We consider the graph $k$-partitioning problem under the min-max objective, termed as Minmax $k$-cut. The input here is a graph $G=(V,E)$ with non-negative edge weights $w:E\rightarrow \mathbb{R}_+$ and an integer $k\geq 2$ and the goal is to partition the vertices into $k$ non-empty parts $V_1, \ldots, V_k$ so as to minimize $\max_{i=1}^k w(δ(V_i))$. Although minimizing the sum objective $\sum_{i=1}^k w(δ(V_i))$, termed as Minsum $k$-cut, has been studied extensively in the literature, very little is known about minimizing the max objective. We initiate the study of Minmax $k$-cut by showing that it is NP-hard and W[1]-hard when parameterized by $k$, and design a parameterized approximation scheme when parameterized by $k$. The main ingredient of our parameterized approximation scheme is an exact algorithm for Minmax $k$-cut that runs in time $(λk)^{O(k^2)}n^{O(1)}$, where $λ$ is value of the optimum and $n$ is the number of vertices. Our algorithmic technique builds on the technique of Lokshtanov, Saurabh, and Surianarayanan (FOCS, 2020) who showed a similar result for Minsum $k$-cut. Our algorithmic techniques are more general and can be used to obtain parameterized approximation schemes for minimizing $\ell_p$-norm measures of $k$-partitioning for every $p\geq 1$.
We study two variants of \textsc{Maximum Cut}, which we call \textsc{Connected Maximum Cut} and \textsc{Maximum Minimal Cut}, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas \textsc{Maximum Cut} on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-wid…
Trevisan [SICOMP 2012] presented an algorithm for Max-Cut based on spectral partitioning techniques. This is the first algorithm for Max-Cut with an approximation guarantee strictly larger than 1/2 that is not based on semidefinite programming. Trevisan showed that its approximation ratio is of at least 0.531. In this paper we improve this bound up to 0.614247. We also define and extend this result for the more general Maximum Colored Cut problem.
An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut δ(S) such that the induced graph G[S] is connected. We present the first non-trivial Ω(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark contrast to the classical max-cut problem, we show that the connecte…
In a second seminal paper on the application of semidefinite programming to graph partitioning problems, Goemans and Williamson showed how to formulate and round a complex semidefinite program to give what is to date still the best-known approximation guarantee of .836008 for Max-$3$-Cut. (This approximation ratio was also achieved independently by De Klerk et al.) Goemans and Williamson left open the problem of how to apply their techniques to Max-$k$-Cut for general $k$. They point out that it does not seem straightforward or even possible to formulate a good quality complex semidefinite pro…
We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for $γ$-stable Max Cut instances with $γ\geq c\sqrt{\log n}\log\log n$ for some absolute constant $c > 0$. Our algorithm is robust: it never returns an incorrect answer; if the instance is $γ$-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not $γ$-stable. We prove that there is no robust polynomial-time algorithm for $γ$-stable instances of Max Cut when $γ< α_{SC}(n/2)$, where $α_{SC}$ is the best approximation factor f…
We propose a fixed-parameter tractable algorithm for the \textsc{Max-Cut} problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithm recursively reduces a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. The \textsc{Max-Cut} problem is then solved on the planar graphs using established polynomial-time algorithms. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for th…
We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithms recursively reduce a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [23]. In the case of non-negative edge weights, we suggest a variant that al…
We give a deterministic algorithm for finding the minimum (weight) cut of an undirected graph on $n$ vertices and $m$ edges using $\text{polylog}(n)$ calls to any maximum flow subroutine. Using the current best deterministic maximum flow algorithms, this yields an overall running time of $\tilde O(m \cdot \min(\sqrt{m}, n^{2/3}))$ for weighted graphs, and $m^{4/3+o(1)}$ for unweighted (multi)-graphs. This marks the first improvement for this problem since a running time bound of $\tilde O(mn)$ was established by several papers in the early 1990s. Our global minimum cut algorithm is obtained as…
Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min $s-t$ Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled $+$ or $-$ and the goal is to produce a clustering that agrees with the labels as much as possible: $+$ edges within clusters and $-$ edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective. We depart from this and study local objectives: minimizing the maximum number of disagr…
We show that for every $\varepsilon > 0$, the degree-$n^\varepsilon$ Sherali-Adams linear program (with $\exp(\tilde{O}(n^\varepsilon))$ variables and constraints) approximates the maximum cut problem within a factor of $(\frac{1}{2}+\varepsilon')$, for some $\varepsilon'(\varepsilon) > 0$. Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut, and hence resolves the extension complexity of approximate Max-Cut for approximation factors close to $\frac{1}{2}$ (up to the function $\varepsilon'(\varepsilon)$). Previously, only semide…
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-tr…
In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-cut-gap. Planarity means here that the union of the supply and demand graph is planar. We first prove that there exists a multiflow of value at least half of the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer one of value at least half of the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing again at most half of the value, in polynomial time, achieving a $1/4$-a…
We give an $O(n \log \log n)$ time algorithm for computing the minimum cut (or equivalently, the shortest cycle) of a weighted directed planar graph. This improves the previous fastest $O(n\log^3 n)$ solution. Interestingly, while in undirected planar graphs both min-cut and min $st$-cut have $O(n \log \log n)$ solutions, in directed planar graphs our result makes min-cut faster than min $st$-cut, which currently requires $O(n \log n)$.
