Graph Decomposition Research Articles

Image segmentation algorithm based on improved fuzzy clustering

Fuzzy clustering algorithm is the main method of image segmentation, but it can’t be widely used in various fields. Therefore, an image segmentation algorithm based on improved fuzzy clustering was proposed in this paper. The fuzzy clustering theory and analysis method were described in detail, and the research of image segmentation algorithm was discussed in this paper. And then the method of sub graph decomposition and region merging was used to improve the clustering method of fuzzy C mean clustering image segmentation algorithm and the algorithm was verified by an example. The results showed that the algorithm was feasible. Compared with other existing algorithms, the algorithm had more advantages in running time and image segmentation accuracy.

Counting designs

We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed method of Randomised Algebraic Construction: we give a simpler proof of a special case of our result on clique decompositions of hypergraphs, namely triangle decompositions of quasirandom graphs.

Graph congruences and what they connote

The algebraic notion of a “congruence” seems to be foreign to contemporary graph theory. We propound that it need not be so by developing a theory of congruences of graphs: a congruence on a graph G = (V, E) being a pair (∼, ) of which ∼ is an equivalence relation on V and is a set of unordered pairs of vertices of G with a special relationship to ∼ and E. Kernels and quotient structures are used in this theory to develop homomorphism and isomorphism theorems which remind one of similar results in an algebraic context. We show that this theory can be applied to deliver structural decompositions of graphs into “factor” graphs having very special properties, such as the result that each graph, except one, is a subdirect product of graphs with universal vertices. In a final section, we discuss corresponding concepts and briefly describe a corresponding theory for graphs which have a loop at every vertex and which we call loopy graphs. They are in a sense more “algebraic” than simple graphs, with their meet-semilattices of all congruences becoming complete algebraic lattices.

Eulerian Cycle Decomposition Conjecture for the line graph of complete graphs

The Eulerian Cycle Decomposition Conjecture, by Chartrand, Jordon and Zhang, states that if the minimum number of odd cycles in a cycle decomposition of an Eulerian graph G of size m is a, the maximum number of odd cycles in such a cycle decomposition is b and ℓ is an integer such that a≤ℓ≤b where ℓ and m are of the same parity, then there is a cycle decomposition of G with exactly ℓ odd cycles. This conjecture is verified for the line graph of the complete graph.

Linear optimal power flow using cycle flows

Linear optimal power flow (LOPF) algorithms use a linearization of the alternating current (AC) load flow equations to optimize generator dispatch in a network subject to the loading constraints of the network branches. Common algorithms use the voltage angles at the buses as optimization variables, but alternatives can be computationally advantageous. In this article we provide a review of existing methods and describe a new formulation that expresses the loading constraints directly in terms of the flows themselves, using a decomposition of the network graph into a spanning tree and closed cycles. We provide a comprehensive study of the computational performance of the various formulations, in settings that include computationally challenging applications such as multi-period LOPF with storage dispatch and generation capacity expansion. We show that the new formulation of the LOPF solves up to 7 times faster than the angle formulation using a commercial linear programming solver, while another existing cycle-base formulation solves up to 20 times faster, with an average speed-up of factor 3 for the standard networks considered here. If generation capacities are also optimized, the average speed-up rises to a factor of 12, reaching up to factor 213 in a particular instance. The speed-up is largest for networks with many buses and decentral generators throughout the network, which is highly relevant given the rise of distributed renewable generation and the computational challenge of operation and planning in such networks.

The Sandwich Problem for Decompositions and Almost Monotone Properties

We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in $${\mathcal {C}}$$ as an induced subgraph is almost monotone if $${\mathcal {C}}$$ is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length $$\equiv j \mod n$$ is almost monotone if and only if $$j \equiv 2 \mod n$$ or $$n \le 2$$ . Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.

