# Questions tagged [max-cut]

For a graph, a maximum cut is a cut whose size is at least the size of any other cut. The problem of finding a maximum cut in a graph is known as the max-cut problem.

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### Positive cut algorithm on bipartite graphs with negative weights

Let $G=(V,E,w)$ be a bipartite graph with weight function $w:E→\{-1,1\}$. Is there an efficient (polynomial) algorithm for finding some positive (not necessarily maximum) cut of $G$, if one exists? If ...
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### Max flow: either saturate an edge or avoids

Is there a way to create a max flow graph such that it satisfies the condition that a flow either saturates an edge or completely avoids it. It can't have half its flow through one edge and half ...
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I am interested in the complexity of the decision problem whether max-cut with positive and negative edge weights has a solution with positive value: Given a graph $G=(V, E)$ and edge weights $w: E \... 13 votes 4 answers 837 views ### A Multi-cut Problem I'm looking for a name or any references to this problem. Given a weighted graph$G = (V, E, w)$find a partition of the vertices into up to$n = |V|$sets$S_1,\ldots,S_n$so as to maximize the ... 5 votes 2 answers 2k views ### Size of MAXCUT from eigenvalues Is there an interpretation of MAXCUT using eigenvalues of the graph that yields constant factor approximation to MAXCUT? Can the estimates provide sharp lower bound to MAXCUT? 0 votes 0 answers 94 views ### Prove that Vertex Cover is NP-Complete by reducing MaxCut to Vertex Cover This is not the most straight forward reduction available on the internet since most people start from the fact that vertex cover is NP-complete and reduce a given vertex cover instance to MaxCut ... 1 vote 0 answers 48 views ### Max Flow Routing Let G = (V,E,S,I,T) be a directed flow network with nodes V, edges E with unit capacity, source nodes S$\subseteq$V, intermediate nodes I$\subseteq$V, and target nodes T$\subseteq$V. The problem ... 8 votes 2 answers 1k views ### Approximation algorithms for MAX-CUT, when sizes of partition sets are fixed The MAX-CUT problem has constant factor approximation, but we can't control the sizes of the sets in resulting partition. What is known about maximizing cut size, if we restrict one part of the ... 1 vote 0 answers 180 views ### Easier famility of graphs for MAXCUT [closed] I would like to know if there are particular family of graphs for which the Goemans-Williamson MAXCUT Approximation Algorithm renders higher than 0.878 approximation ratio. TIA 2 votes 1 answer 264 views ### Minimum cut with size bounds$k\leq |S| \leq |V|-k$It is known by the max flow min cut theorem that the minimum cut problem is in$P$. I am interested in knowing what is known on the complexity of the minimum cut with size$k\leq |S| \leq , |V|- k$. ... 1 vote 1 answer 479 views ### State of the art approximation algorithm for$\text{MAXCUT}$that does better than Goemans and Williamson I had thought that the Goemans-Williamson approximation algorithm was the best for MAXCUT. To quote from Wikipedia: The polynomial-time approximation algorithm for Max-Cut with the best known ... 6 votes 0 answers 515 views ### Fastest exact algorithm for MAXCUT Is the algorithm introduced in the following paper still the fastest exact algorithm for general MAXCUT problems? TIA Ryan Williams, A new algorithm for optimal$2$-constraint satisfaction and its ... 2 votes 0 answers 107 views ### Resource on phase transition in MAXCUT problems Could anyone suggest reading materials on phase transition in MAXCUT problems other than [1]? Thanks. Ref: Coppersmith, Don, David Gamarnik, Mohammad Taghi Hajiaghayi, and Gregory B. Sorkin. "... -4 votes 1 answer 164 views ### Partition vertices of graph into two sets such that there are at least$k$edges between sets [closed] I have to show that for every integer$k$, the problem whether the vertices of input graph can be partitioned into two sets such that there are a least$k$edges between the sets can be solved in ... 