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Complex dynamical systems often contain many unknown parameters and variables that may or may not serve as contributors to interesting behaviors a system may exhibit. For chemical and biochemical systems, which are typically quite large, these include (but are not limited to) behaviors stemming from bifurcations such as oscillations, patterns and multistationarity (multiple steady states). Due to the size and complexity of these systems, a dynamics-preserving reduction scheme that is able to isolate the necessary contributors to these systems to not only reduce their complexity, but to also reduce the level of uncertainty a system may have---such as unknown parameters and variables---is desired. The purpose of this thesis is to develop alternative reduction methods for (bio)chemical systems that are modeled by mass-action kinetics that are unlike other common techniques that exploit timescales in stiff models or that are optimization-based. I instead look to a graph-based approach by representing the model as a bipartite graph and investigating its subnetworks known as fragments, which correspond directly to terms in the characteristic polynomial. In this representation, I preserve key elements of these bipartite graphs---critical fragments---in order to maintain necessary conditions for behaviors such as positive-feedback oscillations and multistationarity. These results are then applied to an existing model for the transcriptional control of Hmp, an NO detoxifying enzyme, by the iron-sulfur protein FNR that displays bistability. The initial model consists of 15 mass-action reactions and 11 species, and the reduced model ends up with 10 reactions and 7 species.

We consider a matching problem in a bipartite graph G where every vertex has a capacity and a strict preference list ranking its neighbors. We assume that G admits a perfect matching, i.e., one that fully matches all vertices. It is only perfect matchings that are feasible here and we seek one that is popular within the set of perfect matchings - it is known that such a matching exists in G and can be efficiently computed. Now we are in the weighted setting, i.e., there is a cost function on the edge set, and we seek a min-cost popular perfect matching in G. We show that such a matching can be computed in polynomial time. Our main technical result shows that every popular perfect matching in a hospitals/residents instance G can be realized as a popular perfect matching in the marriage instance obtained by cloning vertices. Interestingly, it is known that such a mapping does not hold for popular matchings in a hospitals/residents instance.

We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Theor. Comput. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that K_t-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are FPT parameterized by bipartite treewidth. We also provide the following complexity dichotomy when H is a 2-connected graph, for each of the H-Subgraph-Packing, H-Induced-Packing, H-Scattered-Packing, and H-Odd-Minor-Packing problems: if H is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if H is non-bipartite, then the problem is solvable in XP-time. Beyond bipartite treewidth, we define 1-ℋ-treewidth by replacing the bipartite graph class by any graph class ℋ. Most of the technology developed here also works for this more general parameter.

For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan’s Theorem states that for any two simple drawings of the complete graph K_n with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n^{16}). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph K_{m,n} minus two edges and K_{m,n} plus one edge for any m,n ≥ 4, as well as K_n minus a 4-cycle for any n ≥ 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when

We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph G and is denoted by evc(G). It is clear that evc(G) is at least mvc(G), the size of a minimum vertex cover of G. We say that G is Spartan if evc(G) = mvc(G). The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on 2n vertices where every edge belongs to a perfect matching, an easy strategy is to have n guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs.

We investigate the maximum bottleneck matching problem in bipartite graphs. Given a bipartite graph with nonnegative edge weights, the problem is to find a maximum cardinality matching in which the minimum weight of an edge is the maximum. To the best of our knowledge, there are two widely used solvers for this problem based on two different approaches. There exists a third known approach in the literature, which seems inferior to those two which is presumably why there is no implementation of it. We take this third approach, make theoretical observations to improve its behavior, and implement the improved method. Experiments with the existing two solvers show that their run time can be too high to be useful in many interesting cases. Furthermore, their performance is not predictable, and slight perturbations of the input graph lead to considerable changes in the run time. On the other hand, the proposed solver’s performance is much more stable; it is almost always faster than or comparable to the two existing solvers, and its run time always remains low.

A k-role coloring of a graph G is an assignment of colors to the vertices of G such that every color is used at least once and if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. By definition, every graph on n vertices admits an n-role coloring. While for every graph on n vertices, it is trivial to decide if it admits a 1-role coloring, determining whether a graph admits a k-role coloring is a notoriously hard problem for k greater than or equal to 2. In fact, it is known that k-Role coloring is NP-complete for k at least 2 on general graph class. There has been extensive research on the complexity of k-role coloring on various hereditary graph classes. Furthering this direction of research, we show that k-Role coloring is NP-complete on bipartite graphs for k at least 3 (while it is trivial for k=2). We complement the hardness result by characterizing 3-role colorable bipartite chain graphs, leading to a polynomial time algorithm for 3-Role coloring for this class of graphs. We further show that 2-Role coloring is NP-complete for graphs that are d vertices or edges away from the class of bipartite graphs, even when d=1.