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 describe a new approximation algorithm for Max Cut. Our algorithm runs in $\tilde O(n^2)$ time, where $n$ is the number of vertices, and achieves an approximation ratio of $.531$. On instances in which an optimal solution cuts a $1-ε$ fraction of edges, our algorithm finds a solution that cuts a $1-4\sqrtε + 8ε-o(1)$ fraction of edges. Our main result is a variant of spectral partitioning, which can be implemented in nearly linear time. Given a graph in which the Max Cut optimum is a $1-ε$ fraction of edges, our spectral partitioning algorithm finds a set $S$ of vertices and a bipartition $…
We show sublinear-time algorithms for Max Cut and Max E2Lin$(q)$ on expanders in the adjacency list model that distinguishes instances with the optimal value more than $1-\varepsilon$ from those with the optimal value less than $1-ρ$ for $ρ\gg \varepsilon$. The time complexities for Max Cut and Max $2$Lin$(q)$ are $\widetilde{O}(\frac{1}{ϕ^2ρ} \cdot m^{1/2+O(\varepsilon/(ϕ^2ρ))})$ and $\widetilde{O}(\mathrm{poly}(\frac{q}{ϕρ})\cdot {(mq)}^{1/2+O(q^6\varepsilon/ϕ^2ρ^2)})$, respectively, where $m$ is the number of edges in the underlying graph and $ϕ$ is its conductance. Then, we show a sublinea…
Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $δ(U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $Ω$ such that $w(Ω)$ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the…
In this work, we initiate the study of fault tolerant Max Cut, where given an edge-weighted undirected graph $G=(V,E)$, the goal is to find a cut $S\subseteq V$ that maximizes the total weight of edges that cross $S$ even after an adversary removes $k$ vertices from $G$. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures $k$ we present an approximation of $(0.878-ε)$ against an adaptive adversary and of $α_{GW}\approx 0.8786$ against an o…
In an undirected graph, a $k$-cut is a set of edges whose removal breaks the graph into at least $k$ connected components. The minimum weight $k$-cut can be computed in $O(n^{O(k)})$ time, but when $k$ is treated as part of the input, computing the minimum weight $k$-cut is NP-Hard [Holdschmidt and Hochbaum 1994]. For $\operatorname{poly}(m,n,k)$-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi 2017]. Saran and Vazirani [1995] showed that a $(2 - 2/k)$-approximately minimum weight $k$-cut can be computed by $O(k)$ min…
The $k$-cut problem asks, given a connected graph $G$ and a positive integer $k$, to find a minimum-weight set of edges whose removal splits $G$ into $k$ connected components. We give the first polynomial-time algorithm with approximation factor $2-ε$ (with constant $ε> 0$) for the $k$-cut problem in planar and minor-free graphs. Applying more complex techniques, we further improve our method and give a polynomial-time approximation scheme for the $k$-cut problem in both planar and minor-free graphs. Despite persistent effort, to the best of our knowledge, this is the first improvement for the…
An instance of the graph-constrained max-cut (GCMC) problem consists of (i) an undirected graph G and (ii) edge-weights on a complete undirected graph on the same vertex set. The objective is to find a subset of vertices satisfying some graph-based constraint in G that maximizes the total weight of edges in the cut. The types of graph constraints we can handle include independent set, vertex cover, dominating set and connectivity. Our main results are for the case when G is a graph with bounded treewidth, where we obtain a 0.5-approximation algorithm. Our algorithm uses an LP relaxation based …
Given a graph $G=(V, E)$, a connected cut $δ(U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the connected maximum cut problem (CMAX CUT) is to find a connected cut $Ω$ such that $w(Ω)$ is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs. In this paper, we prove that CMAX CUT is polynomial for graphs without $K_5\backslash e$ as a minor. We deduce a quadratic time algorithm for the minimum cut pro…
In 1988, Johnson, Papadimitriou and Yannakakis wrote that "Practically all the empirical evidence would lead us to conclude that finding locally optimal solutions is much easier than solving NP-hard problems". Since then the empirical evidence has continued to amass, but formal proofs of this phenomenon have remained elusive. A canonical (and indeed complete) example is the local max-cut problem, for which no polynomial time method is known. In a breakthrough paper, Etscheid and Röglin proved that the smoothed complexity of local max-cut is quasi-polynomial, i.e., if arbitrary bounded weights …
We investigate the space complexity of two graph streaming problems: Max-Cut and its quantum analogue, Quantum Max-Cut. Previous work by Kapralov and Krachun [STOC `19] resolved the classical complexity of the \emph{classical} problem, showing that any $(2 - \varepsilon)$-approximation requires $Ω(n)$ space (a $2$-approximation is trivial with $\textrm{O}(\log n)$ space). We generalize both of these qualifiers, demonstrating $Ω(n)$ space lower bounds for $(2 - \varepsilon)$-approximating Max-Cut and Quantum Max-Cut, even if the algorithm is allowed to maintain a quantum state. As the trivial a…
We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the c…
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.