Cograph generation with linear delay

Cographs have always been a research target in areas such as coloring, graph decomposition, and spectral theory. In this work, we present an algorithm to generate all unlabeled cographs with n vertices, based on the generation of cotrees. The delay of our algorithm (time spent between two consecutive outputs) is O(n). The time needed to generate the first output is also O(n), which gives an overall O(nMn) time complexity, where Mn is the number of unlabeled cographs with n vertices. The algorithm avoids the generation of duplicates (isomorphic outputs) and produces, as a by-product, a linear ordering of unlabeled cographs with n vertices.

On locally irregular decompositions of subcubic graphs

A graph G is locally irregular if every two adjacent vertices of G have different degrees. A locally irregular decomposition of G is a partition E_1, ..., E_k of E(G) such that each G[E_i] is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przybylo and Wozniak that they decompose into at most 3 locally irregular graphs. Towards that conjecture, it was recently proved by Bensmail, Merker and Thomassen that every decomposable graph decomposes into at most 328 locally irregular graphs. We here focus on locally irregular decompositions of subcubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are subcubic. As a main result, we prove that decomposable subcubic graphs decompose into at most 5 locally irregular graphs, and only 4 when the maximum average degree is less than 12/5. We then consider weaker decompositions, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for subcubic graphs.

Model Decomposition of Timed Event Graphs under Partial Synchronization in Dioids

Timed Event Graphs (TEGs) are a graphical model for decision free and time-invariant Discrete Event Systems (DESs). To express systems with time-variant behaviors, a new form of synchronization, called partial synchronization (PS), has been introduced for TEGs. Unlike exact synchronization, where two transitions t1, t2 can only fire if both transitions are simultaneously enabled, PS of transition t1 by transition t2 means that t1 can fire only when transition t2 fires, but t1 does not influence the firing of t2. Under some assumptions, we can show that the dynamic behavior of a TEG under PS can be decomposed into a time-variant and a time-invariant part. The time-invariant part can be interpreted as a standard TEG. Moreover, it is shown that the tools introduced for standard TEGs can be used to analyze the overall system.

Decompositions of balanced complete bipartite graphs into suns and stars

Decompositions of balanced complete bipartite graphs into suns and stars

Applications of Ear Decomposition to Efficient Heterogeneous Algorithms for Shortest Path/Cycle Problems

Graph algorithms play an important role in several fields of sciences and engineering. Prominent among them are the All-Pairs-Shortest-Paths ( APSP ) and related problems. Indeed there are several efficient implementations for such problems on a variety of modern multi- and many-core architectures. It can be noticed that for several graph problems, parallelism offers only a limited success as current parallel architectures have severe short-comings when deployed for most graph algorithms. At the same time, some of these graphs exhibit clear structural properties due to their sparsity. This calls for particular solution strategies aimed at scalable processing of large, sparse graphs on modern parallel architectures. In this paper, we study the applicability of an ear decomposition of graphs to problems such as all-pairs-shortest-paths and minimum cost cycle basis. Through experimentation, we show that the resulting solutions are scalable in terms of both memory usage and also their speedup over best known current implementations. We believe that our techniques have the potential to be relevant for designing scalable solutions for other computations on large sparse graphs.

Corrigendum: Orthogonal Tree Decompositions of Graphs

The following is a corrigendum to [Orthogonal tree decompositions of graphs, SIAM J. Discrete Math., 32 (2018), pp. 839--863].

Hierarchical modeling of graphs using modular decomposition

Hierarchical modeling of graphs using modular decomposition

Orthogonal Tree Decompositions of Graphs

This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the following classes has a tree decomposition and a linear-sized path decomposition with bounded intersections: (1) every proper minor-closed class, (2) string graphs with a linear number of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface. Here `linear size' means that the total size of the bags in the path decomposition is $O(n)$ for $n$-vertex graphs. We then show that every $n$-vertex graph that has a tree decomposition and a linear-sized path decomposition with bounded intersections has $O(\sqrt{n})$ treewidth. As a corollary, we conclude a new lower bound on the crossing number of a graph in terms of its treewidth. Finally, we consider graph classes that have two path decompositions with bounded intersections. Trees and outerplanar graphs have this property. But for the next most simple class, series parallel graphs, we show that no such result holds.

