Edge Clique Cover Research Articles

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.

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.

Structural parameterizations of Tracking Paths problem

Given a graph G with source and destination vertices s,t∈V(G) respectively, Tracking Paths asks for a minimum set of vertices T⊆V(G), such that the sequence of vertices restricted to T encountered in each simple path from s to t is unique. The problem was proven NP-hard [3] and was found to admit a quadratic kernel when parameterized by the size of the desired solution [8]. Following recent trends, for the first time, we study Tracking Paths with respect to structural parameters of the input graph, parameters that measure how far the input graph is, from an easy instance. We prove that Tracking Paths admits fixed-parameter tractable (FPT) algorithms when parameterized by the following parameters: (i) size of a 2-dominating set, (ii) size of cluster vertex deletion set, and (iii) size of split deletion set. En route we also look at Vertex Cover parameterized by the size of edge clique cover and show it fixed-parameter tractable.

Structural properties of Toeplitz graphs

In this paper, we study structural properties of Toeplitz graphs. We characterize Kq-free Toeplitz graphs for an integer q≥3 and give equivalent conditions for a Toeplitz graph Gn〈t1,t2,…,tk〉 with t1<⋯<tk and n≥tk−1+tk being chordal and equivalent conditions for a Toeplitz graph Gn〈t1,t2〉 being perfect. Then we compute the edge clique cover number and the vertex clique cover number of a chordal Toeplitz graph. Finally, we characterize the degree sequence (d1,d2,…,dn) of a Toeplitz graph with n vertices and show that a Toeplitz graph is a regular graph if and only if it is a circulant graph.

Finding Optimal Triangulations Parameterized by Edge Clique Cover

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (texttt {cc}) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem texttt {cc} is at most the number of taxa, in fractional hypertreewidth texttt {cc} is at most the number of hyperedges, and in treewidth of Bayesian networks texttt {cc} is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2^texttt {cc}, the number of potential maximal cliques is at most 3^texttt {cc}, and these objects can be listed in times O^*(2^texttt {cc}) and O^*(3^texttt {cc}), respectively, even when no edge clique cover is given as input; the O^*(cdot ) notation omits factors polynomial in the input size. These enumeration algorithms imply O^*(3^texttt {cc}) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O^*(4^m) time and O^*(3^m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size texttt {cc}' is given as a part of the input we give O^*(2^{texttt {cc}'}) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O^*(2^n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O^*(9^{texttt {cc}'}) and O^*(9^{texttt {cc}+ O(log ^2 texttt {cc})}) for problems in this framework.

On some FPT problems without polynomial Turing compressions

An impressive hardness theory which can prove compression lower bounds for a large number of FPT problems has been established under the assumption that NP ⊈ coNP/poly. However, there are no problems in FPT for which the existence of polynomial Turing compressions under any widely believed complexity assumptions has been excluded. In this paper, we provide a technique which can be used to prove that some FPT problems have no small Turing compressions under the assumption that there exists a problem in NP which does not have small-sized circuits. These FPT problems, which include edge clique cover parameterized by the number of cliques, integer linear programming and a-choosability parameterized by some structural parameters, etc. have the property that they remain NP-hard under Cook reductions even if their parameter values are small. Moreover, a trade-off between the size of the Turing compression lower bound and the robustness of the complexity assumption is obtained. In particular, we demonstrate that these FPT problems have no polynomial Turing compressions unless every set in NP has quasi-polynomial-sized circuits, and have no 2o(k) Turing compressions unless every set in NP has sub-exponential-sized circuits. Additionally, Turing kernelization lower bounds for these FPT problems are provided under some weaker complexity assumptions. Lastly, compression lower bounds for the above-mentioned FPT problems are proved under some complexity assumptions which are weaker than NP ⊈ coNP/poly, moreover, these results are proved under a method which is different from the previous hardness theory for compression lower bounds.

