Determining the Solution Space of Vertex-Cover by Interactions and Backbones

To solve the combinatorial optimization problems especially the minimal Vertex-cover problem with high efficiency, is a significant task in theoretical computer science and many other subjects. Aiming at detecting the solution space of Vertex-cover, a new structure named interaction between nodes is defined and discovered for random graph, which results in the emergence of the frustration and long-range correlation phenomenon. Based on the backbones and interactions with a node adding process, we propose an Interaction and Backbone Evolution Algorithm to achieve the reduced solution graph, which has a direct correspondence to the solution space of Vertex-cover. By this algorithm, the whole solution space can be obtained strictly when there is no leaf-removal core on the graph and the odd cycles of unfrozen nodes bring great obstacles to its efficiency. Besides, this algorithm possesses favorable exactness and has good performance on random instances even with high average degrees. The interaction with the algorithm provides a new viewpoint to solve Vertex-cover, which will have a wide range of applications to different types of graphs, better usage of which can lower the computational complexity for solving Vertex-cover

Claw-free t-perfect graphs can be recognised in polynomial time

A graph is called t-perfect if its stable set polytope is defined by non-negativity, edge and odd-cycle inequalities. We show that it can be decided in polynomial time whether a given claw-free graph is t-perfect

Approximation Algorithms for Partially Colorable Graphs

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances

Streaming Lower Bounds for Approximating MAX-CUT

We consider the problem of estimating the value of max cut in a graph in the streaming model of computation. At one extreme, there is a trivial $2$-approximation for this problem that uses only $O(\log n)$ space, namely, count the number of edges and output half of this value as the estimate for max cut value. On the other extreme, if one allows $\tilde{O}(n)$ space, then a near-optimal solution to the max cut value can be obtained by storing an $\tilde{O}(n)$-size sparsifier that essentially preserves the max cut. An intriguing question is if poly-logarithmic space suffices to obtain a non-trivial approximation to the max-cut value (that is, beating the factor $2$). It was recently shown that the problem of estimating the size of a maximum matching in a graph admits a non-trivial approximation in poly-logarithmic space. Our main result is that any streaming algorithm that breaks the $2$-approximation barrier requires $\tilde{\Omega}(\sqrt{n})$ space even if the edges of the input graph are presented in random order. Our result is obtained by exhibiting a distribution over graphs which are either bipartite or $\frac{1}{2}$-far from being bipartite, and establishing that $\tilde{\Omega}(\sqrt{n})$ space is necessary to differentiate between these two cases. Thus as a direct corollary we obtain that $\tilde{\Omega}(\sqrt{n})$ space is also necessary to test if a graph is bipartite or $\frac{1}{2}$-far from being bipartite. We also show that for any $\epsilon > 0$, any streaming algorithm that obtains a $(1 + \epsilon)$-approximation to the max cut value when edges arrive in adversarial order requires $n^{1 - O(\epsilon)}$ space, implying that $\Omega(n)$ space is necessary to obtain an arbitrarily good approximation to the max cut value