Editing Open Problems:67
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.
The edit can be undone.
Please check the comparison below to verify that this is what you want to do, and then save the changes below to finish undoing the edit.
| Latest revision | Your text | ||
| Line 1: | Line 1: | ||
{{Header | {{Header | ||
| + | |title=Difficult Instance for Max-Cut in the Streaming Model | ||
|source=bertinoro14 | |source=bertinoro14 | ||
|who=Robert Krauthgamer | |who=Robert Krauthgamer | ||
}} | }} | ||
| + | |||
We are interested in '''Max-Cut''' in the streaming model, and specifically in the tradeoff between approximation and space (storage) complexity. | We are interested in '''Max-Cut''' in the streaming model, and specifically in the tradeoff between approximation and space (storage) complexity. | ||
Formally, in the '''Max-Cut''' problem, the input is a graph $G$, and the goal is to compute the maximum number of edges that cross any single cut $(V_1,V_2)$. This is clearly equivalent to computing the least number of edges that need to be removed to make the graph bipartite. | Formally, in the '''Max-Cut''' problem, the input is a graph $G$, and the goal is to compute the maximum number of edges that cross any single cut $(V_1,V_2)$. This is clearly equivalent to computing the least number of edges that need to be removed to make the graph bipartite. | ||
| Line 8: | Line 10: | ||
Here is a concrete suggestion for a hard input distribution, which is known to be a hard instance for bipartiteness testing in sparse graphs {{cite|GoldreichR-02}}. | Here is a concrete suggestion for a hard input distribution, which is known to be a hard instance for bipartiteness testing in sparse graphs {{cite|GoldreichR-02}}. | ||
| − | Let $ | + | Let $G'$ be a graph consisting of a cycle of length $n$ (where $n$ is even) and a random matching. It is known that with high probability, $G'$ is an expander and at least $0.01 n$ edges have to be removed to turn it into a bipartite graph. Let $G''$ be a graph consisting of a cycle of length $n$ and a random matching, with the constraint that the matching must consist only of ''odd chords'': these are chords that are an odd number of vertices apart on the cycle. It is easy to see that $G''$ is always bipartite. |
| − | The total number of edges in both $ | + | The total number of edges in both $G'$ and $G''$ is exactly $3n/2$. It is easy to see that |
| − | * $ | + | * $G''$ has a cut of size $3n/2$, |
| − | * with high probability, $ | + | * with high probability, $G'$ has no cut of size greater than $(3/2 - 0.01)n$. |
How much space is required to distinguish between these two graphs in the streaming model? Is it $\Omega(n)$? | How much space is required to distinguish between these two graphs in the streaming model? Is it $\Omega(n)$? | ||
And what about the (multi-round) communication complexity of the problem, namely, the edges of the input graph are split between two parties, Alice and Bob, who need to estimate the '''Max-Cut'''? | And what about the (multi-round) communication complexity of the problem, namely, the edges of the input graph are split between two parties, Alice and Bob, who need to estimate the '''Max-Cut'''? | ||
| − | |||
| − | |||
| − | |||
