We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erd o s bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size m 2 + (n-1) 4 + k in time 2^O(k)n^4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed several times over the past 15 years. Our algorithm is asymptotically optimal, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size.
For variants of Max-Cut , the boundary of tractability'' above guaranteed values has also been investigated in the setting of parameterized complexity. For instance, in Max-Bisection we seek a cut such that the number of vertices in both sides of the bipartition is as equal as possible, the tight lower bound on the bisection size is only @math ; fixed-parameter tractability of Max-Bisection above @math was recently shown by Gutin and Yeo @cite_20 and Mnich and Zenklusen @cite_8 .
@cite_20 @cite_8
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We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erd o s bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size m 2 + (n-1) 4 + k in time 2^O(k)n^4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed several times over the past 15 years. Our algorithm is asymptotically optimal, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size. In a graph G=(V,E), a bisection (X,Y) is a partition of V into sets X and Y such that |X|=<|Y|=<|X|+1. The size of (X,Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G=(V,E) and are required to find a bisection of maximum size. It is not hard to see that @?|E| 2@? is a tight lower bound on the maximum size of a bisection of G. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (Max-Bisec-ATLB): decide whether a graph G=(V,E) has a bisection of size at least @?|E| 2@?+k, where k is the parameter. We show that this parameterized problem has a kernel with O(k^2) vertices and O(k^3) edges, i.e., every instance of Max-Bisec-ATLB is equivalent to an instance of Max-Bisec-ATLB on a graph with at most O(k^2) vertices and O(k^3) edges. 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 O(n) space, then a near-optimal solution to the max cut value can be obtained by storing an 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 [EQUATION] 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 1 2-far from being bipartite, and establishing that [EQUATION] space is necessary to differentiate between these two cases. Thus as a direct corollary we obtain that [EQUATION] space is also necessary to test if a graph is bipartite or 1 2-far from being bipartite. We also show that for any e > 0, any streaming algorithm that obtains a (1 + e)-approximation to the max cut value when edges arrive in adversarial order requires n1−O(e) space, implying that Ω(n) space is necessary to obtain an arbitrarily good approximation to the max cut value. A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most one, and the size of the bisection is the number of edges which go across the two parts. Every graph with m edges has a bisection of size at least ⌈m 2 ⌉, and this bound is sharp for infinitely many graphs. Therefore, Gutin and Yeo considered the parameterized complexity of deciding whether an input graph with m edges has a bisection of size at least ⌈m 2 ⌉+k, where k is the parameter. They showed fixed-parameter tractability of this problem, and gave a kernel with O(k2) vertices. Here, we improve the kernel size to O(k) vertices. Under the Exponential Time Hypothesis, this result is best possible up to constant factors.