Matching Edges Research Articles

The Largest Crossing Number of Tanglegrams

A tanglegram $\mathcal{T}$ consists of two rooted binary trees with the same number of leaves, and a perfect matching between the two leaf sets. In a layout, the tanglegrams is drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines are drawn as plane trees, and the perfect matching is drawn in straight line segments inside the strip. The tanglegram crossing number ${\rm cr}({\mathcal{T}})$ of $\mathcal{T}$ is the smallest number of crossings of pairs of matching edges, over all possible layouts of $\mathcal{T}$. The size of the tanglegram is the number of matching edges, say $n$. An earlier paper showed that the maximum of the tanglegram crossing number of size $n$ tanglegrams is $<\frac{1}{2}\binom{n}{2}$; but is at least $\frac{1}{2}\binom{n}{2}-\frac{n^{3/2}-n}{2}$ for infinitely many $n$. Now we make better bounds: the maximum crossing number of a size $n$ tanglegram is at most $\frac{1}{2}\binom{n}{2}-\frac{n}{4}$, but for infinitely many $n$, at least $\frac{1}{2}\binom{n}{2}-\frac{n\log_2 n}{4}$. The problem shows analogy with the Unbalancing Lights Problem of Gale and Berlekamp.

Fabricable Multi‐Scale Wang Tiles

AbstractWang tiles, also known as Wang dominoes, are a jigsaw puzzle system with matching edges. Due to their compactness and expressiveness in representing variations, they have become a popular tool in the procedural synthesis of textures, height fields, 3D printing and representing other large and non‐repetitive data. Multi‐scale tiles created from low‐level tiles allow for a higher tiling efficiency, although they face the problem of combinatorial explosion. In this paper, we propose a generation method for multi‐scale Wang tiles that aims at minimizing the amount of needed tiles while still resembling a tiling appearance similar to low‐level tiles. Based on a set of representative multi‐scale Wang tiles, we use a dynamic generation algorithm for this purpose. Our method can be used for rapid texture synthesis and image halftoning. Respecting physical constraints, our tiles are connected, lightweight, independent of the fabrication scale, able to tile larger areas with image contents and contribute to “mass customization”.

Social Group Query Based on Multi-Fuzzy-Constrained Strong Simulation

Traditional social group analysis mostly uses interaction models, event models, or other social network analysis methods to identify and distinguish groups. This type of method can divide social participants into different groups based on their geographic location, social relationships, and/or related events. However, in some applications, it is necessary to make more specific restrictions on the members and the interactions between members of the group. Generally, Graph Pattern Matching (GPM) technique is used to solve this problem. However, the existing GPM methods rarely consider the rich contextual information of nodes and edges to measure the credibility between members. In this article, first, a social group query problem that needs to consider the trust between members of the group is proposed. Then, to solve this problem, a multi-fuzzy-constrained strong simulation matching model is proposed based on multi-constrained simulation, and a Strong Simulation GPM algorithm (NTSS) based on the exploration of pattern Node Topological ordered sequence is proposed. Aiming at the inefficiency of the NTSS algorithm when pattern graph with multiple nodes with zero in-degree and the problem of repeated calculation of matching edges shared by multiple matching subgraphs, two optimization strategies are proposed. Finally, we conduct verification experiments on the effectiveness and efficiency of the NTSS algorithm and the algorithms with the optimization strategies on four social network datasets in real applications. Experimental results show that the NTSS algorithm is significantly better than the existing multi-constrained GPM algorithm, and the NTSS_Inv_EdgC algorithm, which combines two optimization strategies, greatly improves the efficiency of the NTSS algorithm.

The bipartite distance matrix of a nonsingular tree

Let G be a labeled, connected, bipartite graph with the bi-partition (L={l1,…,lk},R={r1,…,rp}) of the vertex set V. Let D be the usual distance matrix of G, where rows and columns are indexed by l1,…,lk,r1,…,rp. For X,Y⊆V, let us define DG[X,Y] as the submatrix of D induced by the rows indexed in X and columns indexed in Y. Let us call DG[L,R] the bipartite distance matrix of G. If G has a unique perfect matching, then k=p and we assume that the bi-partition is canonical, that is, [li,ri] are matching edges. For a nonsingular tree T, let us denote the bipartite distance matrix of T by B(T).We observe that det⁡B(T) is always a multiple of 2p−1. This is similar to the well known result of Graham and Pollak (1971) [1] which tells that the determinant of the usual distance matrix D is a multiple of 2n−2. Call the number bd(T):=det⁡B(T)/(−2)p−1 the bipartite distance index of T. We supply a recursive formula to compute this index. We show that this index satisfies an interesting inclusion-exclusion type of principle at any matching edge of the tree. Even more interestingly, we show that the index is completely characterized by the structure of T via what we call the f-alternating sums, that is, the sum f(T):=∑[d(u)−2][d(v)−2]S|Puv|/2, where the sum is taken over all u-v-alternating paths Puv, and S is the sequence (1,1,3,3,5,5,…).A well known result by Graham, Hoffman and Hosoya (1977) [2] is that the determinant of the distance matrix of a graph only depends on the blocks and is independent of how they are assembled. Such a result does not hold true for B(T). However, we identify some basic elements and a merging operation and show that each of the trees that can be constructed from a given set of elements, sequentially using this operation, have the same det⁡B(T), independent of the order in which the sequence is followed. For the class of trees that can be obtained in this way, we give a surprisingly simple way to evaluate the determinant of B(T).

