Polynomial-time approximation scheme for Max-Cut problem

最大化匹配割

Maximizing Matching Cuts

本文综述了最大匹配割问题，即寻找图中最大的匹配割，并比较了相关匹配割变体，探讨了与最大割和最小割的关系。

Disconnected Cuts in Claw-free Graphs

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several homomorphism and contraction problems, and fits in an extensive line of research on vertex cuts with additional properties. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms are known for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to t…

Approximating Max-Cut under Graph-MSO Constraints

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.

A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer $d\ge 3$, Matching Cut is NP-complete in the class of graphs of diameter $d$. This resolves an open problem posed by Borowiecki and Jesse-Józefczyk in [Matching cutsets in graphs of diameter $2$, Theoretical Computer Science 407 (200…

Max-Cut and Max-Bisection are NP-hard on unit disk graphs

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}$.

量子最大割问题是NP难的近似问题

Quantum Max-Cut is NP hard to approximate

本文无条件证明在常数有界度无权图上，量子最大割问题的常数乘法近似是NP难的，并通过两阶段归约及PTAS归约证明了Max-Cut_k对所有常数k是APX完全的。

Clique cover and graph separation: New incompressibility results

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. Although a framework for proving kernelization lower bounds has been discovered in 2008 and successfully applied multiple times over the last three years, establishing kernelization complexity of many important problems remains open. In this paper we show that, unless NP is a subset of coNP/poly and the polynomial hierarchy collapses up to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduce…

全局标签最小割问题的紧准多项式界

A tight quasi-polynomial bound for Global Label Min-Cut

研究证明全局标签最小割问题的准多项式时间复杂度可能最优，其下界与指数时间假设相矛盾。

On The Complexity of Matching Cut for Graphs of Bounded Radius and $H$-Free Graphs

For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most $d$ if $d\leq 2$ and NP-complete if $d\geq 3$. We prove the same dichotomy for graphs of bounded radius. For a graph $H$, a graph is $H$-free if it does not contain $H$ as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for $P_6$-free graphs, extending a recent result of Feghali for $P_5$-free g…

Max-Cut Parameterized Above the Edwards-Erdős Bound

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…

Approximation Complexity of Max-Cut on Power Law Graphs

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 $β>2$.

ReLU网络训练的多项式时间解：基于Max-Cut和Zonotope的复杂性分类

Polynomial-Time Solutions for ReLU Network Training: A Complexity Classification via Max-Cut and Zonotopes

本文研究了带权重衰减正则化的两层ReLU网络训练复杂性，证明了其近似难度与Max-Cut问题对应，并针对三类数据集首次给出了多项式时间近似保证与近似难度结果。

分离 MAX 2-AND、MAX DI-CUT 与 MAX CUT

Separating MAX 2-AND, MAX DI-CUT and MAX CUT

本文在唯一博弈猜想下改进 MAX DI-CUT 的近似算法并给出新的 UGC 困难性结果，证明其最佳近似比严格低于 MAX CUT，同时反驳 MAX 2-AND 与 MAX DI-CUT 近似比相同的猜想，完成三个问题的分离。

Improving the smoothed complexity of FLIP for max cut problems

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 …

MAX-3-CUT与量子MAX-CUT的强难解性

Sharp Hardness for MAX-3-CUT and Quantum MAX-CUT

假设唯一游戏猜想，该研究证明MAX-3-CUT和量子MAX-CUT的近似问题分别在对应因子下是NP难的，解决了相关猜想并推广了稳定定理。

以树宽和割宽为参数时Tutte多项式在整数点上求值复杂度的细粒度分类

A Fine-Grained Classification of the Complexity of Evaluating the Tutte Polynomial on Integer Points Parameterized by Treewidth and Cutwidth

本文对以树宽和割宽为参数时Tutte多项式在整数点上的求值复杂度进行了细粒度分类，通过改进现有归约并引入矩阵秩界新贡献，明确了不同整数点下的多项式与指数级时间复杂度边界及ETH下界。

最大匹配割的二分性：$H$-无关性、有界直径、有界半径

Dichotomies for Maximum Matching Cut: $H$-Freeness, Bounded Diameter, Bounded Radius

研究 Maximum Matching Cut 的计算复杂性，在有界直径、有界半径和无 $H$ 图上给出完整二分性，并据此证明 Disconnected Perfect Matching 在无 $(P_6+sP_2)$ 图上多项式可解。

Finding cuts of bounded degree: complexity, FPT and exact algorithms, and kernelization

A matching cut is a partition of the vertex set of a graph into two sets $A$ and $B$ such that each vertex has at most one neighbor in the other side of the cut. The MATCHING CUT problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [IPEC 2018], we introduce a natural generalization of this problem, which we call $d$-CUT: for a positive integer $d$, a $d$-cut is a bipartition of the vertex set of a graph into two sets $A$ and $B$ such that each vertex has at most $d$ neighbors across the cut. We …

Smoothed complexity of local Max-Cut and binary Max-CSP

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…

Approximating Max-Cut on Bounded Degree Graphs: Tighter Analysis of the FKL Algorithm

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.

On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes

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…

Near-optimal approximation algorithm for simultaneous Max-Cut

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 facto…

平面图和环面图的最大割算法与上界

Maximum Cut Algorithms and Upper Bounds for Planar and Toroidal Graphs

提出将任意权重平面图最大割映射到绝对对偶图的最小T-join问题，推广了Hadlock算法；证明环面图应用平面算法可得最大割上界，并基于此推导GSet基准中所有环面图的上界，部分问题达到上界；引入基于平面图算法的环面图最大割启发式算法，复现所有已知最佳结果并发现新最佳值。

Maximum Cut Parameterized by Crossing Number

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…

最大割问题的SDP精确性识别的复杂性

The Complexity of Recognizing SDP Exactness for the Maximum Cut Problem

本文解决最大割问题SDP精确性的计算复杂性，证明加权图的SDP精确性判定为强NP难，还将该难解性扩展到简单无权图，所用几何锁定机制解耦了连续界与组合难解性。

Node Max-Cut and Computing Equilibria in Linear Weighted Congestion Games

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…

Computing the Largest Bond and the Maximum Connected Cut of a Graph

The cut-set $\partial(S)$ of a graph $G=(V,E)$ is the set of edges that have one endpoint in $S\subset V$ and the other endpoint in $V\setminus S$, and whenever $G[S]$ is connected, the cut $[S,V\setminus S]$ of $G$ is called a connected cut. A bond of a graph $G$ is an inclusion-wise minimal disconnecting set of $G$, i.e., bonds are cut-sets that determine cuts $[S,V\setminus S]$ of $G$ such that $G[S]$ and $G[V\setminus S]$ are both connected. Contrasting with a large number of studies related to maximum cuts, there exist very few results regarding the largest bond of general graphs. In this…