Computation of the Lovász Theta Function for Circulant Graphs
2003
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Abstract
The Lovász theta function θ(G) of a graph G has attracted a lot of attention for its connection with diverse issues, such as communicating without errors and computing large cliques in graphs. Indeed this function enjoys the remarkable property of being computable in polynomial time, despite being sandwitched between clique and chromatic number, two well known hard to compute quantities.
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How unique is Lovász's theta function?
The famous Lovász's ϑ function is computable in polynomial time for every graph, as a semi-definite program (Grötschel, Lovász and Schrijver, 1981 [5]). The chromatic number and the clique number of every perfect graph G are computable in polynomial time, since they are equal to f ϑ (G) = ϑ(G). Despite numerous efforts since the last three decades, recently stimulated by the Strong Perfect Graph Theorem (Chudnovsky, Robertson, Seymour and Thomas, 2006 [2]), no combinatorial proof of this result is known. In this work, we try to understand why the "key properties" of Lovász's ϑ function make it so "unique". We introduce an infinite set of convex functions, which includes the clique number ω and f ϑ. This set includes a sequence of linear programs which are monotone increasing and converging to f ϑ. We provide some evidences that f ϑ is the unique function in this setting allowing to compute the chromatic number of perfect graphs in polynomial time.
Hardness Results and Spectral Techniques for Combinatorial Problems on Circulant Graphs
IEEE Transactions on Computers, 1998
We show that computing (and even approximating) MAXIMUM CLIQUE and MINI-
An Alternative Proof of the Exponential Monotone Complexity of the Clique Function
2013
Lastly, we point out that the new methodology has potential to apply to a general circuit computing cliques due to the dynamic selection of t and z, and to improve the Alon-Boppana bound exp(\Omega(n / \log n)^{1/3})).
On the complexity of the circular chromatic number
Journal of Graph Theory, 2004
Circular chromatic number, χ c is a natural generalization of chromatic number. It is known that it is NP-hard to determine whether or not an arbitrary graph G satisfies χ(G) = χ c (G). In this paper we prove that this problem is NP-hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k ≥ 2 and n ≥ 3, for a given graph G with χ(G) = n, it is NP-complete to verify if χ c (G) ≤ n − 1 k .
On the Number of Cliques and Cycles in Graphs
2013
We give a new recursive method to compute the number of cliques and cycles of a graph. This method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. In particular, let $G$ be a graph and let $overline {G}$ be its complement, then given the chromatic polynomial of $overline {G}$, we give a recursive method to compute the number of cliques of $G$. Also given the adjacency matrix $A$ of $G$ we give a recursive method to compute the number of cycles by computing the sum of permanent function of the principal minors of $A$. In both cases we confront to a new computable parameter which is defined as the number of disjoint cliques in $G$.
of Mathematics And its Applications On the Star Coloring of Circulant Graphs
2017
Let G = (V,E) be an undirected simple graph. The star chromatic number of a graph G is the least number of colors needed to color the path on four vertices with three distinct colors. The aim of this paper is to determine the star chromatic number of some Circulant graphs.
Graph reconstruction in the congested clique
Journal of Computer and System Sciences, 2020
The congested clique model is a message-passing model of distributed computation where the underlying communication network is the complete graph of n nodes. In this paper we consider the situation where the joint input to the nodes is an nnode labeled graph G, i.e., the local input received by each node is the indicator function of its neighborhood in G. Nodes execute an algorithm, communicating with each other in synchronous rounds and their goal is to compute some function that depends on G. In every round, each of the n nodes may send up to n − 1 different b-bit messages through each of its n − 1 communication links. We denote by R the number of rounds of the algorithm. The product Rb, that is, the total number of bits received by a node through one link, is the cost of the algorithm. The most difficult problem we could attempt to solve is the reconstruction problem, where nodes are asked to recover all the edges of the input graph G. Formally, given a class of graphs G, the problem is defined as follows: if G / ∈ G, then every node must reject; on the other hand, if G ∈ G, then every node must end up, after the R rounds, knowing all the edges of G. It is not difficult to see that the cost Rb of any algorithm that solves this problem (even with public coins) is at least Ω(log |G n |/n), where G n is the subclass of all n-node labeled graphs in G. In this paper we prove that previous bound is tight and that it is possible to achieve it with only R = 2 rounds. More precisely, we exhibit (i) a one-round algorithm that achieves this bound for hereditary graph classes; and (ii) a two-round algorithm that achieves this bound for arbitrary graph classes. Later, we show that the bound Ω(log |G n |/n) cannot be achieved in one-round for arbitrary graph classes, and we give tight algorithms for that case. From (i) we recover all known results concerning the reconstruction of graph classes in one round and bandwidth O(log n): forests, planar graphs, cographs, etc. But we also get new one-round algorithms for other hereditary graph classes such as unit disc graphs, interval graphs, etc. From (ii), we can conclude that any problem restricted to a class of graphs of size 2 O(n log n) can be solved in the congested clique model in two rounds, with bandwidth O(log n). Moreover, our general two-round algorithm is valid for any set of labeled graphs, not only for graph classes (which are sets of labeled graphs closed under isomorphims).