Strongly Stable and Maximum Weakly Stable Noncrossing Matchings

In IWOCA 2019, Ruangwises and Itoh introduced stable noncrossing matchings, where participants of each side are aligned on each of two parallel lines, and no two matching edges are allowed to cross each other. They defined two stability notions, strongly stable noncrossing matching (SSNM) and weakly stable noncrossing matching (WSNM), depending on the strength of blocking pairs. They proved that a WSNM always exists and presented an O(n^{2})-time algorithm to find one for an instance with n men and n women. They also posed open questions of the complexities of determining existence of an SSNM and finding a largest WSNM. In this paper, we show that both problems are solvable in polynomial time. Our algorithms are applicable to extensions where preference lists may include ties, except for one case which we show to be NP-complete. This NP-completeness holds even if each person's preference list is of length at most two and ties appear in only men's preference lists. To complement this intractability, we show that the problem is solvable in polynomial time if the length of preference lists of one side is bounded by one (but that of the other side is unbounded).

Online Dynamic B-Matching

This paper initiates the study of online algorithms for the maximum weight b-matching problem, a generalization of maximum weight matching where each node has at most b≥1 adjacent matching edges. The problem is motivated by emerging optical technologies which allow to enhance datacenter networks with reconfigurable matchings, providing direct connectivity between frequently communicating racks. These additional links may improve network performance, by leveraging spatial and temporal structure in the workload. We show that the underlying algorithmic problem features an intriguing connection to online paging (a.k.a. caching), but introduces a novel challenge. Our main contribution is an online algorithm which is O(b)- competitive; we also prove that this is asymptotically optimal. We complement our theoretical results with extensive trace-driven simulations, based on real-world datacenter workloads as well as synthetic traffic traces.

On flips in planar matchings

In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of 2n points that are placed equidistantly on the unit circle. A flip operation on such a matching replaces two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, and the flip is called centered if the quadrilateral contains the center of the unit circle. The graph Gn has those matchings as vertices, and an edge between any two matchings that differ in a flip, and it is known to have many interesting properties. In this paper we focus on the spanning subgraph Hn of Gn obtained by taking all edges that correspond to centered flips, omitting edges that correspond to non-centered flips. We show that the graph Hn is connected for odd n, but has exponentially many small connected components for even n, which we characterize and count via Catalan and generalized Narayana numbers. For odd n, we also prove that the diameter of Hn is linear in n. Furthermore, we determine the minimum and maximum degrees of Hn for all n, and characterize and count the corresponding vertices. Our results imply the non-existence of certain rainbow cycles in Gn, and they resolve several open questions and conjectures raised in a recent paper by Felsner, Kleist, Mütze, and Sering.

Strongly self-inverse weighted graphs

Let G be a connected, bipartite graph. Let Gw denote the weighted graph obtained from G by assigning weights to its edges using the positive weight function w : E(G) ! (0;1). In this article we consider a class Hnmc of bipartite graphswith unique perfect matchings and the family WG of weight functions with weight 1 on the matching edges, and characterize all pairs G in Hnmc and w in WG such that Gw is strongly self-inverse.

High-Performance Vision-Based Navigation on SoC FPGA for Spacecraft Proximity Operations

Future autonomous spacecraft rendezvous with uncooperative or unprepared objects will be enabled by vision-based navigation, which imposes great computational challenges. Targeting short duration missions in low Earth orbit, this paper develops high-performance avionics supporting custom computer vision algorithms of increased complexity for satellite pose tracking. At algorithmic level, we track 6D pose by rendering a depth image from an object mesh model and robustly matching edges detected in the depth and intensity images. At system level, we devise an architecture to exploit the structure of commercial system-on-chip FPGAs, i.e., Zynq7000, and the benefits of tightly coupling VHDL accelerators with CPU-based functions. At implementation level, we employ our custom HW/SW co-design methodology and an elaborate combination of digital circuit design techniques to optimize and map efficiently all functions to a compact embedded device. Providing significant performance per watt improvement, the resulting VBN system achieves a throughput of 10-14 FPS for 1 Mpixel images, with only 4.3 watts mean power and 1U size, while tracking ENVISAT in real-time with only 0.5% mean positional error.

