion and the parameter is the chromatic number, we determine the computational complexity of the associated problem for cographs and complete multipartite graphs. Our results answer several open questions stated in Diner et al. (Theor Comput Sci 746:49–72, 2012, https://doi.org/10.1016/j.tcs.2018.06.023).

Improved Approximation Algorithms for Bin Packing with Conflicts

Given a set of items, and a conflict graph defined on the item set, the problem of bin packing with conflicts asks for a partition of items into a minimum number of independent sets so that the total size of items in each independent set does not exceed the bin capacity. As a generalization of both classic bin packing and classic vertex coloring, it is hard to approximate the problem on general graphs. We present new approximation algorithms for bipartite graphs and split graphs. The absolute approximation ratios are shown to be [Formula: see text] and [Formula: see text] respectively, both improving the existing results.

Trade-Offs in Dynamic Coloring for Bipartite and General Graphs

The dynamic coloring problem has gained attention in the recent past. The focus has largely been on obtaining efficient update time algorithms using \(\Delta +1\) or more colors and the trade-offs between update time and query time. Another important parameter in dynamic coloring is the number of recolorings per update which is addressed by the works of Barba et al. in WADS’17, and Solomon and Wein in ESA’18. In SODA’18, Bhattacharya et al. presented a randomized algorithm that uses \(\Delta +1\) colors and achieves amortized \(O(\log \Delta )\) update time. In STACS’20, Henzinger and Peng presented a randomized \((\Delta +1)\)-coloring algorithm with amortized O(1) update time. Independently on arXiv, Bhattacharya et al. also presented a randomized \((\Delta +1)\)-coloring algorithm with amortized O(1) update time. While works of Bhattacharya et al., and Henzinger and Peng are very efficient in terms of update time, they do not address the number of recolorings per update. We bridge this gap by providing efficient update time algorithms with constant number of recolorings. Moreover our algorithm is deterministic as opposed to the works of Bhattacharya et al. in SODA’18, and Henzinger and Peng in STACS’20. Next, we study bipartite graphs which can be optimally colored in the static setting. We show that even in the incremental setting (where edges are added to the graph and no edge can be deleted), there is a bad update sequence which forces the update time to be at least \(\Omega (\log {n})\) in the amortized setting and \(\Omega (n)\) in the worst case. This possibly explains the lack of any results on dynamic coloring specific to bipartite graphs. We circumvent this lower bound by proposing two approaches. Firstly, we allow the use of more than two colors and obtain significantly better update time. Second, we introduce the idea of implicit coloring. If the color of a vertex is explicitly stored in a data structure and updated at end of every update then we call such an algorithm as explicit coloring algorithm. All prior work on dynamic coloring uses explicit coloring algorithms. We show that using implicit coloring we can obtain near constant update time and query time for incremental coloring for bipartite case. We also bound the number of recolorings to near constant. We also show an efficient implicit fully dynamic algorithm for bipartite graphs. All our algorithms are deterministic and use simple data structures. Hence, we believe that our algorithms are of practical importance.

Approximability of the independent feedback vertex set problem for bipartite graphs

Given an undirected graph G with n vertices, the independent feedback vertex set problem is to find a vertex subset F of G with the minimum number of vertices such that F is both an independent set and a feedback vertex set of G, if it exists. This problem is known to be NP-hard for bipartite planar graphs of maximum degree four. In this paper, we study the approximability of the problem. We first show that, for any fixed ε>0, unless P=NP, there exists no polynomial-time n1−ε-approximation algorithm even for bipartite planar graphs. We then give an α(Δ−1)/2-approximation algorithm for bipartite graphs G of maximum degree Δ, which runs in t(α,G)+O(Δn) time, under the assumption that there is an α-approximation algorithm for the original feedback vertex set problem on bipartite graphs which runs in t(α,G) time. This algorithmic result also yields a polynomial-time (exact) algorithm for the independent feedback vertex set problem on bipartite graphs of maximum degree three.

Analysis of a Canonical Labeling Algorithm for the Alignment of Correlated Erdős-Rényi Graphs

Graph alignment in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact informationtheoretic threshold for graph alignment in correlated Erdős-Rényi graphs. However, very little is known about the existence of efficient algorithms to achieve graph alignment without seeds. In this work we identify a region in which a straightforward O(n11/5 logn)-time canonical labeling algorithm, initially introduced in the context of graph isomorphism, succeeds in aligning correlated Erdős-Rényi graphs. The algorithm has two steps. In the first step, all vertices are labeled by their degrees and a trivial minimum distance alignment (i.e., sorting vertices according to their degrees) matches a fixed number of highest degree vertices in the two graphs. Having identified this subset of vertices, the remaining vertices are matched using a alignment algorithm for bipartite graphs.

