Bipartite density of triangle-free subcubic graphs

We consider the question of whether adaptivity can improve the complexity of property testing algorithms in the dense graphs model. It is known that there can be at most a quadratic gap between adaptive and non-adaptive testers in this model, but it was not known whether any gap indeed exists. In this work we reveal such a gap.Specifically, we focus on the well studied property of bipartiteness. Bogdanov and Trevisan (IEEE Symposium on Computational Complexity, 2004) proved a lower bound of Ω(1/ε 2) on the query complexity of non-adaptive testing algorithms for bipartiteness. This lower bound holds for graphs with maximum degree O(εn). Our main result is an adaptive testing algorithm for bipartiteness of graphs with maximum degree O(εn) whose query complexity is \(\tilde{O}(1/\epsilon^{3/2})\). A slightly modified version of our algorithm can be used to test the combined property of being bipartite and having maximum degree O(εn). Thus we demonstrate that adaptive testers are stronger than non-adaptive testers in the dense graphs model.We note that the upper bound we obtain is tight up-to polylogarithmic factors, in view of the Ω(1/ε 3/2) lower bound of Bogdanov and Trevisan for adaptive testers. In addition we show that \(\tilde{O}(1/\epsilon^{3/2})\) queries also suffice when (almost) all vertices have degree \(\Omega(\sqrt \epsilon \cdot n)\). In this case adaptivity is not necessary.KeywordsQuery ComplexityTesting AlgorithmAdaptive TesterDense GraphNeighbor QueryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Provided that one is willing to use randomness and to tolerate an approximate answer, many computational problems admit ultrafast algorithms that run in less than linear time in the length of the input. In many interesting cases, even algorithms that run in constant time are known, whose efficiency depends only on the accuracy of the approximation and not on the length of the inputs. Algorithms for graph problems on dense graphs are especially efficient and simple. I will describe an algorithm that estimates the size of the maximum cut in a dense graph, and its specialization to the task of distinguishing bipartite dense graphs from dense graphs that are far from bipartite. Results explaining the simplicity of such algorithms will also be discussed. Some sublinear-time algorithms are also known for graph problems in sparse graphs, but they are typically more elaborate. I will describe a simple but very clever algorithm that approximates the number of connected components of a given graph, and its generalization to the problem of approximating the weight of the minimum spanning tree of a given weighted graph. The algorithm runs in time dependent only on the maximum degree, the required quality of approximation, and the range of weights, but the running time is independent of the number of vertices.

Given a bipartite graph, the maximum balanced biclique (MBB) problem, discovering a mutually connected while disjoint sets of equal size with the maximum cardinality, plays a significant role for mining the bipartite graph and has numerous applications. Despite the NP-hardness of the MBB problem, in this paper, we show that an exact MBB can be discovered extremely fast in bipartite graphs for real applications. We propose two exact algorithms dedicated for small dense and large sparse bipartite graphs respectively. For dense bipartite graphs, an O*(1.3803n) algorithm is proposed. This algorithm in fact can find an MBB very fast for small dense bipartite graphs that are common for applications such as VLSI design. This is because, using our proposed novel techniques, the search can fast converge to sufficiently dense bipartite graphs which we prove to be polynomial-time solvable. For large sparse bipartite graphs typical for applications such as biological data analysis, an O*(1.3803 δ) algorithm is proposed, where δ is only a few hundred for large sparse bipartite graphs with millions of vertices. The indispensible optimization that leads to this time complexity is: we transform a large sparse bipartite graph into a limited number of dense subgraphs such that each of the dense subgraphs has up to δ vertices and then apply our proposed algorithm for dense bipartite graphs on each of the subgraphs. To further speed up this algorithm, tighter upper bounds, faster heuristics and more effective reductions are proposed, allowing an MBB to be discovered within a few seconds for bipartite graphs with millions of vertices. Extensive experiments are conducted on synthetic and real large bipartite graphs to demonstrate the efficiency and effectiveness of our proposed algorithms and techniques.

List star edge coloring of sparse graphs

Enumeration of graphs with a heavy-tailed degree sequence

A graph is a mathematical structure consisting of a set of objects, which we call vertices, and links between pairs of objects, which we call edges. Graphs are used to model many problems arising in areas such as physics, sociology, and computer science. It is partially because of the simplicity of the definition of a graph that the concept can be so widely used. Nevertheless, when applied to a particular task, it is not always necessary to study graphs in all their generality, and it can be convenient to studying them from a restricted point of view. Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure. A collection of graphs satisfying a fixed restriction forms a class of graphs. Many important classes of graphs satisfy that graphs belonging to it cannot have many edges in comparison with the number of vertices. Such is the case of classes with an upper bound on the maximum degree, and of classes excluding a fixed minor. Recently, the notion of classes with bounded expansion was introduced by Neˇsetˇril and Ossona de Mendez [62], as a generalisation of many important types of sparse classes. In this thesis we study chromatic and structural properties of classes with bounded expansion. We say a graph is k-degenerate if each of its subgraphs has a vertex of degree at most k. The degeneracy is thus a measure of the density of a graph. This notion has been generalised with the introduction, by Kierstead and Yang [47], of the generalised colouring numbers. These parameters have found applications in many areas of Graph Theory, including a characterisation of classes with bounded expansion. One of the main results of this thesis is a series of upper bounds on the generalised colouring numbers, for different sparse classes of graphs, such as classes excluding a fixed complete minor, classes with bounded genus and classes with bounded tree-width. We also study the following problem: for a fixed positive integer p, how many colours do we need to colour a given graph in such a way that vertices at distance exactly p get different colours? When considering classes with bounded expansion, we improve dramatically on the previously known upper bounds for the number of colours needed. Finally, we introduce a notion of addition of graph classes, and show various cases in which sparse classes can be summed so as to obtain another sparse class.