Analogies between the Crossing Number and the Tangle Crossing Number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straight-line drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum number of crossings over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts.Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $\gamma(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}$. For an arbitrary tanglegram $T$, the tangle crossing number, $\mathrm{crt}(T)$, is NP-hard to compute (Fernau et al. 2005). We provide an algorithm which lower bounds $\mathrm{crt}(T)$ and runs in $O(n^4)$ time. To demonstrate the strength of the algorithm, simulations on tanglegrams chosen uniformly at random suggest that the tangle crossing number is at least $0.055n^2$ with high probabilty, which matches the result that the tangle crossing number is $\Theta(n^2)$ with high probability (Czabarka et al. 2017).

Versatile and accurate schemes of discretization for the electromagnetic scattering analysis of arbitrarily shaped piecewise homogeneous objects

The discretization by the method of moments (MoM) of integral equations in the electromagnetic scattering analysis most often relies on divergence-conforming basis functions, such as the Rao–Wilton–Glisson (RWG) set, which preserve the normal continuity of the expanded currents across the edges arising from the discretization of the target boundary. Although for such schemes the boundary integrals become free from hypersingular kernel-contributions, which is numerically advantageous, their practical implementation in real-life scenarios becomes particularly cumbersome. Indeed, the application of the normal continuity condition on composite objects becomes elaborate and convoluted at junction-edges, where several regions intersect. Also, such edge-based schemes cannot even be applied to nonconformal meshes, where adjacent facets may not share single matching edges. In this paper, we present nonconforming schemes of discretization for the scattering analysis of complex objects based on the expansion of the boundary unknowns, electric or magnetic currents, with the facet-based monopolar-RWG set. We show with examples how these schemes exhibit great flexibility when handling composite piecewise homogeneous objects with junctions or targets modeled with nonconformal meshes. Furthermore, these schemes offer improved near- and far-field accuracy in the scattering analysis of electrically small single sharp-edged dielectric targets with moderate or high dielectric contrasts.

WANG TILES LOCAL TILINGS CREATED BY MERGING EDGE-COMPATIBLE FINITE ELEMENT MESHES

In this contribution, we present the concept of Wang Tiles as a surrogate of the periodic unit cell method (PUC) for modelling of materials with heterogeneous microstructures and for synthesis of micro-mechanical fields. The concept is based on a set of specifically designed cells that compresses the stochastic microstructure into a small set of statistical volume elements – tiles. Tiles are placed side by side according to matching edges like in a game of domino. Opposite to the repeating pattern of PUC the Wang Tiles method with the stochastic tiling algorithm preserves the randomness for reconstructed microstructures. The same process is applied to obtain the micro-mechanical response of domains where the evaluation as one piece would be time consuming. Therefore the micro-mechanical quantities are evaluated only on tiles (with surrounding layers of tiles of each addressed tile included into the evaluation) and then synthesized to the micro-mechanical field of whole domain.

On some graphs which satisfy reciprocal eigenvalue properties

We consider only simple graphs. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. On the class Hg of such graphs G the equivalence of the following statements are known.i)The reciprocal of the spectral radius of the adjacency matrix A(G) is the least positive eigenvalue of the adjacency matrix.ii)The graph is isomorphic to its inverse, where the inverse of a graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s.iii)The graph has the reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G).iv)The graph has the strong reciprocal eigenvalue property, that is, the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities.v)The graph is a corona graph, that is, it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex. It is known that the statements iii) and iv) are not equivalent, if we allow nonbipartite graphs. However, the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings, where we relax the condition of bipartiteness of the contracted graph.In this article, we establish the equivalence of the first four statements for a class larger than Hg. We then supply a constructive structural description of those graphs.

Inverses of bicyclic graphs

A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite. In [C.D. Godsil. Inverses of trees. Combinatorica, 5(1):33–39, 1985.], Godsil proved that such graphs are invertible. He posed the question of characterizing the bipartite graphs with unique perfect matchings possessing inverses. In this article, Godsil’s question for the class of bicyclic graphs is answered.

