Clique Cover Research Articles

Invariant correlation under marginal transforms

A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent samples. In this paper, we study this invariance property on the Pearson correlation coefficient and its applications. A multivariate random vector is said to have an invariant correlation if its pairwise correlation coefficients remain unchanged under any common marginal transforms. For a bivariate case, we characterize all models of such a random vector via a certain combination of comonotonicity—the strongest form of positive dependence—and independence. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fréchet copulas. In the general multivariate case, we characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model that admits any prescribed invariant correlation matrix. Finally, we show that all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.

Some results and problems on clique coverings of hypergraphs

AbstractFor a ‐uniform hypergraph we consider the parameter , the minimum size of a clique cover of the edge set of . We derive bounds on for belonging to various classes of hypergraphs.

On conflict-free cuts: Algorithms and complexity

One way to define the Matching Cut problem is: Given a graph G, is there an edge-cut M of G such that M is an independent set in the line graph of G? We propose the more general Conflict-Free Cut problem: Together with the graph G, we are given a so-called conflict graph Gˆ on the edges of G, and we ask for an edge-cutset M of G that is independent in Gˆ. Since conflict-free settings are popular generalizations of classical optimization problems and Conflict-Free Cut was not considered in the literature so far, we start the study of the problem. We show that the problem is NP-complete even when the maximum degree of G is 5 and Gˆ is 1-regular. The same reduction implies an exponential lower bound on the solvability based on the Exponential Time Hypothesis. We also give parameterized complexity results: We show that the problem is fixed-parameter tractable with the vertex cover number of G as a parameter, and we show W[1]-hardness even when G has a feedback vertex set of size one, and the clique cover number of Gˆ is the parameter. Since the clique cover number of Gˆ is an upper bound on the independence number of Gˆ and thus the solution size, this implies W[1]-hardness when parameterized by the cut size. We list polynomial-time solvable cases and interesting open problems. At last, we draw a connection to a symmetric variant of SAT.

A graph based named entity disambiguation using clique partitioning and semantic relatedness

Disambiguating name mentions in texts is a crucial task in Natural Language Processing, especially in entity linking. The credibility and efficiency of such systems depend largely on this task. For a given name entity mention in a text, there are many potential candidate entities that may refer to it in the knowledge base. Therefore, it is very difficult to assign the correct candidate from the whole set of candidate entities of this mention. To solve this problem, collective entity disambiguation is a prominent approach. In this paper, we present a novel algorithm called CPSR for collective entity disambiguation, which is based on a graph approach and semantic relatedness. A clique partitioning algorithm is used to find the best clique that contains a set of candidate entities. These candidate entities provide the answers to the corresponding mentions in the disambiguation process. To evaluate our algorithm, we carried out a series of experiments on seven well-known datasets, namely, AIDA/CoNLL2003-TestB, IITB, MSNBC, AQUAINT, ACE2004, Cweb, and Wiki. The Kensho Derived Wikimedia Dataset (KDWD) is used as the knowledge base for our system. From the experimental results, our CPSR algorithm outperforms both the baselines and other well-known state-of-the-art approaches.

Graph-partitioning entity resolution for resolving noisy product names in OCR scans of retail receipts

In business intelligence for retail, it is critical to ensure consistent and unambiguous product dimension information. This is challenging, especially if an organization does not have full control over the source of either transaction or master data. Such lack of control is the case when brands rely on data provided directly by consumers through images of receipts. Product name strings obtained from the digitization of receipts often contain substitution, insertion, and deletion errors. These errors prevent product names from serving as a useful dimension for further analysis. This paper proposes a clustering-based approach to link error-laden product names to underlying SKUs to remove this noise. The problem can be modeled as an entity resolution problem: each digitized product name is a reference to an underlying entity SKU. The entity resolution problem can further be modeled as a clique-partitioning problem that can be solved in a reasonable time with an agglomerative clustering heuristic. The results of clustering a synthetic data set show that the approach can successfully resolve product references to reveal coarse-grained (i.e., category, generic product) groupings. Future work may be done on implementing blocking strategies, optimizing the model parameters, and understanding the limits of the model for fine-grained (i.e., size variation) groupings.

Exact algorithms for restricted subset feedback vertex set in chordal and split graphs

The Restricted Subset Feedback Vertex Set problem (R-SFVS) takes a graph G=(V,E), a terminal set T⊆V, and an integer k as the input. The task is to determine whether there exists a subset S⊆V∖T of at most k vertices, after deleting which no terminal in T is contained in a cycle in the remaining graph. R-SFVS is NP-complete even when the input graph is restricted to split graphs. In this paper, we mainly show that R-SFVS in chordal and split graphs can be solved in O(1.1550|V|) time and exponential space or in O(1.1605|V|) time and polynomial space, significantly improving all previous results. As a by-product, we show that the Maximum Independent Set problem parameterized by the edge clique cover number is fixed-parameter tractable. Furthermore, by using a simple reduction from R-SFVS to Vertex Cover, we obtain an O⁎(1.2738k)-time parameterized algorithm and a tight O(k2)-kernel for R-SFVS in chordal and split graphs.

