Abstract In this paper, the percolation properties of higher-order networks that have non-trivial clustering and subgraph-based assortative mixing (the tendency of vertices to connect to other vertices based on subgraph joint degree) are examined. Our analytical method is based on generating functions and is exact for the networks we model. We also propose a Monte Carlo graph generation algorithm to draw random networks from the ensemble of graphs with fixed statistics. The proposed model is used to understand the effect that network microstructure has, through the arrangement of inter-subgraph clustering, on the global connective properties of the network. We find that even in k-regular networks, with fixed joint degree distributions and clustering coefficients, the arrangement of clustering has a non-trivial influence on the percolation properties of the network. We find that subgraph disassortativity increases the percolation threshold, whilst assortativity among subgraphs decreases and broadens the transition. Finally, we use an edge disjoint clique cover to represent empirical networks using our formulation, finding the resultant model offers a significant improvement over edge-based theory.

We consider scheduling deadline-constrained packets in multihop wireless networks. Packets with arbitrary deadlines and weights arrive at and are destined to different nodes. The goal is to design online admission, routing, and scheduling algorithms in order to maximize the cumulative weight of packets that reach their destinations within their deadlines. Under a general interference graph model of the wireless network, we provide online algorithms that are <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\gamma ,\mathrm {R})$</tex-math> </inline-formula>-competitive, i.e., they achieve at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/\gamma $</tex-math> </inline-formula> fraction of the value of the optimal offline algorithm, and do not exceed the capacity by more than a factor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {R}\geq 1$</tex-math> </inline-formula>. In particular, our algorithm can achieve <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma =O({\psi }^{\star } \log (\Delta \rho L)/\mathrm {R})$</tex-math> </inline-formula> when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {R}\mathrm {C} = \Omega ({\psi }^{\star } \log (\Delta \rho L))$</tex-math> </inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rho $</tex-math> </inline-formula> is the ratio of maximum weight to minimum weight of packets, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> is the length of the longest route of packets, and C is the minimum link capacity or the number of channels. Here, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta $</tex-math> </inline-formula> is the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">maximum degree</i> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {\psi }^{\star } $</tex-math> </inline-formula> is the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local clique cover number</i> of the interference graph. Our results translate directly to many networks of interest, for example, in one-hop interference networks, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {\psi }^{\star } =2$</tex-math> </inline-formula>, and in the case of wired networks (no interference), <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {\psi }^{\star } =1$</tex-math> </inline-formula>. We further provide lower bounds that show that our results are asymptotically optimal in many settings. Finally, we present extensive simulations that show our algorithms provide significant improvement over the prior approaches.

Dynamic index coding extends traditional index coding by handling real-time traffic flows instead of static data batches. In this paper, we propose a dynamic index coding scheme that operates over an arbitrary side information graph. The scheme leverages time-sharing across an optimal clique covering set of the side information graph, achieving coding gains with a guaranteed lower bound. Additionally, convex optimization is employed to refine the time-sharing coefficients, minimizing the transmission delay. Numerical results show that the optimization process can achieve a typical 10% reduction in the transmission delay, demonstrating the efficiency of the proposed approach.

Clique Cover on L-EPG representations of graphs

The reconfiguration graph of the $k$-colourings of a graph $G$, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two vertices of $\mathcal{R}_k(G)$ are joined by an edge if the colourings of $G$ they correspond to differ in colour on exactly one vertex. A $k$-colouring of a graph $G$ is called frozen if for every vertex $v \in V(G)$, $v$ is adjacent to a vertex of every colour different from its colour. A clique partition is a partition of the vertices of a graph into cliques. A clique partition is called a $k$-clique-partition if it contains at most $k$ cliques. Clearly, a $k$-colouring of a graph $G$ corresponds precisely to a $k$-clique-partition of its complement, $\overline{G}$. A $k$-clique-partition $\mathcal{Q}$ of a graph $H$ is called frozen if for every vertex $v \in V(H)$, $v$ has a non-neighbour in each of the cliques of $\mathcal{Q}$ other than the one containing $v$. The complement of the cycle on four vertices, $C_4$, is called $2K_2$. We give several infinite classes of $2K_2$-free graphs with frozen colourings. We give an operation that transforms a $k$-chromatic graph with a frozen $(k+1)$-colouring into a $(k+1)$-chromatic graph with a frozen $(k+2)$-colouring. The operation requires some restrictions on the graph, the colouring, and the frozen colouring. The operation preserves being $2K_2$-free. Using this we prove that for all $k \ge 4$, there is a $k$-chromatic $2K_2$-free graph with a frozen $(k+1)$-colouring. We prove these results by studying frozen clique partitions in $C_4$-free graphs. We say a graph $G$ is recolourable if $R_{\ell}(G)$ is connected for all $\ell$ greater than the chromatic number of $G$. We prove that every 3-chromatic $2K_2$-free graph $G$ is recolourable and that for all $\ell$ greater than the chromatic number of $G$, the diameter of $R_{\ell}(G)$ is at most $14n$ where $n$ is the number of vertices of $G$.

The clique partitioning problem (CPP) aims to find a partition of vertices of a complete graph in order to maximize the sum of edge weights within each partition (clique), which has been proven to be NP-hard and has wide real-world applications. In this paper, we propose an elite-guided weighted simulated annealing algorithm called EWSA to solve the CPP. First, EWSA employs two specific configurations and alternates between them via an oscillation strategy, which balances the exploitation and exploration of the search. Second, a weighting strategy is introduced to improve the scoring function in traditional simulated annealing, which is able to guide the search to explore diverse solutions. Finally, a partition restriction strategy is adopted to reduce search space and increase the search efficiency. Experiments on 255 instances demonstrate the competitiveness of EWSA. For 130 open instances, EWSA discovers new upper bounds in 32 cases and matches the best known results for the others. For the remaining 125 closed instances, EWSA achieves the best known objective values within a short computational time.