Structural Parameterizations of Clique Coloring
2020
A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with $q$ colors. For fixed $q \ge 2$, we give an $\mathcal{O}^{\star}(q^{tw})$-time algorithm when the input graph is given together with one of its tree decompositions of width $tw$. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width.
The clique operator on circular-arc graphs
Discrete Applied Mathematics, 2010
A circular-arc graph G is the intersection graph of a collection of arcs on the circle and such a collection is called a model of G. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graph K(G) of a graph G is the intersection graph of its cliques. The iterated clique graph K i (G) of G is defined by K 0 (G) = G and K i+1 (G) = K(K i (G)). In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph K-converges, if it is Kconvergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.
Complexity of networks II: The set complexity of edge-colored graphs
Complexity, 2012
We previously introduced the concept of ' 'set-complexity,' ' based on a context-dependent measure of information, and used this concept to describe the complexity of gene interaction networks. In a previous paper of this series we analyzed the set-complexity of binary graphs. Here, we extend this analysis to graphs with multicolored edges that more closely match biological structures like the gene interaction networks. All highly complex graphs by this measure exhibit a modular structure. A principal result of this work is that for the most complex graphs of a given size the number of edge colors is equal to the number of ' 'modules' ' of the graph. Complete multipartite graphs (CMGs) are defined and analyzed. The relation between complexity and structure of these graphs is examined in detail. We establish that the mutual information between any two nodes in a CMG can be fully expressed in terms of entropy, and present an explicit expression for the set complexity of CMGs (Theorem 3). An algorithm for generating highly complex graphs from CMGs is described. We establish several theorems relating these concepts and connecting complex graphs with a variety of practical network properties. In exploring the relation between symmetry and complexity we use the idea of a similarity matrix and its spectrum for highly complex graphs.
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2006
We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n f( k)), where f(k) ≤3 for the inerlace polynomial, f(k) ≤2k+1 for the matching polynomial and f(k) ≤3 2k + 2 for the chromatic polynomial.
The algorithmic complexity of the minus clique-transversal problem
Applied Mathematics and Computation, 2007
A minus clique-transversal function of a graph G ¼ ðV ; EÞ is a function f : V ! fÀ1; 0; 1g such that P u2V ðCÞ f ðuÞ P 1 for every clique C of G. The weight of a minus clique-transversal function is f ðV Þ ¼ P f ðvÞ, over all vertices v 2 V . The minus clique-transversal number of a graph G, denoted s À c ðGÞ, equals the minimum weight of a minus clique-transversal function of G. The upper minus clique-transversal number of a graph G, denoted T À c ðGÞ, equals the maximum weight of a minimal minus clique-transversal function of G. In this paper, we show that the minus clique-transversal problem (respectively, upper minus clique-transversal problem) is NP-complete even when restricted to chordal graphs. On the other hand, we give a linear algorithm to solve the minus clique-transversal problem for block graphs.