Scheduling an Agile Multipayload Earth-Observing Satellite

A highly integrated Earth-observing satellite can possess several maneuverable payloads to perform different missions simultaneously, which brings some challenges to the method of task scheduling. This paper addresses the selection and scheduling problem of an agile satellite with several independently maneuverable optical payloads. Some differences compared to the traditional scheduling problem of agile satellites are presented and considered in a constrained optimization model. A two-stage method is proposed to accomplish the scheduling of the satellite and payloads in different stages. Clusters are generated from preprocessed tasks by a clique partition algorithm, and their centers are used to calculate the pointing direction of the satellite in the first stage. A multiobjective local search algorithm is introduced to schedule tasks in each selected cluster in the second stage. Considering the time-dependent property of the transition time, the problem of determining the start observation time is transformed into linear programming in a proposed insertion operator that guarantees the feasibility of generated solutions. Two types of instances are created and tested to demonstrate the effectiveness of the proposed method, and some analyses are conducted based on the experimental results.

Subnetwork constraints for tighter upper bounds and exact solution of the clique partitioning problem

We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning problem, being NP-hard in the general case of having edge weights of different signs. We propose a new method of estimating an upper bound of the objective function that we combine with the classical branch-and-bound technique to find the exact solution. We evaluate our approach on a broad range of random graphs and real-world networks. The proposed approach provided tighter upper bounds and achieved significant convergence speed improvements compared to known alternative methods.

Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph

Using the notions of clique partitions and edge clique covers of graphs, we consider the corresponding incidence structures. This connection furnishes lower bounds on the negative eigenvalues and their multiplicities associated with the adjacency matrix, bounds on the incidence energy, and on the signless Laplacian energy for graphs. For the more general and well-studied set S(G) of all real symmetric matrices associated with a graph G, we apply an extended version of an incidence matrix tied to an edge clique cover to establish several classes of graphs that allow two distinct eigenvalues.

The role of vaccination and public awareness in forecasts of Mpox incidence in the United Kingdom

Beginning in May 2022, Mpox virus spread rapidly in high-income countries through close human-to-human contact primarily amongst communities of gay, bisexual and men who have sex with men (GBMSM). Behavioural change arising from increased knowledge and health warnings may have reduced the rate of transmission and modified Vaccinia-based vaccination is likely to be an effective longer-term intervention. We investigate the UK epidemic presenting 26-week projections using a stochastic discrete-population transmission model which includes GBMSM status, rate of formation of new sexual partnerships, and clique partitioning of the population. The Mpox cases peaked in mid-July; our analysis is that the decline was due to decreased transmission rate per infected individual and infection-induced immunity among GBMSM, especially those with the highest rate of new partners. Vaccination did not cause Mpox incidence to turn over, however, we predict that a rebound in cases due to behaviour reversion was prevented by high-risk group-targeted vaccination.

How Many Cliques Can a Clique Cover Cover?

This work examines the problem of clique enumeration on a graph by exploiting its clique covers. The principle of inclusion/exclusion is applied to determine the number of cliques of size $r$ in the graph union of a set $\mathcal{C} = \{c_1, \ldots, c_m\}$ of $m$ cliques. This leads to a deeper examination of the sets involved and to an orbit partition, $\Gamma$, of the power set $\mathcal{P}(\mathcal{N}_{m})$ of $\mathcal{N}_{m} = \{1, \ldots, m\}$. Applied to the cliques, this partition gives insight into clique enumeration and yields new results on cliques within a clique cover, including expressions for the number of cliques of size $r$ as well as generating functions for the cliques on these graphs. The quotient graph modulo this partition provides a succinct representation to determine cliques and maximal cliques in the graph union. The partition also provides a natural and powerful framework for related problems, such as the enumeration of induced connected components, by drawing upon a connection to extremal set theory through intersecting sets.

Sharp bounds on the least eigenvalue of a graph determined from edge clique partitions

Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove that the least eigenvalue of the n-Queens graph {mathcal {Q}}(n) is equal to -4 for every n ge 4 and it is also proven that the multiplicity of this eigenvalue is (n-3)^2. Finally, edge clique partitions of additional infinite families of connected graphs and their relations with the least eigenvalues are presented.

Problems and results on 1-cross-intersecting set pair systems

Abstract The notion of cross-intersecting set pair system of size $m$ , $ (\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m )$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne \emptyset$ , was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le\binom{a+b}{a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$ . Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$ . Such a system is called $1$ -cross-intersecting. We show that these systems are related to perfect graphs, clique partitions of graphs, and finite geometries. We prove that their maximum size is at least $5^{n/2}$ for $n$ even, $a=b=n$ , equal to $\bigl (\lfloor \frac{n}{2}\rfloor +1\bigr )\bigl (\lceil \frac{n}{2}\rceil +1\bigr )$ if $a=2$ and $b=n\ge 4$ , at most $|\cup _{i=1}^m A_i|$ , asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ( $|A_i\cap A_j|\le 1$ for $i\ne j$ ), asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.

