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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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Deciding if max-cut with negative edge weights has a solution with positive value

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 \...
badboul's user avatar
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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 ...
Chaithanya's user avatar
1 vote
0 answers
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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 ...
sripurva's user avatar
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
Omar Shehab's user avatar
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 ...
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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 ...
Omar Shehab's user avatar
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. "...
Omar Shehab's user avatar
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 $k\leq |S| \leq , |V|- k$. ...
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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 ...
Majstro's user avatar
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 $f$ ...
learning's user avatar
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 ...
Freiburger0's user avatar
4 votes
1 answer
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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 ...
Aryeh's user avatar
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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 : ...
Mathieu Mari's user avatar
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 ...
allrtaken's user avatar
3 votes
1 answer
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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 ...
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1 answer
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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'...
Burak Yıldıran Stodolsky's user avatar
9 votes
1 answer
336 views

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 ...
curl-up's user avatar
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2 votes
2 answers
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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"-...
Zack's user avatar
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7 votes
0 answers
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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 ...
taninamdar's user avatar
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 $1-\sqrt{\epsilon}$...
Lior Eldar's user avatar
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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 ...
Ross Churchley's user avatar
3 votes
2 answers
892 views

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 ...
Quanquan Liu's user avatar
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 ...
Jeremy Kun's user avatar
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21 votes
2 answers
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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 $G=(V,E)$, ...
Yonatan's user avatar
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-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 ...
Victor Miller's user avatar
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 ...
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?
Turbo's user avatar
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2 votes
1 answer
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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 ...
user16035's user avatar
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....
sd234's user avatar
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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 (...
Dimitris's user avatar
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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-...
Standa Zivny's user avatar
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 ...
Ben's user avatar
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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 ...
singhsumit's user avatar
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 ...
N27's user avatar
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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 ...
Milos Hasan's user avatar
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 ...
mkatkov's user avatar
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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 = ...
Saeed's user avatar
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11 votes
1 answer
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$\epsilon$-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}\...
Aaron Roth's user avatar
  • 9,900
4 votes
0 answers
219 views

Complexity of Max Bisection on cubic planar graphs?

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 ...
Mohammad Al-Turkistany's user avatar
8 votes
1 answer
803 views

Is MAX CUT approximation resistant?

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 ...
Mohammad Al-Turkistany's user avatar
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 ...
Aaron Roth's user avatar
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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 ...
Grigory Yaroslavtsev's user avatar
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 ...
Aaron Roth's user avatar
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