9 votes 1 answer 4k views ### What's the approximation factor of this Max k-Cut approximation? I'm thinking about an approximation algorithm for Max k-Cut. One simple and more involved approximation algorithms can be found here. The Max k-Cut problem is defined as follows. Input is a graph G = ... 2 votes 1 answer 209 views ### Sum-of-Squares Certificates We say that$f$has a degree$2d$sum-of-squares certificate if$f=\sum_{i=1}^r (g_i(x))^2$, where for each$i\in[r]$, we have that$g_i$is a polynomial of degree at most$d$. Thus showing that$f$... 2 votes 1 answer 362 views ### MaxCut instance with smallest max cut Let us look at all 4-regular undirected graphs with$n$nodes and edge weight equals to 1 for all edges. Out of these graphs, I would like to find the MaxCut instance with least number of edges in its ... 4 votes 1 answer 458 views ### Intuitive explanation behind Goemans-Williamson randomized rounding A very simple randomized cut algorithm achieves$1/2$of the optimal value: just choose each vertex to be in the cut with probability$1/2$, independently. Goemans-Williamson does something more ... 2 votes 1 answer 94 views ### Weighted Min-Cut in bounded-genus graphs What is the status of the following decision problem ? Input : A graph$G=(V,E)$embedded in a torus (or more generally a surface of genus$g$), a weight function$w:E \rightarrow \{-1,1\}$Output : ... 4 votes 1 answer 248 views ### Complexity of finding Exact Size Cut-Sets in Bipartite Graphs I am interested in the problem of deciding if a cut-set of a given size$k$(i.e. the number of edges crossing the partitions is$k$) exists in a given bipartite graph (both the graph and$k$are part ... 3 votes 1 answer 182 views ### SOS hardness of$Max-2-Lin(\mathbb{Z}_2)$? Do we know of instances of$Max-2-Lin(\mathbb{Z}_2)$which have a integrality gaps w.r.t to high degree (> 4) SOS relaxations? Or if we specialize to Max-CUT do we know of graphs whose Max-CUT ... -2 votes 1 answer 1k views ### Is the max cut problem still NP-Complete for graphs with unit weights on the edges? [closed] We know that finding a max cut for weighted graphs is NP-Complete. I am trying to find a proof showing that even for graphs with just unit weights (every edge has weight 1) it is still NP-Complete. I'... 39 votes 3 answers 5k views ### Max-cut with negative weight edges Let$G = (V, E, w)$be a graph with weight function$w:E\rightarrow \mathbb{R}$. The max-cut problem is to find: $$\arg\max_{S \subset V} \sum_{(u,v) \in E : u \in S, v \not \in S}w(u,v)$$ If the ... 2 votes 2 answers 109 views ### Name of graph partition that balances edges between sets with edges remaining within sets Is there a common name for this problem: Let G=(V, E) be an undirected graph. Partition V into sets$S_1$,$S_2$, ...,$S_k$, such that (the number of edges between sets) + (the number of "non"-... 7 votes 0 answers 3k views ### Reduction from Vertex Cover to Max-Cut? [closed] I am referring to Computational Complexity by Arora and Barak for my course. In the section on NP-completeness reductions, the book has a diagram that is represents how one NP-complete problem ... 5 votes 0 answers 121 views ### The distribution on the solution space induced by randomized rounding Consider the Goemans-Williamson algorithm for the MAX-CUT problem. It is known, that if$maxcut(G) \geq 1-\epsilon$, then the algorithm returns a cut$S$of fractional size at least$1-\sqrt{\epsilon}$... 8 votes 0 answers 111 views ### Approximating a max-cut's intersection with other cuts For the purposes of this question, a cut in a graph$G$is the edge-set$\delta (S)\subseteq E(G)$between some vertex-set$S$and its complement. A max cut is one with at least as many edges as any ... 9 votes 1 answer 486 views ### A purely graph-theoretic explanation of the reduction from Unique Label Cover to Max-Cut I am studying the Unique Games Conjecture and the famous reduction to Max-Cut of Khot et al. From their paper and elsewhere on the internet, most authors use (what to me is) an implicit equivalence ... 