Factor graph of an instance of a constraint satisfaction problem with n variables and m constraints is the bipartite graph between [m] and [n] describing which variable appears in which constraints. Thus, an instance of a CSP is completely defined by its factor graph and the list of predicates. We show inapproximability of Max-3-LIN over non-abelian groups (both in the perfect completeness case and in the imperfect completeness case), with the same inapproximability factor as in the general case, even when the factor graph is fixed. Along the way, we also show that these optimal hardness results hold even when we restrict the linear equations in the Max-3-LIN instances to the form x · y · z = g, where x, y, z are the variables and g is a group element. We use representation theory and Fourier analysis over non-abelian groups to analyze the reductions., Part of proceedings: ISBN 978-3-95977-217-4QC 20221019

This paper addresses the classical problem of characterizing degree sequences that can be realized by a bipartite graph. For the simpler variant of the problem, where a partition of the sequence into the two sides of the bipartite graph is given as part of the input, a complete characterization was given by Gale and Ryser over 60 years ago. However, the general question, in which both the partition and the realizing graph need to be determined, is still open. This paper provides an overview of some of the known results on this problem in interesting special cases, including realizations by bipartite graphs and bipartite multigraphs.

We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices in A rank their neighbors in a strict order of preference while vertices in B are allowed to have weak rankings, i.e., ties are allowed in their preferences. Stable matchings always exist in G and are easy to find, however popular matchings need not exist and it is NP-complete to decide if one exists. This motivates the "approximately popular" matching problem. A well-known measure of approximate popularity is low unpopularity factor. We show that when each tie in G has length at most k, there always exists a stable matching whose unpopularity factor is at most k. Our proof is algorithmic and we compute such a stable matching in polynomial time. Our result can be considered to be a generalization of Gärdenfors' result (1975) which showed that when rankings are strict, every stable matching is popular. There are several applications where the size of the matching is its most important attribute. What one seeks here is a maximum matching M such that there is no maximum matching more popular than M. When rankings are weak, it is NP-hard to decide if G admits such a matching. When ties are one-sided and of length at most k, we show a polynomial time algorithm to find a maximum matching whose unpopularity factor within the set of maximum matchings is at most 2k.

We consider the problem of characterizing degree sequences that can be realized by a bipartite graph. If a partition of the sequence into the two sides of the bipartite graph is given as part of the input, then a complete characterization has been established over 60 years ago. However, the general question, in which a partition and a realizing graph need to be determined, is still open. We investigate the role of an important class of special partitions, called High-Low partitions, which separate the degrees of a sequence into two groups, the high degrees and the low degrees. We show that when the High-Low partition exists and satisfies some natural properties, analysing the High-Low partition resolves the bigraphic realization problem. For sequences that are known to be not realizable by a bipartite graph or that are undecided, we provide approximate realizations based on the High-Low partition.

We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is half its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on n vertices, it is polynomial-time solvable on graphs with minimum degree n-3 and NP-hard on (n-4)-regular graphs. We prove that it is polynomial-time 4/3-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree n-t for any constant t ≥ 4.

We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices have strict preferences over their neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M is at least the number that prefer N; thus M does not lose a head-to-head election against any matching where vertices are voters. It is easy to find popular matchings; however when there are edge costs, it is NP-hard to find (or even approximate) a min-cost popular matching. This hardness motivates relaxations of popularity. Here we introduce fairly popular matchings. A fairly popular matching may lose elections but there is no good matching (wrt popularity) that defeats a fairly popular matching. In particular, any matching that defeats a fairly popular matching does not occur in the support of any popular mixed matching. We show that a min-cost fairly popular matching can be computed in polynomial time and the fairly popular matching polytope has a compact extended formulation. We also show the following hardness result: given a matching M, it is NP-complete to decide if there exists a popular matching that defeats M. Interestingly, there exists a set K of at most m popular matchings in G (where |E| = m) such that if a matching is defeated by some popular matching in G then it has to be defeated by one of the matchings in K.