Analysis of a Canonical Labeling Algorithm for the Alignment of Correlated Erdos-Rényi Graphs

Graph alignment in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact information-theoretic threshold for graph alignment in correlated Erdös-Rényi graphs. However, very little is known about the existence of efficient algorithms to achieve graph alignment without seeds. In this work we identify a region in which a straightforward O(n 11/5 log n)-time canonical labeling algorithm, initially introduced in the context of graph isomorphism, succeeds in aligning correlated Erdos-Rényi graphs. The algorithm has two steps. In the first step, all vertices are labeled by their degrees and a trivial minimum distance alignment (i.e., sorting vertices according to their degrees) matches a fixed number of highest degree vertices in the two graphs. Having identified this subset of vertices, the remaining vertices are matched using a alignment algorithm for bipartite graphs. Finally, we show that the implementation of a variant of this algorithm allows for the efficient alignment of large graphs under limited noise.

CrosstalkNet: A Visualization Tool for Differential Co-expression Networks and Communities.

Variations in physiological conditions can rewire molecular interactions between biological compartments, which can yield novel insights into gain or loss of interactions specific to perturbations of interest. Networks are a promising tool to elucidate intercellular interactions, yet exploration of these large-scale networks remains a challenge due to their high dimensionality. To retrieve and mine interactions, we developed CrosstalkNet, a user friendly, web-based network visualization tool that provides a statistical framework to infer condition-specific interactions coupled with a community detection algorithm for bipartite graphs to identify significantly dense subnetworks. As a case study, we used CrosstalkNet to mine a set of 54 and 22 gene-expression profiles from breast tumor and normal samples, respectively, with epithelial and stromal compartments extracted via laser microdissection. We show how CrosstalkNet can be used to explore large-scale co-expression networks and to obtain insights into the biological processes that govern cross-talk between different tumor compartments.Significance: This web application enables researchers to mine complex networks and to decipher novel biological processes in tumor epithelial-stroma cross-talk as well as in other studies of intercompartmental interactions. Cancer Res; 78(8); 2140-3. ©2018 AACR.

Photo Stylistic Brush: Robust Style Transfer via Superpixel-Based Bipartite Graph

With the rapid development of social network and multimedia technology, customized image and video stylization have been widely used for various social-media applications. In this paper, we explore the problem of exemplar-based photo style transfer, which provides a flexible and convenient way to invoke fantastic visual impression. Rather than investigating some fixed artistic patterns to represent certain styles as was done in some previous works, our work emphasizes styles related to a series of visual effects in the photograph ( e.g. , color, tone, and contrast). We propose a photo stylistic brush, an automatic robust style transfer approach based on Super pixel-based BI partite G raph (SuperBIG). A two-step bipartite graph algorithm with different granularity levels is employed to aggregate pixels into superpixels and find their correspondences. In the first step, with the extracted hierarchical features, a bipartite graph is constructed to describe the content similarity for pixel partition to produce superpixels. In the second step, superpixels in the input/reference image are rematched to form a new superpixel-based bipartite graph, and superpixel-level correspondences are generated by bipartite matching. Finally, the refined correspondence guides SuperBIG to perform the transformation in a decorrelated color space. Extensive experimental results demonstrate the effectiveness and robustness of the proposed method for transferring various styles of exemplar images, even for some challenging cases, such as night images.

An all-pairs shortest path algorithm for bipartite graphs

Abstract Bipartite graphs are widely used for modeling of complex structures in biology, engineering, and computer science. The search for shortest paths in such structures is a highly demanded procedure that requires optimization. This paper presents a variant of the all-pairs shortest path algorithm for bipartite graphs. The method is based on the distance matrix product and improves the general algorithm by exploiting the graph topology. The space complexity is reduced by a factor of at least four and the time complexity decreased by almost an order of magnitude when compared with the basic APSP algorithm.

Improved approximation algorithms for the Max Edge-Coloring problem

The Max Edge-Coloring problem asks for a proper edge-coloring of an edge-weighted graph minimizing the sum of the weights of the heaviest edges in the color classes. In this paper we present a PTAS for trees and a 1.74-approximation algorithm for bipartite graphs; we also adapt the last algorithm to one for general graphs of the same, asymptotically, approximation ratio.

Variations on Muchnik’s Conditional Complexity Theorem

Muchnik’s theorem about simple conditional descriptions states that for all strings a and b there exists a program p transforming a to b that has the least possible length and is simple conditional on b. In this paper we present two new proofs of this theorem. The first one is based on the on-line matching algorithm for bipartite graphs. The second one, based on extractors, can be generalized to prove a version of Muchnik’s theorem for space-bounded Kolmogorov complexity. Another version of Muchnik’s theorem is proven for a resource-bounded variant of Kolmogorov complexity based on Arthur–Merlin protocols.