The Decomposition Problem for the Set of Paths in a Directed Graph and Its Application

We consider the problem of decomposing the set of paths in a directed graph and its application to reducing the dimension of an applied problem on the assignment and transportation of locomotives. On a given set of paths and a set of strongly connected subgraphs, we define a special table. To solve the graph decomposition problem, we develop a heuristic algorithm based on the idea of quicksorting the constructed table. We estimate of the complexity of the resulting algorithm. The obtained results were used to reduce the dimension of the above-mentioned applied problem. We also show the results of computational experiments.

On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs

Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition’s width. Following the work of Allender et al. [5], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [18], we prove that the question is closely related to a conjecture that the L ongest C ommon S ubsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we extend the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth . We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition’s depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.

Deep optimization for spectrum repacking

Over 13 months in 2016--17 the U.S. Federal Communications Commission conducted an "incentive auction" to repurpose radio spectrum from broadcast television to wireless internet. In the end, the auction yielded $19.8 bn, $10.05 bn of which was paid to 175 broadcasters for voluntarily relinquishing their licenses across 14 Ultra High Frequency (UHF) channels. Stations that continued broadcasting were assigned potentially new channels to fit as densely as possible into the channels that remained. The government netted more than $7 bn (used to pay down the national debt) after covering costs (including retuning). A crucial element of the auction design was the construction of a solver, dubbed SAT-based Feasibility Checker (SATFC), that determined whether sets of stations could be "repacked" in this way; it needed to run every time a station was given a price quote. This paper describes the process by which we built SATFC. We adopted an approach we dub "deep optimization," taking a data-driven, highly parametric, and computationally intensive approach to solver design. More specifically, to build SATFC we designed software that could pair both complete and local-search SAT-encoded feasibility checking with a wide range of domain-specific techniques, such as constraint graph decomposition and novel caching mechanisms that allow for reuse of partial solutions from related, solved problems. We then used automatic algorithm configuration techniques to construct a portfolio of 8 complementary algorithms to be run in parallel, aiming to achieve good performance on instances that arose in proprietary auction simulations. To evaluate the impact of our solver in this paper, we built an open-source reverse auction simulator. We found that within the short time budget required in practice, SATFC solved more than 95% of the problems it encountered. Furthermore, the incentive auction paired with SATFC produced nearly optimal allocations in a restricted setting and substantially outperformed other alternatives at national scale.

Decomposition techniques applied to the Clique-Stable set separation problem

In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K∩S=∅, there exists a cut (W,W′) in C such that K⊆W and S⊆W′. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis (Yannakakis, 1991) asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive and negative results related to this question were given recently. Here we show how graph decomposition can be used to prove that a class of graphs admits a polynomial CS-separator. We apply this method to apple-free graphs and cap-free graphs.

Minimal path decomposition of complete bipartite graphs

This paper deals with the subject of minimal path decomposition of complete bipartite graphs. A path decomposition of a graph is a decomposition of it into simple paths such that every edge appears in exactly one path. If the number of paths is the minimum possible, the path decomposition is called minimal. Algorithms that derive such decompositions are presented, along with their proof of correctness, for the three out of the four possible cases of a complete bipartite graph.

Accurate assembly of transcripts through phase-preserving graph decomposition.

We introduce Scallop, an accurate reference-based transcript assembler that improves reconstruction of multi-exon and lowly expressed transcripts. Scallop preserves long-range phasing paths extracted from reads, while producing a parsimonious set of transcripts and minimizing coverage deviation. On 10 human RNA-seq samples, Scallop produces 34.5% and 36.3% more correct multi-exon transcripts than StringTie and TransComb, and respectively identifies 67.5% and 52.3% more lowly expressed transcripts. Scallop achieves higher sensitivity and precision than previous approaches over a wide range of coverage thresholds.