Quantitative linguistics studies human language using statistical methods. It aims to build general theories from the statistical laws observed in a wide variety of languages. As part of the scientific method, these theories should be able to make novel predictions. This thesis is based on a family of models of human language. These models have shown to reproduce language laws, such as Zipf's law. They have also been used to make predictions, such as the biases present in child word learning. This family of models is based on the minimization of a cost function. The cost function is defined using a combination of information theoretic measures on a bipartite graph of associations between words (or, more generally, forms) and meanings (more generally, counterparts). It balances between the entropy of words and the mutual information of words and meanings. Entropy is a measure of surprisal, the cost of the speaker, and should be minimized. Mutual information is the amount of information obtained from a meaning while observing a word, the cost of the listener, and it should be maximized. The model is then optimized with a Markov Chain Monte Carlo method at zero temperature. This thesis is centered on two models belonging to this family, the "internal model" and the "external model". This thesis makes several contributions in relation to these models. The mathematical equations defining them are derived, including dynamic equations which reduce the computational complexity of the optimization process. In addition, several techniques are introduced which aim to reduce the significant problem of numerical error due to floating point arithmetic without compromising efficiency. Another contribution is the replication of results obtained by previous models based on this family which had been published originally with replicability issues. The models go through an optimization process. After this process the linguistic laws they can predict are examined, as well as to which d

We consider a matching problem in a marriage instance G. Every node has a strict preference order ranking its neighbors. There is a set C of prioritized or critical nodes and we are interested in only those matchings that match as many critical nodes as possible. Such matchings are useful in several applications and we call them critical matchings. A stable matching need not be critical. We consider a well-studied relaxation of stability called popularity. Our goal is to find a popular critical matching, i.e., a weak Condorcet winner within the set of critical matchings where nodes are voters. We show that popular critical matchings always exist in G and min-size/max-size such matchings can be efficiently computed.

We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.

Let G be a bipartite graph where every node has a strict ranking of its neighbors. For any node, its preferences over neighbors extend naturally to preferences over matchings. A maximum matching M in G is a popular max-matching if for any maximum matching N in G, the number of nodes that prefer M is at least the number that prefer N. Popular max-matchings always exist in G and they are relevant in applications where the size of the matching is of higher priority than node preferences. Here we assume there is also a cost function on the edge set. So what we seek is a min-cost popular max-matching. Our main result is that such a matching can be computed in polynomial time. We show a compact extended formulation for the popular max-matching polytope and the algorithmic result follows from this. In contrast, it is known that the popular matching polytope has near-exponential extension complexity and finding a min-cost popular matching is NP-hard.

We design and implement an efficient parallel algorithm for finding a perfect matching in a weighted bipartite graph such that weights on the edges of the matching are large. This problem differs from the maximum weight matching problem, for which scalable approximation algorithms are known. It is primarily motivated by finding good pivots in scalable sparse direct solvers before factorization. Due to the lack of scalable alternatives, distributed solvers use sequential implementations of maximum weight perfect matching algorithms, such as those available in MC64. To overcome this limitation, we propose a fully parallel distributed memory algorithm that first generates a perfect matching and then iteratively improves the weight of the perfect matching by searching for weight-increasing cycles of length 4 in parallel. For most practical problems the weights of the perfect matchings generated by our algorithm are very close to the optimum. An efficient implementation of the algorithm scales up to 256 nodes (17,408 cores) on a Cray XC40 supercomputer and can solve instances that are too large to be handled by a single node using the sequential algorithm.

We consider the maximum cardinality matching problem in bipartite graphs. There are a number of exact, deterministic algorithms for this purpose, whose complexities are high in practice. There are randomized approaches for special classes of bipartite graphs. Random 2-out bipartite graphs, where each vertex chooses two neighbors at random from the other side, form one class for which there is an O(m+nlog n)-time Monte Carlo algorithm. Regular bipartite graphs, where all vertices have the same degree, form another class for which there is an expected O(m + nlog n)-time Las Vegas algorithm. We investigate these two algorithms and turn them into practical heuristics with randomization. Experimental results show that the heuristics are fast and obtain near optimal matchings. They are also more robust than the state of the art heuristics used in the cardinality matching algorithms, and are generally more useful as initialization routines.