Exact exponential-time algorithms for finding bicliques

Due to a large number of applications, bicliques of graphs have been widely considered in the literature. This paper focuses on non-induced bicliques. Given a graph G = ( V , E ) on n vertices, a pair ( X , Y ) , with X , Y ⊆ V , X ∩ Y = ∅ , is a non-induced biclique if { x , y } ∈ E for any x ∈ X and y ∈ Y . The NP-complete problem of finding a non-induced ( k 1 , k 2 ) -biclique asks to decide whether G contains a non-induced biclique ( X , Y ) such that | X | = k 1 and | Y | = k 2 . In this paper, we design a polynomial-space O ( 1.6914 n ) -time algorithm for this problem. It is based on an algorithm for bipartite graphs that runs in time O ( 1.30052 n ) . In deriving this algorithm, we also exhibit a relation to the spare allocation problem known from memory chip fabrication. As a byproduct, we show that the constraint bipartite vertex cover problem can be solved in time O ( 1.30052 n ) .

On the max-weight edge coloring problem

We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges, so that the total weight of the partition is minimized. We explore the frontier between polynomial and NP-hard variants of the problem, with respect to the class of the underlying graph, as well as the approximability of NP-hard variants. In particular, we present polynomial algorithms for bounded degree trees and star of chains, as well as an approximation algorithm for bipartite graphs of maximum degree at most twelve which beats the best known approximation ratios.

A two-stage hardware scheduler combining greedy and optimal scheduling

Greedy scheduling heuristics provide a low complexity and scalable albeit particularly sub-optimal strategy for hardware-based crossbar schedulers. In contrast, the maximum matching algorithm for Bipartite graphs can be used to provide optimal scheduling for crossbar-based interconnection networks with a significant complexity and scalability cost. In this paper, we show how maximum matching can be reformulated in terms of Boolean operations rather than the more traditional formulations. By leveraging the inherent parallelism available in custom hardware design, we reformulate maximum matching in terms of Boolean operations rather than matrix computations and introduce three maximum matching implementations in hardware. Specifically, we examine a Pure Logic Scheduler with three dimensions of parallelism, a Matrix Scheduler with two dimensions of parallelism and a Vector Scheduler with one dimension of parallelism. These designs reduce the algorithmic complexity for an N × N network from O ( N 3 ) to O ( 1 ) , O ( K ) , and O ( K N ) , respectively, where K is the number of optimization steps. While an optimal scheduling algorithm requires K = 2 N − 1 steps, by starting with our hardware-based greedy strategy to generate an initial schedule, our simulation results show that the maximum matching scheduler can achieve 99% of the optimal schedule when K = 9 . We examine hardware and time complexity of these architectures for crossbar sizes of up to N = 1024 . Using FPGA synthesis results, we show that a greedy schedule for crossbars, ranging from 8×8 to 256×256, can be optimized in less than 20 ns per optimization step. For crossbars reaching 1024×1024 the scheduling can be completed in approximately 10 μs with current technology and could reach under 90 ns with future technologies.

The Ferry Cover Problem

In this paper we define and study a family of optimization problems called Ferry problems, which may be viewed as generalizations of the classical wolf-goat-cabbage puzzle. We present the Ferry Cover problem (FC), where the objective is to determine the minimum required boat size to safely transport n items represented by a graph G and demonstrate a close connection with Vertex Cover which leads to hardness and approximation results. We also completely solve the problem on trees. Then we focus on a variation of the same problem with the added constraint that only 1 round-trip is allowed (FC1). We present a reduction from MAX-NAE-{3}-SAT which shows that this problem is NP-hard and APX-hard. We also provide an approximation algorithm for bipartite graphs with a factor asymptotically equal to $\frac{4}{3}$and a 1.56-approximation algorithm for planar graphs. Finally, we generalize the above problem to define FCm , where at most m round-trips are allowed, and MFTk , which is the problem of minimizing the number of round-trips when the boat capacity is k. We present some preliminary lemmata for both, which provide bounds on the value of the optimal solution, and relate them to FC.

Matching Algorithms Are Fast in Sparse Random Graphs

We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every non-maximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like the Hopcroft–Karp algorithm for bipartite graphs and the Micali–Vazirani algorithm for general graphs, which have a worst case running time of \(O(m\sqrt{n})\), run in time O(m log n) with high probability, where m is the number of edges in the graph. Motwani proved these results for random graphs when the average degree is at least ln(n) [Average Case Analysis of Algorithms for Matchings and Related Problems, Journal of the ACM, 41(6), 1994]. Our results hold, if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.

Another Simple Algorithm for Edge-Coloring Bipartite Graphs

A new edge-coloring algorithm for bipartite graphs is presented. This algorithm, based on the framework of the O(mlogd + (m/d)log(m/d)log d) algorithm by Makino--Takabatake--Fujishige and the O(mlogm) one by Alon, finds an optimal edge-coloring of a bipartite graph with m edges and maximum degree d in O(mlogd + (m/d)log(m/d)) time. This algorithm does not require elaborate data structures, which the best known O(mlogd) algorithm due to Cole--Ost--Schirra depends on.