For a graph G, let a(G) denote the maximum size of a subset of vertices that induces a forest. Suppose that G is connected with n vertices, e edges, and maximum degree Δ. Our results include: (a) if Δ ≤ 3, and G ≠ K4, then a(G) ≥ n - e-4 - 1-4 and this is sharp for all permissible e ≡ 3 (mod 4); and (b) if Δ ≥ 3, then a(G) ≥ α(G) + (n - α(G))-(Δ - 1)2. Several problems remain open. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 113–123, 2001

The firefighter problem: Further steps in understanding its complexity

In this paper we prove the following almost optimal theorem. For any δ > 0, there exist constants c and n0 such that, if n [ges ] n0, T is a tree of order n and maximum degree at most cn/log n, and G is a graph of order n and minimum degree at least (1/2 + δ)n, then T is a subgraph of G.

This work is an analysis of the character networks in two science fiction television series: Stargate and Star Trek. These networks are constructed on the basis of scene co-occurrence between characters to indicate the presence of a connection. Global network structure measures such as the average path length, graph density, network diameter, average degree, median degree, maximum degree, and average clustering coefficient are computed as well as individual node centrality scores. The two fictional networks constructed are found to be quite similar in structure which is astonishing given that Stargate only ran for 18 years in comparison to the 48 years for Star Trek.

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.

We study the complexity of local search in the max-cut problem with FLIP neighborhood, in which exactly one node changes the partition. We introduce a technique of constructing instances which enforce certain sequences of improving steps. Using our technique we can show that already graphs with maximum degree four satify the following two properties.There are instances with initial solutions for which every local search takes exponential time to converge to a local optimum. The problem of computing a local optimum reachable from a given solution by a sequence of improving steps is PSPACE-complete. Schaffer and Yannakakis (JOC '91) showed via a so called “tight” PLS-reduction that the properties (1) and (2) hold for graphs with unbounded degree. Our improvement to the degree four is the best possible improvement since Poljak (JOC '95) showed for cubic graphs that every sequence of improving steps has polynomial length, whereby his result is easily generalizable to arbitrary graphs with maximum degree three. In his paper Poljak also asked whether (1) holds for graphs with maximum degree four, which is settled by our result. Many tight PLS-reductions in the literature are based on the max-cut problem. Via some of them our constructions carry over to other problems and show that the properties (1) and (2) already hold for very restricted sets of feasible inputs of these problems. Since our paper provides the two results that typically come along with tight PLS-reductions it does naturally put the focus on the question whether it is even PLS-complete to compute a local optimum on graphs with maximum degree four – a question that was recently asked by Ackermann et al. We think that our insights might be helpful for tackling this question.

As our main result, we prove that for every multigraph G = (V, E) which has no loops and is of order n, size m, and maximum degree $\Delta < {{{10}}^{-{{3}}}{{m}}\over \sqrt{{{8}}{{n}}}}$ there is a mapping ${{f}}:{{V}}\cup {{E}}\to \big\{{{1}},{{2}},\ldots,\big\lceil{{{m}}+{{2}}\over {{3}}}\big\rceil\big\}$ such that ${{f}}({{u}})+{{f}}({{uv}})+{{f}}({{v}})\not={{f}}({{u}}')+{{f}}({{u}}'{{v}}')+{{f}}({{v}}')$ for every ${{uv}},{{u}}'{{v}}'\in {{E}}$ with ${{uv}}\not={{u}}'{{v}}'$. Functions with this property were recently introduced and studied by Bača et al. and were called edge irregular total labelings. Our result confirms a recent conjecture of Ivančo and Jendrol′ about such labelings for dense graphs, for graphs where the maximum and minimum degree are not too different in terms of the order, and also for large graphs of bounded maximum degree. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 333–343, 2008

We consider ad-hoc networks consisting of $n$ wireless nodes that are located on the plane. Any two given nodes are called neighbors if they are located within a certain distance (communication range) from one another. A given node can be directly connected to any one of its neighbors and picks its connections according to a unique topology control algorithm that is available at every node. Given that each node knows only the indices (unique identification numbers) of its one- and two-hop neighbors, we identify an algorithm that preserves connectivity and can operate without the need of any synchronization among nodes. Moreover, the algorithm results in a sparse graph with at most $5n$ edges and a maximum node degree of $10$. Existing algorithms with the same promises further require neighbor distance and#x002F;or direction information at each node. We also evaluate the performance of our algorithm for random networks. In this case, our algorithm provides an asymptotically connected network with $n(1#x002B;o(1))$ edges with a degree less than or equal to $6$ for $1-o(1)$ fraction of the nodes. Numerical results confirm our analytical findings.

Homomorphisms from sparse graphs with large girth