Matchings with Lower Quotas: Algorithms and Complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G= (A, dot{cup }, P, E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between textsf {NP}-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota u_{max } as basis, and we prove that this dependence is necessary unless textsf {FPT}= textsf {W}[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee u_{max }+1, which is asymptotically best possible unless textsf {P}= textsf {NP}. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas.

Synchronizing Files From a Large Number of Insertions and Deletions

Developing efficient algorithms to synchronize between different versions of files is an important problem with numerous applications. We consider the interactive synchronization protocol introduced by Yazdi and Dolecek, based on an earlier synchronization algorithm by Venkataramanan et al. Unlike preceding synchronization algorithms, Yazdi and Dolecek’s algorithm is specifically designed to handle a number of deletions linear in the length of the file. We extend this algorithm in three ways. First, we handle nonbinary files. Second, these files contain symbols chosen according to nonuniform distributions. Finally, the files are modified by both insertions and deletions. We take into consideration the collision entropy of the source and refine the matching graph developed by Yazdi and Dolecek by appropriately placing weights on the matching graph edges. We compare our protocol with the widely used synchronization software rsync, and with the synchronization protocol by Venkataramanan et al. In addition, we provide tradeoffs between the number of rounds of communication and the total amount of bandwidth required to synchronize the two files under various implementation choices of the baseline algorithm. Finally, we show the robustness of the protocol under imperfect knowledge of the properties of the edit channel, which is the expected scenario in practice.

Maximum Matchings of a Digraph Based on the Largest Geometric Multiplicity

Matching theory is one of the most forefront issues of graph theory. Based on the largest geometric multiplicity, we develop an efficient approach to identify maximum matchings in a digraph. For a given digraph, it has been proved that the number of maximum matched nodes has close relationship with the largest geometric multiplicity of the transpose of the adjacency matrix. Moreover, through fundamental column transformations, we can obtain the matched nodes and related matching edges. In particular, when a digraph contains a cycle factor, the largest geometric multiplicity is equal to one. In this case, the maximum matching is a perfect matching and each node in the digraph is a matched node. The method is validated by an example.

Near optimal algorithms for online maximum edge-weighted b-matching and two-sided vertex-weighted b-matching

This paper studies the online maximum edge-weighted b-matching problem. The input of the problem is a weighted bipartite graph G=(L,R,E,w). Vertices in R arrive online, and each vertex in L can be matched to at most b vertices in R. The objective is to maximize the total weight of the matching edges. We give a randomized algorithm Greedy-RT for this problem, and show that its competitive ratio is Ω(1∏j=1log⁎⁡wmax−1log(j)⁡wmax) where wmax is an upper bound on the edge weights, which may not be known ahead of time. We can improve the competitive ratio to Ω(1log⁡wmax) if wmax is known to the algorithm when it starts. We also derive an upper bound O(1log⁡wmax) on the competitive ratio, suggesting that Greedy-RT is near optimal. We also consider deterministic algorithms; we present a near optimal algorithm Greedy-D which has competitive ratio 11+2ξ(wmax+1)1ξ, where ξ=min⁡{b,⌈ln⁡(1+wmax)⌉}.We also study a variant of the problem called online maximum two-sided vertex-weighted b-matching problem, and give a modification of the randomized algorithm Greedy-RT called Greedy-vRT for this variant. We show that the competitive ratio of Greedy-vRT is also near optimal.

A Guide-and-Observe Hyper-Heuristic Approach to the Eternity II Puzzle

The present paper considers the optimisation version of the Eternity II puzzle problem and unsigned edge matching puzzles in general. The goal of this optimisation problem is to maximise the number of matching edges in the puzzle. In 2010, the META Eternity II contest awarded the best performing metaheuristic approach to this hard combinatorial optimisation problem. The winning hyper-heuristic of the contest is subject of this paper. Heuristic design decisions are motivated based on the results of extensive experiments. Furthermore, new results for the Eternity II puzzle problem are presented. The main contribution of this paper is the description of a novel guide-and-observe search mechanism combining a set of objectives. The approach significantly outperforms search methods guided by the default objective only.

Bipartite Matching in the Semi-streaming Model

We present the first deterministic 1+ε approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O((1/ε)5) passes over the input stream. In this model, the input graph G=(V,E) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O(npolylog n) bits, where \(n = \lvert {V}\rvert \). The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005), which uses Ω((1/ε)1/ε) passes. We show that even for bipartite graphs, McGregor’s algorithm needs Ω(1/ε)Ω(1/ε) passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization.We repeatedly grow an initial matching using augmenting paths up to a length of 2k+1 for k=⌈2/ε⌉. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k, that guarantees a 1+ε approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits: when a matching edge is the ith matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i. This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand.