# 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.

36 questions
Filter by
Sorted by
Tagged with
118 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 $k\leq |S| \leq , |V|- k$. ...
103 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 ...
93 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 $f$ ...
168 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 ...
227 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 ...
83 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 : ...
146 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 ...
177 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 ...
738 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'...
195 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 ...
90 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"-...
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 ...
116 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 $1-\sqrt{\epsilon}$...
107 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 ...
606 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 ...
354 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 ...
897 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 $G=(V,E)$, ...
197 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 ...
305 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 ...
1k 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?
111 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 ...
290 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....
816 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 (...
665 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-...
209 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 ...
1k 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 ...
154 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 ...
473 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 ...
417 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 ...