We describe a factor-revealing convex optimization problem for the integrality gap of the maximum-cut semidefinite programming relaxation: for each $n \geq 2$ we present a convex optimization problem whose optimal value is the largest possible ratio between the value of an optimal rank-$n$ solution to the relaxation and the value of an optimal cut. This problem is then used to compute lower bounds for the integrality gap.
We give a simple classical algorithm which provably achieves the performance in the title. The algorithm is a simple modification of the Gaussian wave process.
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…
Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $ε> 0$, we develop randomized approximation algorithms achieving a ratio of $(1-ε)$ to the optimum for Max-Cut on bipartite graphs…
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…
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 problem. Let $G = (V,E)$ be a graph with adjacency matrix $(a_{ij})_{i,j=1}^{n}$. Burer, Monteiro & Zhang proposed to find, for $n$ angles $\left\{θ_1, θ_2, \dots, θ_n\right\} \subset [0, 2π]$, minima of the energy $$ f(θ_1, \dots, θ_n) = \sum_{i,j=1}^{n} a_{ij} \cos{(θ_i - θ_j)}$$ because configurations achieving a global minimum leads to a partition of size 0.878 Max-Cut(G). This approach is known to be computationally viable and leads to very good results in practice. We prove that by replacing $\cos{(θ_i - θ_j)}$ with an explicit function $g_{\varepsilon}(θ_i - θ_j)…
Finding locally optimal solutions for max-cut and max-$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 …
For a graph $G$, let $f(G)$ denote the size of the maximum cut in $G$. The problem of estimating $f(G)$ as a function of the number of vertices and edges of $G$ has a long history and was extensively studied in the last fifty years. In this paper we propose an approach, based on semidefinite programming (SDP), to prove lower bounds on $f(G)$. We use this approach to find large cuts in graphs with few triangles and in $K_r$-free graphs.
We consider the problem of deterministically enumerating all minimum $k$-cut-sets in a given hypergraph for any fixed $k$. The input here is a hypergraph $G = (V, E)$ with non-negative hyperedge costs. A subset $F$ of hyperedges is a $k$-cut-set if the number of connected components in $G - F$ is at least $k$ and it is a minimum $k$-cut-set if it has the least cost among all $k$-cut-sets. For fixed $k$, we call the problem of finding a minimum $k$-cut-set as Hypergraph-$k$-Cut and the problem of enumerating all minimum $k$-cut-sets as Enum-Hypergraph-$k$-Cut. The special cases of Hypergraph-$k…
The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph $G=(V,E)$ and two disjoint sets of its vertices $S,T$, we design a fixed-parameter tractable algorithm which counts minimum edge $(S,T)$-cuts parameterized by their size $p$. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most $n=\left| V \right|$ successive minimum $(S,T)$-cuts $Z_i$. We prove that any minimum $(S,T)$-cut $X$ contains edges of at least one cut $Z…
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 $β>2$.
The Max-Cut problem is a fundamental NP-hard problem, which is attracting attention in the field of quantum computation these days. Regarding the approximation algorithm of the Max-Cut problem, algorithms based on semidefinite programming have achieved much better approximation ratios than combinatorial algorithms. Therefore, filling the gap is an interesting topic as combinatorial algorithms also have some merits. In sparse graphs, there is a linear-time combinatorial algorithm with the approximation ratio $\frac{1}{2}+\frac{n-1}{4m}$ [Ngoc and Tuza, Comb. Probab. Comput. 1993], which is know…
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…
There are many applications of max flow with capacities that depend on one or more parameters. Many of these applications fall into the "Source-Sink Monotone" framework, a special case of Topkis's monotonic optimization framework, which implies that the parametric min cuts are nested. When there is a single parameter, this property implies that the number of distinct min cuts is linear in the number of nodes, which is quite useful for constructing algorithms to identify all possible min cuts. When there are multiple Source-Sink Monotone parameters and the vector of parameters are ordered in th…