The clique partitioning problem is a combinatorial optimisation problem which has many applications. At present, the most promising exact algorithms are those that are based on an understanding of the associated polytope. We present two new families of valid inequalities for that polytope, and show that the inequalities define facets under certain conditions.

Satellite communication (SatCom) systems play a vital role in providing global connectivity and enable a wide range of applications, including Internet of Things (IoT) connectivity for remote areas, such as forests and oceans. Two crucial resource allocation challenges in SatCom are beam placement (BP) and frequency assignment (FA) problems, which involve the clique covering (CC) and graph coloring (GC) problems, respectively. Conventional solutions for these problems incur excessive computational cost, which is intractable for classical computers. A promising approach is to formulate these problems using the Ising model, construct their Hamiltonians, and then solve them efficiently by a quantum computer. However, the current quantum computers have very limited hardware and can only handle rather small inputs. To overcome this limitation, we propose a hybrid-quantum-classical-computational pipeline where an efficient hamiltonian reduction method is the key for solving large CC/GC instances. Through experiments on real quantum computers, our reduction method outperforms commercial solutions, allowing quantum annealers to handle significantly larger BP/FA instances while maintaining high probability to achieve feasible solutions and near-optimal performance. Although the inherent hardness of the CC/GC problems cannot be overcome by quantum computing, our research contributes to the early exploration of quantum computing in the context of the complex optimization problems in SatCom systems, particularly in the realm of IoT connectivity for remote areas.

Given a graph G = (V, E), the Edge Clique Cover Problem (ECCP) asks for a minimum number of cliques so that every edge e ∈ E belongs to at least one of the subgraphs induced by the selected cliques. Rather than fine tuned ECCP algorithms, exact or heuristic, we investigate conceptual frameworks for designing them. In particular we focus on frameworks that are based on the formulation of ECCP as a Set Covering Problem (SCP). This formulation typically contains an exponential number of variables that are in a one-to-one relation with the distinct maximal cliques of G. Our frameworks firstly generate reduced SCP formulations that contain a conveniently small number of variables and then solve them to obtain feasible solutions to ECCP. Variables for these formulations are selected via two distinct clique sampling criteria. One of them relies on random considerations for selecting the cliques (variables) while the other is based on Linear Programming reduced costs. The latter one, in particular, is further refined here into a procedure that implements local search within a column generation environment and is straightforward to adapt to numerous other problems.

The index coding problem is concerned with broadcasting encoded information to a collection of receivers in a way that enables each receiver to discover its required data based on its side information, which comprises the data required by some of the others. Given the side information map, represented by a graph in the symmetric case and by a digraph otherwise, the goal is to devise a coding scheme of minimum broadcast length. We present a general method for developing efficient algorithms for approximating the index coding rate for prescribed families of instances. As applications, we obtain polynomial-time algorithms that approximate the index coding rate of graphs and digraphs on n vertices to within factors of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n/\log ^{2} n)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n/\log n)$ </tex-math></inline-formula> respectively. This improves on the approximation factors of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n/\log n)$ </tex-math></inline-formula> for graphs and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n \cdot \log \log n/\log n)$ </tex-math></inline-formula> for digraphs achieved by Blasiak, Kleinberg, and Lubetzky. For the family of quasi-line graphs, we exhibit a polynomial-time algorithm that approximates the index coding rate to within a factor of 2. This improves on the approximation factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n^{2/3})$ </tex-math></inline-formula> achieved by Arbabjolfaei and Kim for graphs on n vertices taken from certain sub-families of quasi-line graphs. Our approach is applicable for approximating a variety of additional graph and digraph quantities to within the same approximation factors. Specifically, it captures every graph quantity sandwiched between the independence number and the clique cover number and every digraph quantity sandwiched between the maximum size of an acyclic induced sub-digraph and the directed clique cover number.

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.

Abstract For 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.

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

In a graph G, a clique partition of G is a partition P = {V1,V2, . . . ,Vq} of V(G) such that the induced subgraph G[Vi] is a clique (called a clique of P) for each i ? [q]. If a clique partition P also satisfies that |G[Vi]| = t for each i ? [q], the graph G is called a Kt-partitionable graph. A Kt-partition edge-fault set of G is a subset F of E(G) such that the deletion of F results in a graph where no Kt-partitions exist. The Kt-partition edge-fault number of G, denoted by ft(G), is the smallest size among all Kt-partition edge-fault sets of G. The Kt-preclusion number of G, denoted by 1t(G), is the minimum size of an edge subset A such that there exists at least one vertex in G not contained in any clique Kt ? G ? A. In this paper, we prove that arrangement graphs and data center networks are clique partitionable. Furthermore, arrangement graphs are shown to be clique decomposable. We determine the exact value of fn?k+1 for the arrangement graphs An,k and establish bounds for ft(An,k) and ft(Kn) for specific values of t. Additionally, we derive the exact values of 13 for maximal planar graphs, 1r for Tur?an graphs T(n, r), and ft for graphs obtained from the arrangement graphs An,k by shrinking a partition R, for specific values of t.

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

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.

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.

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.