Let G = (A ∪ B,E) be a bipartite graph where A consists of agents or main players and B consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M to N is at least the number that prefer N to M. Popular matchings always exist in G since every stable matching is popular. A matching M is A-popular if for any matching N, the number of agents (i.e., vertices in A) that prefer M to N is at least the number of agents that prefer N to M. Unlike popular matchings, A-popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A-popular matching and compute one, if so. We consider the problem of deciding if G admits a matching that is both popular and A-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when A is the more important side - so along with overall popularity, we would like to maintain "popularity within the set A". A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O (log2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the Ω (log n) lower bound. The previous best strategy known achieved amortized O (√n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than the trivial O (n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O (log2 n) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load L, then the SAP protocol makes amortized O (min{L log2 n, √n log n}) reassignments. We also show that this is close to tight, because Ω (min{L, √n}) reassignments can be necessary.

Evaluation of the Jacobian is the most computationally expensive operation while solving a non-linear system. Knowledge of the sparsity pattern in advance reduces the computational cost. Bi-directional partitioning to determine non-zeroes in the sparse matrix works better than unidirectional partitioning for dense rows and dense columns. We have developed a bidirectional coloring algorithm that determines all the non-zeroes of a sparse Jacobian matrix. Our algorithm is inspired by complete direct cover. Several numerical experiments have been carried out on standard data sets. Test results ensure that our proposed algorithm works better than existing algorithms. We have implemented our algorithm using the data structures and partitioning algorithms defined in software tool kit DSJM (Determine Sparse Jacobian Matrices). We have added new procedures in DSJM, which facilitates bi-directional partitioning.

The gonality sequence (dr)r>1 of a smooth algebraic curve comprises the minimal degrees dr of linear systems of rank r. We explain two approaches to compute the gonality sequence of smooth curves in P1 ×P1: a tropical and a classical approach. The tropical approach uses the recently developed Brill–Noether theory on tropical curves and Baker’s specialization of linear systems from curves to metric graphs [1]. The classical one extends the work [12] of Hartshorne on plane curves to curves on P1 × P1, SCOPUS: ar.j, DecretOANoAutActif, info:eu-repo/semantics/published

This is the peer reviewed version of the following article: Coulson, M, Perarnau, G. Rainbow matchings in Dirac bipartite graphs. Random Struct Alg. 2019; 55: 271– 289., which has been published in final form at https://doi.org/10.1002/rsa.20835. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions, We show the existence of rainbow perfect matchings in µn-bounded edge colorings of Dirac bipartite graphs, for a sufficiently small µ¿>¿0. As an application of our results, we obtain several results on the existence of rainbow k-factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree., Postprint (author's final draft)

In this paper, we consider the Hospital Residents problem (HR) and the Hospital Residents problem with Lower Quotas (HRLQ). In this model with two sided preferences, stability is a well accepted notion of optimality. However, in the presence of lower quotas, a stable and feasible matching need not exist. For the HRLQ problem, our goal therefore is to output a good feasible matching assuming that a feasible matching exists. Computing matchings with minimum number of blocking pairs (Min-BP) and minimum number of blocking residents (Min-BR) are known to be NP-Complete. The only approximation algorithms for these problems work under severe restrictions on the preference lists. We present an algorithm which circumvents this restriction and computes a popular matching in the HRLQ instance. We show that on data-sets generated using various generators, our algorithm performs very well in terms of blocking pairs and blocking residents. Yokoi [Yokoi, 2017] recently studied envy-free matchings for the HRLQ problem. We propose a simple modification to Yokoi's algorithm to output a maximal envy-free matching. We observe that popular matchings outperform envy-free matchings on several parameters of practical importance, like size, number of blocking pairs, number of blocking residents. In the absence of lower quotas, that is, in the Hospital Residents (HR) problem, stable matchings are guaranteed to exist. Even in this case, we show that popularity is a practical alternative to stability. For instance, on synthetic data-sets generated using a particular model, as well as on real world data-sets, a popular matching is on an average 8-10% larger in size, matches more number of residents to their top-choice, and more residents prefer the popular matching as compared to a stable matching. Our comprehensive study reveals the practical appeal of popular matchings for the HR and HRLQ problems. To the best of our knowledge, this is the first study on the empirical evaluation of popular m

Our input is a bipartite graph G=(A\cup B,E) where each vertex in A\cup B has a preference list strictly ranking its neighbors. The vertices in A and in B are called students and courses, respectively. Each student a seeks to be matched to cap(a)\geq 1 many courses while each course b seeks cap(b)\geq 1 many students to be matched to it. The Gale-Shapley algorithm computes a pairwise-stable matching (one with no blocking edge) in G in linear time. We consider the problem of computing a popular matching in G - a matching M is popular if M cannot lose an election to any matching where vertices cast votes for one matching versus another. Our main contribution is to show that a max-size popular matching in G can be computed by the 2-level Gale-Shapley algorithm in linear time. This is an extension of the classical Gale-Shapley algorithm and we prove its correctness via linear programming.

