In the simultaneous Max-Cut problem, we are given $k$ weighted graphs on the same set of $n$ vertices, and the goal is to find a cut of the vertex set so that the minimum, over the $k$ graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of $1/2 - o(1)$ for this problem (and an approximation factor of $1/2 + ε_k$ in the unweighted case, where $ε_k \rightarrow 0$ as $k \rightarrow \infty$).
In this work, we give a polynomial time approximation algorithm for simultaneous Max-Cut with an approximation factor of $0.8780$ (for all constant $k$). The natural SDP formulation for simultaneous Max-Cut was shown to have an integrality gap of $1/2+ε_k$ in [BKS15]. In achieving the better approximation guarantee, we use a stronger Sum-of-Squares hierarchy SDP relaxation and a rounding algorithm based on Raghavendra-Tan [RT12], in addition to techniques from [BKS15].
We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk graphs. We also show that $λ$-precision graphs are planar for $λ$ > 1 / \sqrt{2}$.
We show that the smoothed complexity of the FLIP algorithm for local Max-Cut is at most $\smash{ϕn^{O(\sqrt{\log n})}}$, where $n$ is the number of nodes in the graph and $ϕ$ is a parameter that measures the magnitude of perturbations applied on its edge weights. This improves the previously best upper bound of $ϕn^{O(\log n)}$ by Etscheid and Röglin. Our result is based on an analysis of long sequences of flips, which shows~that~it is very unlikely for every flip in a long sequence to incur a positive but small improvement in the cut weight. We also extend the same upper bound on the smoothed…
In this note, we describe a $α_{GW} + \tildeΩ(1/d^2)$-factor approximation algorithm for Max-Cut on weighted graphs of degree $\leq d$. Here, $α_{GW}\approx 0.878$ is the worst-case approximation ratio of the Goemans-Williamson rounding for Max-Cut. This improves on previous results for unweighted graphs by Feige, Karpinski, and Langberg and Florén. Our guarantee is obtained by a tighter analysis of the solution obtained by applying a natural local improvement procedure to the Goemans-Williamson rounding of the basic SDP strengthened with triangle inequalities.
Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this paper, we give FPT algorithms for the maximum cardinality cut problem in classes of graphs containing proper interval graphs and mixed unit interval graphs when parameterized by some new parameters that we introduce. These new parameters are related to a generalization of the so-called bubble representations of proper interval graphs and mixed unit interval graphs and to clique-width decompositions…
Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number $k$ of crossings in a given drawing of $G$. Our algorithm achieves a running time of $O(2^k \cdot p(n + k))$, where $p$ is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs (i.e., at most one crossing per edge) and its dependency on $k$ is of order $3^k$ . A d…
Finding locally optimal solutions for max-cut and max-$k$-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei showed that the smoothed complexity of …
We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erdős 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 computabl…
We consider the max-cut and max-$k$-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a $\frac{1}{2}$-approximation algorithm for this class of problems.
In this paper we study the MAX-CUT problem on power law graphs (PLGs) with power law exponent $β$. We prove some new approximability results on that problem. In particular we show that there exist polynomial time approximation schemes (PTAS) for MAX-CUT on PLGs for the power law exponent $β$ in the interval $(0,2)$. For $β>2$ we show that for some $ε>0$, MAX-CUT is NP-hard to approximate within approximation ratio $1+ε$, ruling out the existence of a PTAS in this case. Moreover we give an approximation algorithm with improved constant approximation ratio for the case of $β>2$.
Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-ferromagnetic XY model) and 0.498 (anti-f…
In this work, we seek a more refined understanding of the complexity of local optimum computation for Max-Cut and pure Nash equilibrium (PNE) computation for congestion games with weighted players and linear latency functions. We show that computing a PNE of linear weighted congestion games is PLS-complete either for very restricted strategy spaces, namely when player strategies are paths on a series-parallel network with a single origin and destination, or for very restricted latency functions, namely when the latency on each resource is equal to the congestion. Our results reveal a remarkabl…