The Inductive Graph Dimension from the Minimum Edge Clique Cover

In this paper we prove that the inductively defined graph dimension has a simple additive property under the join operation. The dimension of the join of two simple graphs is one plus the sum of the dimensions of the component graphs: \(\mathrm {dim\,} (G_1+ G_2) = 1 +\mathrm {dim\,} G_1+ \mathrm {dim\,} G_2\). We use this formula to derive an expression for the inductive dimension of an arbitrary finite simple graph from its minimum edge clique cover. A corollary of the formula is that any arbitrary finite simple graph whose maximal cliques are all of order N has dimension \(N{-}1\). We finish by finding lower and upper bounds on the inductive dimension of a simple graph in terms of its clique number.

Network clique cover approximation to analyze complex contagions through group interactions

Contagion processes have been proven to fundamentally depend on the structural properties of the interaction networks conveying them. Many real networked systems are characterized by clustered substructures representing either collections of all-to-all pair-wise interactions (cliques) and/or group interactions, involving many of their members at once. In this work, focusing on interaction structures represented as simplicial complexes, we present a discrete-time microscopic model of complex contagion for a susceptible-infected-susceptible dynamics. Introducing a particular edge clique cover and a heuristic to find it, the model accounts for the higher-order dynamical correlations among the members of the substructures (cliques/simplices). The analytical computation of the critical point reveals that higher-order correlations are responsible for its dependence on the higher-order couplings. While such dependence eludes any mean-field model, the possibility of a bi-stable region is extended to structured populations.

Edge clique covers in graphs with independence number two

AbstractThe edge clique cover number of a graph is the size of the smallest collection of complete subgraphs whose union covers all edges of . Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if is claw‐free, then is bounded above by its order (denoted ). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in for for such graphs, improving upon the bou nd of due to Javadi, Maleki, and Omoomi. More precisely we prove that is at most the minimum of and , where is the minimum degree of . In the fractional version of the problem, we improve these upper bounds to . We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for ) equals , and when contains no induced subgraph isomorphic to where is any fixed graph of order 4.

Static beam placement and frequency plan algorithms for LEO constellations

SummaryLEO satellites are growing rapidly in number, but also in configurability of the payload, which presents new challenges of how to allocate the satellite's resources. We present an algorithm to solve two subproblems of the resource allocation problem for multi‐beam satellites in large LEO constellations. Our approach produces a static beam placement and a static frequency plan, taking into account the user demands and interference conditions, respectively. We provide an easy to implement and fast approximation for the beam placement problem, which is converted into an Edge Clique Cover problem using graph theory. The frequency assignment problem is translated into a constraint satisfaction problem, which is later solved with a first‐fit heuristic. We apply our algorithm to a scenario based on SpaceX's Starlink constellation, and when compared to two different baselines, we show that our method requires 40% fewer beams while being able to provide service to 2% more users.

Sample Complexity Bounds for RNNs with Application to Combinatorial Graph Problems (Student Abstract)

Learning to predict solutions to real-valued combinatorial graph problems promises efficient approximations. As demonstrated based on the NP-hard edge clique cover number, recurrent neural networks (RNNs) are particularly suited for this task and can even outperform state-of-the-art heuristics. However, the theoretical framework for estimating real-valued RNNs is understood only poorly. As our primary contribution, this is the first work that upper bounds the sample complexity for learning real-valued RNNs. While such derivations have been made earlier for feed-forward and convolutional neural networks, our work presents the first such attempt for recurrent neural networks. Given a single-layer RNN with a rectified linear units and input of length b, we show that a population prediction error of ε can be realized with at most Õ(a4b/ε2) samples.1 We further derive comparable results for multi-layer RNNs. Accordingly, a size-adaptive RNN fed with graphs of at most n vertices can be learned in Õ(n6/ε2), i.,e., with only a polynomial number of samples. For combinatorial graph problems, this provides a theoretical foundation that renders RNNs competitive.