Using majorization, we give new bounds for the resolvent energy of general bipartite graphs. We also find more specific lower bounds in the particular case of trees.

Let S ⊂ Z be a set of integers. A graph G is said to have the S-property if there exists an S-edge-weighting w : E(G) → S such that any two adjacent vertices have different sums of incident edge-weights. In this paper we characterise all bridgeless bipartite graphs and all trees without the {0,1}-property. In particular this problem belongs to P for these graphs while it is NP-complete for all graphs.

Peer Reviewed, Postprint (published version)

Studies of ecosystems containing millions of microbial organisms, so called microbiomes, using DNA sequencing became very popular. The main step involves fast clustering of sequences belonging to the same operational taxonomic units (OTU) that represent the same or taxonomically related organisms. This step, known as OTU picking, can be performed by several techniques and significantly affects the final interpretation of the data. In this paper, a novel R/Biconductor package for microbiome data presentation using bipartite graphs is introduced while its benefits are demonstrated during comparison of 3 OTU picking techniques.

We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called Contact Representation of Word Networks (Crown) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Crown is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, Max-Crown, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-complete on bipartite graphs of bounded maximum degree. © 2016, Springer Science+Business Media New York.

Studies of ecosystems containing millions of microbial organisms, so called microbiomes, using DNA sequencing became very popular. The main step involves fast clustering of sequences belonging to the same operational taxonomic units (OTU) that represent the same or taxonomically related organisms. This step, known as OTU picking, can be performed by several techniques and significantly affects the final interpretation of the data. In this paper, a novel R/Biconductor package for microbiome data presentation using bipartite graphs is introduced while its benefits are demonstrated during comparison of 3 OTU picking techniques.

Studies of ecosystems containing millions of microbial organisms, so called microbiomes, using DNA sequencing became very popular. The main step involves fast clustering of sequences belonging to the same operational taxonomic units (OTU) that represent the same or taxonomically related organisms. This step, known as OTU picking, can be performed by several techniques and significantly affects the final interpretation of the data. In this paper, a novel R/Biconductor package for microbiome data presentation using bipartite graphs is introduced while its benefits are demonstrated during comparison of 3 OTU picking techniques.

Studies of ecosystems containing millions of microbial organisms, so called microbiomes, using DNA sequencing became very popular. The main step involves fast clustering of sequences belonging to the same operational taxonomic units (OTU) that represent the same or taxonomically related organisms. This step, known as OTU picking, can be performed by several techniques and significantly affects the final interpretation of the data. In this paper, a novel R/Biconductor package for microbiome data presentation using bipartite graphs is introduced while its benefits are demonstrated during comparison of 3 OTU picking techniques.

En este trabajo de nimos las coloraciones de un grafo que lo resuelven por vecindad. Notamos NL ( G ) al cardinal m nimo de una de estas coloraciones consideradas del grafo G . Estudiamos propiedades generales de estas coloraciones, valores concretos del par ametro NL ( G ) en algunas familias de grafos y cotas ajustadas., Postprint (author's final draft)

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in O(m\sqrt{n}) time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.

The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases., Peer Reviewed, Postprint (published version)

We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2 + ϵ)-approximate maximum matching in general graphs with O(poly(log n, 1/ϵ)) update time. (2) An algorithm that maintains an αk approximation of the value of the maximum matching with O(n2/K) update time in bipartite graphs, for every sufficiently large constant positive integer K. Here, 1 ≤ αk ≤ 2 is a constant determined by the value of K. Result (1) is the first deterministic algorithm that can maintain an o(log n)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph., QC 20161207

Let k=2 be an integer. Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k-1 contains k vertex-disjoint cycles. Recently Bai, Li and Li proved this conjecture for bipartite digraphs. In this paper we prove that every bipartite tournament with minimum out-degree at least 2k-2, minimum in-degree at least 1 and partite sets of cardinality at least 2k contains k vertex-disjoint 4-cycles whenever k=3. Finally, we show that every bipartite tournament with minimum degree d=min(d+,d-) at least 1.5k-1 contains at least k vertex-disjoint 4-cycles., Peer Reviewed, Postprint (author's final draft)

In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M' to M outnumber those that prefer M to M'. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M_1 + M_2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time.