Hardness of computing clique number and chromatic number for Cayley graphs
European Journal of Combinatorics, 2017
Computing the clique number and chromatic number of a general graph are well-known NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl., 285(1-3): [123][124][125][126][127][128][129][130][133][134][135][136][137][138][139][140][141][142] 1998) showed that computing clique number and chromatic number are still NP-Hard problems for the class of circulant graphs. We show that computing clique number is NP-Hard for the class of Cayley graphs for the groups G n , where G is any fixed finite group (e.g., cubelike graphs). We also show that computing chromatic number cannot be done in polynomial time (under the assumption P = NP) for the same class of graphs. Our presentation uses free Cayley graphs. The proof combines free Cayley graphs with quotient graphs and Goppa codes.
Computing Cliques is Intractable
2013
The class P is in fact a proper sub-class of NP. We explore topological properties of the Hamming space 2^[n] where [n]=1, 2,..., n. With the developed theory, we show: (i) a theorem that is closely related to Erdos and Rado's sunflower lemma, and claims a stronger statement in most cases, (ii) a new approach to prove the exponential monotone circuit complexity of the clique problem, (iii) NC NP through the impossibility of a Boolean circuit with poly-log depth to compute cliques, based on the construction of (ii), and (iv) P NP through the exponential circuit complexity of the clique problem, based on the construction of (iii). Item (i) leads to the existence of a sunflower with a small core in certain families of sets, which is not an obvious consequence of the sunflower lemma. In (iv), we show that any Boolean circuit computing the clique function CLIQUE_n,k (k=n^1/4) has a size exponential in n. Thus, we will separate P/poly from NP also. Razborov and Rudich showed strong ev...
Computation of Chromatic Polynomials Using Triangulations and Clique Trees
2005
In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G. Our method is based on the addition of edges and contraction of non-edges of G, the base case of the recursion being chordal graphs. The set of edges to be considered is taken from a triangulation of G. To achieve our goal, we use the properties of triangulations and clique-trees with respect to the previous operations, and guide our algorithm to efficiently divide the original problem. Furthermore, we give some lower bounds of the general complexity of our method, and provide experimental results for several families of graphs.
The complexity of clique graph recognition
Theoretical Computer Science, 2009
Consider finite, simple and undirected graphs. V and E denote the vertex set and the edge set of the graph G, respectively. A complete set of G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. The clique family of G is denoted by C(G). The clique graph of G is the intersection graph of C(G). The clique operator, K, assigns to each graph G its clique graph which is denoted by K(G). On the other hand, say that G is a clique graph if G belongs to the image of the clique operator, i.e. if there exists a graph H such that G = K (H). Clique operator and its image were widely studied. First articles focused on recognizing clique graphs [20,36], In [4,13], graphs for which the clique graph changes whenever a vertex is removed are considered. Graphs fixed under the operator K or fixed under the iterated clique operator, I<n, for some positive integer n; and the behavior under these operators of parameters such as the number of vertices or diameter were studied in [5,8,9,12,26,30] and more recently in [7,14,21-23, 29], For several classes of graphs, the image of the class under the clique operator was characterized [10,18,19,24,34,37]; and, in some cases, also the inverse image of the class [16,28,35], Results of the previous bibliography can be found in the survey [39], Clique graphs have been much studied as intersection graphs and are included in several books [11,25,33], In this paper we are concerned with the time complexity of the problem of recognizing clique graphs, this is the time complexity of the following decision problem.
On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász's Theta Function
Computational Optimization and Applications, 2005
We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the theta function.
On the generalized θ-number and related problems for highly symmetric graphs
2021
This paper is an in-depth analysis of the generalized θ-number of a graph. The generalized θ-number, θk(G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of θk(G), such as that the series (θk(G))k is increasing and bounded above by the order of the graph G. We study θk(G) when G is the graph strong, disjunction and Cartesian product of two graphs. We provide closed form expressions for the generalized θ-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs.
On the complexity of fixed parameter clique and dominating set
Theoretical Computer Science, 2004
We provide simple, faster algorithms for the detection of cliques and dominating sets of ÿxed order. Our algorithms are based on reductions to rectangular matrix multiplication. We also describe an improved algorithm for diamonds detection.
On the Lovász theta function and some variants
Discrete Optimization
The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Grötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Grötschel et al. scheme typically yields a stronger bound than the Lovász one.
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