On the triangle clique cover and [formula omitted] clique cover problems

An edge clique cover of a graph is a set of cliques that covers all edges of the graph. We generalize this concept to Ktclique cover, i.e. a set of cliques that covers all complete subgraphs on t vertices of the graph, for every t≥1. In particular, we extend a classical result of Erdös et al. (1966) on the edge clique cover number (t=2), also known as the intersection number, to the case t=3. The upper bound is tight, with equality holding only for the Turán graph T(n,3). As part of the proof, we obtain new upper bounds on the classical intersection number, which may be of independent interest. We also extend an algorithm of Scheinerman and Trenk (1999) to solve a weighted version of the Kt clique cover problem on a superclass of chordal graphs. We also prove that the Kt clique cover problem is NP-hard.

Large-scale clique cover of real-world networks

The edge clique cover (ecc ) problem deals with discovering a set of (possibly overlapping) cliques in a given graph that covers each of the graph's edges. This problem finds applications ranging from social networks to compiler optimization and stringology. We consider several variants of the ecc problem, using classical quality measures (like the number of cliques) and new ones. We describe efficient heuristic algorithms, the fastest one taking O(mdG) time for a graph with m edges, degeneracy dG (also known as k-core number). For large real-world networks with millions of nodes, like social networks, an algorithm should have (almost) linear running time to be practical: Our algorithm for finding eccs of large networks has linear-time performance in practice because dG is small, as our experiments show, on real-world networks with thousands to several million nodes.

Q-Rung Orthopair Fuzzy Competition Graphs with Application in the Soil Ecosystem

The q-rung orthopair fuzzy set is a powerful tool for depicting fuzziness and uncertainty, as compared to the Pythagorean fuzzy model. The aim of this paper is to present q-rung orthopair fuzzy competition graphs (q-ROFCGs) and their generalizations, including q-rung orthopair fuzzy k-competition graphs, p-competition q-rung orthopair fuzzy graphs and m-step q-rung orthopair fuzzy competition graphs with several important properties. The study proposes the novel concepts of q-rung orthopair fuzzy cliques and triangulated q-rung orthopair fuzzy graphs with real-life characterizations. In particular, the present work evolves the notion of competition number and m-step competition number of q-rung picture fuzzy graphs with algorithms and explores their bounds in connection with the size of the smallest q-rung orthopair fuzzy edge clique cover. In addition, an application is illustrated in the soil ecosystem with an algorithm to highlight the contributions of this research article in practical applications.

Bounds for the boxicity of Mycielski graphs

A box in Euclidean k-space is the Cartesian product of k closed intervals on the real line. The boxicity of a graph G, denoted by box(G), is the minimum nonnegative integer k such that G can be isomorphic to the intersection graph of a family of boxes in Euclidean k-space. Mycielski introduced an interesting graph operation that extends a graph G to a new graph M(G), called the Mycielski graph of G. In this paper we observe the behavior of boxicity of Mycielski graphs. We see that box(M(G)) is at least box(G) for a graph G, and hence we are interested in whether the boxicity of Mycielski graph of G is more than that of G or not. Here we give bounds for the boxicity of Mycielski graphs in terms of the number of universal vertices of G and the edge clique cover number of the complement of G. Further observations determine the boxicity of the Mycielski graph M(G) if G has no universal vertices or odd universal vertices and box(G) is equal to the edge clique cover number of the complement of G. We also present relations between the Mycielski graph M(G) and its generalizations M3(G) and Mr(G) in the context of boxicity, which will encourage us to calculate the boxicity of M(G) and M3(G).