The incidence matrix of a simple undirected graph is used as an information table. Then, rough set notions are applied to it: approximations, membership function, positive region and discernibility matrix. The particular cases of complete and bipartite graphs are analyzed. The symmetry induced in graphs by the indiscernibility relation is studied and a new concept of generalized discernibility matrix is introduced.

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain (q + 1)-regular graphs of girth 7 and order 2q(3) + q(2) + 2q for each even prime power q >= 4, and of order 2q(3) + 2q(2) q + 1 for each odd prime power q >= 5., Postprint (published version)

We analyze the Odd/odd vertex removal game introduced by P. Ottaway. We prove that every bipartite graph has Grundy value 0 or 1 only depending on the parity of the number of edges in the graph, which is a generalization of a conjecture of K. Shelton.

We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called Contact Representation of Word Networks (Crown) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Crown is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, Max-Crown, in which realizing each desired adjacency yields a certain profit. We show that the problem is APX-complete on bipartite graphs of bounded maximum degree. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we consider several planar graph classes (stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. © 2014 Springer-Verlag Berlin Heidelberg.

A dominating set S of a graph G is a locating-dominating-set, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S . Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number, ¿(G)¿(G). An LD-set S of a graph G is global if it is an LD-set for both G and its complement, View the MathML sourceG¯. One of the main contributions of this work is the definition of the LD-graph, an edge-labeled graph associated to an LD-set, that will be very helpful to deduce some properties of location-domination in graphs. Concretely, we use LD-graphs to study the relation between the location-domination number in a bipartite graph and its complement., Peer Reviewed, Postprint (published version)