IMeP: Impedance Matching Enhanced Power-Delivered-to-Load Optimization for Magnetic MIMO Wireless Power Transfer System

Recently, multiple-input multiple-output (MIMO) technology has been introduced into magnetic resonant coupling (MRC) enabled wireless power transfer (WPT) systems for concurrent charging of multiple devices. However, impedance mismatching phenomena caused by strong TX-RX, TX-TX, or RX-RX coupling greatly affect the power delivered to load (PDL) in practical charging systems. To solve this issue, we propose an effective scheduling algorithm for Impedance Matching–enhanced PDL optimization in MIMO MRC-WPT systems (called IMeP ), which integrates the transmitter scheduling together with the impedance matching techniques, i.e., adjusting TX coils for tuning TX-RX/TX-TX coupling and grouping RXs to separate strongly coupled RX pairs. We formulate this as a joint optimization problem and decouple it into three sub-problems, i.e., current scheduling, coil adjustment, and RX grouping. We then solve them through alternating direction method of multipliers–based, randomized beamforming–based, and graph clique cover–based algorithms, respectively. Extensive experiments are performed on a prototype testbed, and the results demonstrate the effectiveness of our solution. Compared with the state-of-the-art power transfer efficiency maximization solution, the proposed algorithm IMeP achieves a 74.7× performance improvement of PDL on average.

Total coloring conjecture on certain classes of product graphs

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G , denoted by χ ″( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G , Δ ( G )+1 ≤ χ ″( G )≤ Δ ( G )+2 , where Δ ( G ) is the maximum degree of G . In this paper, we prove the Behzad and Vizing conjecture for Indu - Bala product graph, Skew and Converse Skew product graph, Cover product graph, Clique cover product graph and Comb product graph.

A polyhedral study of lifted multicuts

Fundamental to many applications in data analysis are the decompositions of a graph, i.e. partitions of the node set into component-inducing subsets. One way of encoding decompositions is by multicuts, the subsets of those edges that straddle distinct components. Recently, a lifting of multicuts from a graph G=(V,E) to an augmented graph Ĝ=(V,E∪F) has been proposed in the field of image analysis, with the goal of obtaining a more expressive characterization of graph decompositions in which it is made explicit also for pairs F⊆V2∖E of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in RE∪F whose vertices are precisely the characteristic vectors of multicuts of Ĝ lifted from G, connecting it, in particular, to the rich body of prior work on the clique partitioning and multilinear polytope.

Fixed set search applied to the clique partitioning problem

The Clique Partitioning Problem (CPP) seeks to decompose a set of vertices into disjoint subsets (cliques) maximizing the sum of edge weights over all the cliques. The interest for the CPP comes from the fact that it represents well many practical problems from areas like data mining, engineering, bio-informatics, etc. To address this NP-hard combinatorial optimization problem, the novel fixed set search (FSS) metaheuristic is applied. This is performed, firstly, by developing a greedy randomized adaptive search procedure (GRASP) using a new type of neighborhood in the local search. The GRASP is further extended to the FSS by adding a learning mechanism. The FSS application to the CPP provides a new approach for generating fixed sets for problems with solution symmetries. This approach opens the application of the FSS to new families of combinatorial optimization problems. The conducted computational experiments show that the FSS significantly outperforms state-of-the-art metaheuristics for the CPP. The proposed approach obtains a large number of new best solutions for commonly used CPP test instances.

A stepped tabu search method for the clique partitioning problem

Given an undirected graph, a clique is a subset of vertices in which the induced subgraph is complete; that is, all pairs of vertices of this subset are adjacent. Clique problems in graphs are very important due to their numerous applications. One of these problems is the clique partitioning problem (CPP), which consists of dividing the set of vertices of a graph into the smallest number of cliques possible. The CPP is an NP-hard problem with many application fields (timetabling, manufacturing, scheduling, telecommunications, etc.). Despite its great applicability, few recent studies have focused on proposing specific resolution methods for the CPP. This article presents a resolution method that combines multistart strategies with tabu search. The most novel characteristic of our method is that it allows unfeasible solutions to be visited, which facilitates exploration of the solution space. The computational tests show that our method performs better than previous methods proposed for this problem. In fact, our method strictly improves the results of these methods in most of the instances considered while requiring less computation time.

On the Clique Partitioning of a Graph

On the Clique Partitioning of a Graph

Graph coloring with physics-inspired graph neural networks

We show how graph neural networks can be used to solve the canonical graph coloring problem. We frame graph coloring as a multi-class node classification problem and utilize an unsupervised training strategy based on the statistical physics Potts model. Generalizations to other multi-class problems such as community detection, data clustering, and the minimum clique cover problem are straightforward. We provide numerical benchmark results and illustrate our approach with an end-to-end application for a real-world scheduling use case within a comprehensive encode-process-decode framework. Our optimization approach performs on par or outperforms existing solvers, with the ability to scale to problems with millions of variables.