On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph $G=(V,E)$, the complete width of $G$ is the minimum $k$ such that there exist $k$ independent sets $\mathtt{N}_i\subseteq V$, $1\le i\le k$, such that the graph $G'$ obtained from $G$ by adding some new edges between certain vertices inside the sets $\mathtt{N}_i$, $1\le i\le k$, is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most $k$ or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on $3K_2$-free bipartite graphs and polynomially solvable on $2K_2$-free bipartite graphs and on $(2K_2,C_4)$-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on $\overline{3K_2}$-free co-bipartite graphs and polynomially solvable on $C_4$-free co-bipartite graphs and on $(2K_2, C_4)$-free graphs. We also give a characterization for $k$-probe complete graphs which implies that the complete width problem admits a kernel of at most $2^k$ vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most $2^k$ vertices. Finally we determine all graphs of small complete width $k\le 3$.

Edge clique covering sum of graphs

The edge clique cover sum number (resp. edge clique partition sum number) of a graph G, denoted by scc(G) (resp. scp(G)), is defined as the smallest integer k for which there exists a collection of complete subgraphs of G, covering (resp. partitioning) all edges of G such that the sum of sizes of the cliques is at most k. By definition, scc(G) \({\leqq}\) scp(G). Also, it is known that for every graph G on n vertices, scp(G) \({\leqq n^{2}/2}\). In this paper, among some other results, we improve this bound for scc(G). In particular, we prove that if G is a graph on n vertices with no isolated vertex and the maximum degree of the complement of G is d − 1, for some integer d, then scc(G) \({\leqq cnd\left\lceil\log \left(({n-1})/(d-1)\right)\right\rceil}\), where c is a constant. Moreover, we conjecture that this bound is best possible up to a constant factor. Using a well-known result by Bollobas on set systems, we prove that this conjecture is true at least for d = 2. Finally, we give an interpretation of this conjecture as an interesting set system problem which can be viewed as a multipartite generalization of Bollobas’ two families theorem.

Known Algorithms for Edge Clique Cover are Probably Optimal

In the Edge Clique Cover (ECC) problem, given an undirected graph $G$ and an integer $k$, we ask whether one can choose $k$ cliques in $G$ such that each edge of $G$ is contained in at least one of the chosen cliques. Gramm et al. [ACM J. Exp. Algorithmics, 13 (2008)] have shown a set of simple rules that reduce the number of vertices of $G$ to $2^k$ while preserving the answer to the instance at hand, that is, they have shown a kernel for the problem with at most $2^k$ vertices. No algorithm is known with significantly better running time bound than a brute-force search on this kernel. In this paper, we show that the approach of Gramm et al. is essentially optimal: we present a polynomial-time algorithm that reduces an arbitrary instance of $3$-CNF-SAT with $n$ variables and $m$ clauses to an equivalent ECC instance $(G,k)$ with $k = \mathcal{O}(\log n)$ and $|V(G)| = \mathcal{O}(n + m)$. Consequently, there is no $2^{2^{o(k)}}{\rm poly}(n)$ time algorithm for the ECC problem, unless the Exponential Time Hypothesis fails. Moreover, our reduction also implies that, unless P $ = $ NP, the ECC problem does not admit a subexponential kernel, i.e., a kernel of size $2^{o(k)}$.

Clique Cover and Graph Separation

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. In this article, we show that, unless the polynomial hierarchy collapses to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter: ---Edge Clique Cover, parameterized by the number of cliques, ---Directed Edge/Vertex Multiway Cut, parameterized by the size of the cutset, even in the case of two terminals, ---Edge/Vertex Multicut, parameterized by the size of the cutset, and --- k -Way Cut, parameterized by the size of the cutset.

Competitively Tight Graphs

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph G is related to the edge clique cover number θ E (G) of the graph G via θ E (G) − |V(G)| + 2 ≤ k(G) ≤ θ E (G). We first show that for any positive integer m satisfying 2 ≤ m ≤ |V(G)|, there exists a graph G with k(G) = θ E (G) − |V(G)| + m and characterize a graph G satisfying k(G) = θ E (G). We then focus on what we call competitively tight graphs G which satisfy the lower bound, i.e., k(G) = θ E (G) − |V(G)| + 2. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.