Alon 和Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) 證明了如果一個圖是ε -非二部圖，那麼階數為Ỡ(1/ε) 的隨機導出于圖以很大概率是非二部圖。我們進一步猜想，這個導出子圖以很大概率是Ω(ε)-非二部圖。Gonen 和Ron (RANDOM 2007) 證明了當圖的最大度不超過O (εn )時猜想成立。我們將對更一般的情形給出證明，對於任意d，所有d 正則(或幾乎d 正則)的圖猜想成立。, 假設猜想成立的情況下，我們證明二分屬性是可以被單側誤差在O(1/ε^c ) 時間內檢驗的，其中c 是一個嚴格小於2 的常數，而這個結果也改進了Alon 和Krivelevich 的檢驗算法。由於己知對二分屬性的非適應性的檢驗算法需要Ω(1 /ε²) 的複雜性(Bogdanov 和Trevisan ， CCC 2004) ，我們的結果也得出，假設猜想成立，適應性對檢驗二分屬性是有幫助的。, 另外一個有很多屬性檢驗問題被廣泛研究的領域是布爾函數。對布爾函數單調性的檢驗也是屬性檢驗的經典問題。給定對布爾函數f: {0,1}{U+207F} → {0,1} 的訪問，在[18]中，證明了檢驗算法複雜性的下界是Ω(√n) 。另一方面，在[21]中，作者們構造了一個複雜性為O(n) 的算法。在本文中，我們刻畫一些單調布爾函數的本質，設計算法并分析其對於一些困難例子的效果。最近，在[14] 中， 一個類似的算法被證明是非適應性，單側誤差，複雜性為Ỡ (n⁵/⁶ ε⁻⁵/³) 的算法。, Alon and Krivelevich (SIAM J. Discrete Math. 15(2): 211-227 (2002)) show that if a graph is ε-far from bipartite, then the subgraph induced by a random subset of Ỡ (1/ε) vertices is not bipartite with high probability. We conjecture that the induced subgraph is Ω(ε)-far from bipartite with high probability. Gonen and Ron (RANDOM 2007) proved this conjecture in the case when the degrees of all vertices are at most O(εn). We give a more general proof that works for any d-regular (or almost d-regular) graph for arbitrary degree d., Assuming this conjecture, we prove that bipartiteness is testable with one-sided error in time O(1=ε{U+1D9C}), where c is a constant strictly smaller than two, improving upon the tester of Alon and Krivelevich. As it is known that non-adaptive testers for bipartiteness require Ω(1/ε²) queries (Bogdanov and Trevisan, CCC2004), our result shows, assuming the conjecture, that adaptivity helps in testing bipartiteness., The other area in which various properties have been well studied is boolean function. Testing monotonicity of Boolean functions is a classical question in property testing. Given oracle access to a Boolean function f : {0, 1}{U+207F} →{0, 1}, in [18], it is shown a lower bound of testing is Ω(√n). On the other hand, in [21], the authors introduced an algorithm to test monotonicity using O(n) queries. In this paper, we characterize some nature of monotone functions, design a tester and analyze the performance on some generalizations of the hard case. Recently, in [14], a similar tester is shown to be a non-adaptive, one-sided error tester making Ỡ (n⁵/⁶ ε⁻⁵/³) queries., Detailed summary in vernacular field only., Li, Fan., Thesis (M.Phil.)--Chinese University of Hong Kong, 2013., Includes bibliographical references (leaves 76-79)., s also in Chinese., p.i, Chapter 1 --- Introduction --- p.1, Chapter 1.1 --- Property Testing --- p.1, Chapter 1.2 --- Testing Bipartiteness --- p.4, Chapter 1.3 --- Testing Monotonicity --- p.7, Chapter 2 --- Testing Bipartiteness --- p.11, Chapter 2.1 --- Background --- p.11, Chapter 2.1.1 --- Our result --- p.11, Chapter 2.1.2 --- The algorithms of Gonen and Ron --- p.13, Chapter 2.1.3 --- Our proof --- p.16, Chapter 2.1.4 --- Notation --- p.19, Chapter 2.2 --- Splitting the vertices by degree --- p.19, Chapter 2.3 --- The algorithm for high degree vertices --- p.20, Chapter 2.4 --- Eliminating the high degree vertices --- p.22, Chapter 2.5 --- From an XOR game to a bipartite graph --- p.33, Chapter 2.6 --- Proof of the main theorem --- p.35, Chapter 2.7 --- Proof of the conjecture for regular graphs --- p.37, Chapter 3 --- Testing Monotonicity --- p.40, Chapter 3.1 --- Towards an improved tester --- p.40, Chapter 3.1.1 --- Properties of Distribution D --- p.42, Chapter 3.1.2 --- An Alternative Representation of D --- p.46, Chapter 3.1.3 --- Performance of D on Decreasing Functions --- p.51, Chapter 3.1.4 --- Functions Containing Ω(2{U+207F}) Disjoint Violating Edges --- p.54, Chapter 3.2 --- A o(n) Monotonicity Tester [14] and Some Improvements --- p.62, Chapter 3.2.1 --- A o(n) Monotonicity Tester [14] --- p.62, Chapter 3.2.2 --- An Alternative Proof of Theorem 3.2.2 --- p.65, Chapter 4 --- Other Related Results --- p.67, Chapter 4.1 --- Short Odd Cycles in Graphs that are Far From Bipartiteness --- p.67, Chapter 4.2 --- Fourier Analysis on Boolean Functions --- p.69, Bibliography --- p.76, http://library.cuhk.edu.hk/record=b5549331, Use of this resource is governed by the terms and conditions of the Creative Commons “Attribution-NonCommercial-NoDerivatives 4.0 International” License (http://creativecommons.org/licenses/by-nc-nd/4.0/)

We describe some new conceptual tools for the rigorous, mathematical description of the “set-complexity” of graphs. This set-complexity has been shown previously to be a useful measure for analyzing some biological networks, and in discussing biological information in a quantitative fashion. The advances described here allow us to define some significant relationships between the set-complexity measure and the structure of graphs, and of their component sub-graphs. We show here that modular graph structures tend to maximize the set-complexity of graphs. We point out the relationship between modularity and redundancy, and discuss the significance of set-complexity in this regard. We specifically discuss the relationship between complexity and entropy in the case of complete-bipartite graphs, and present a new method for constructing highly complex, binary graphs. These results can be extended to the case of ternary graphs, and to other multi-edge graphs, which are fundamentally more relevant to biological structures and systems. Finally, our results lead us to an approach for extracting high complexity modular graphs from large, noisy graphs with low information content. We illustrate this approach with two examples.

We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes that are optimal with respect to the bouns. We prove that BIBD-graphs are characterized by their eigenvalues. Finally we present a new bound on the expansion coefficient of (c,d)-regular bipartite graphs and compare that with aclassical bound.