21 votes 2 answers 1k views ### Max-Cut algorithm that shouldn't work, unclear why OK, this might seem like a homework question and, in a sense, it is. As a homework assignment in an undergraduate algorithms class, I gave the following classic: Given an undirected graph$G=(V,E)$, ... -2 votes 2 answers 219 views ### MAX Cut with an oracle Suppose that I have a MAX CUT problem on a weighted undirected Graph$G$, but there is an oracle that tells me what the value of the MAX CUT is, but not which edges produce it. Does this make the ... 5 votes 1 answer 327 views ### Best approximation for a HYPERGRAPH-MAXDICUT problem Consider a$(c^a,(c+d)^a,1)$-regular directed hypergraph$\mathcal{H}(a)$on$n^a$vertices with fixed$n\geq c+d+1$, fixed$c\geq 2$, fixed$d\geq 0$and variable parameter$a\geq 1$(meaning every ... 2 votes 1 answer 121 views ### partition to min the max number of intersections Given$n$items and$m$customers, each of whom is interested in some subset of the items, partition the set of items among$k$different stores so that the maximum number of customers visiting any ... 3 votes 1 answer 336 views ### What is this matrix column-selection problem, and how hard is it to approximate? I ran across the following simple-to-state problem involving selection of a subset of columns simultaneously for a number of matrices. I suspect it might be well known, though I can't seem to place it.... 8 votes 3 answers 995 views ### Hardness of MAX-CUT on sparse graphs Let a weighted graph$G(V,E)$, where the weights are real (positve and negative). Assume that$G$has$\mathcal{O}(n\log n)$edges. How fast can we compute MAX-CUT on this graph? Can we compute (... 9 votes 1 answer 739 views ### Is Max-Cut APX-complete on triangle-free graphs? In the Max-Cut problem, one seeks a subset S of vertices of a given simple undirected graph such that the number of edges between S and the complement of S is as large as possible. Max-Cut is APX-... 3 votes 0 answers 225 views ### The Quality of SDP relaxation on MaxCut My question is: given a maxcut instance, if it costs too much to solve it to optimal practically but we can get an optimal solution of SDP relaxation quickly, can we assess the quality of this SDP ... 3 votes 1 answer 2k views ### Max-cut via linear programming or sdp I am looking for a linear programming formulation for the max-cut problem. My interest is to know about the primal - dual algorithm for max-cut. It would be nice if someone can tell me that what is ... 3 votes 0 answers 159 views ### Gram matrix of Max-Cut relaxation It seems that Goemans and Williamson give a unique representation for each graph of the semidefinite relaxation (elements$y_{ij}$of Y). However, semidefinite programming may give the same maximum ... 12 votes 2 answers 524 views ### Euclidean-squared max-cut in low dimensions Let$x_1, \ldots, x_n$be points in the plane$\mathbb{R}^2$. Consider a complete graph with the points as vertices and with edge weights of$\|x_i - x_j\|^2$. Can you always find a cut of weight that ... 11 votes 1 answer 454 views ### Examples of hard instances for Goemans and Williamson algorithm I'm interested in the explicit examples of graphs for which application of Goemans and Williamson algorithm for approximating maximum cuts results in 0.878…-approximation factor. The algorithm to ... 11 votes 1 answer 1k views ###$\epsilon$-nets with respect to the cut norm The cut norm$||A||_C$of a real matrix$A = (a_{i,j}) \in \mathcal{R}^{n\times n}$is the maximum over all$I \subseteq [n], J \subseteq [n]$of the quantity$\left|\sum_{i \in I, j \in J}a_{i,j}\...
Max Bisection problem is to partition the set of nodes into two equal size sets such that the number of crossing edges is maximum. Max Bisection is $NP$-complete on cubic graphs and also on planar ...
CSP optimization problem is approximation resistant if it is $NP$-hard to beat the approximation factor of a random assignment. For instance, MAX 3-LIN is approximation resistant since a random ...