Difference between revisions of "Open Problems:67"
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| − | The | + | The construction presented below is a hard instance for bipartiteness testing in sparse graphs {{cite|GoldreichR-02}}. In the '''MaxCut''' problem the goal is to approximate the size of the maximum cut in the graph. More formally, the goal is to approximate the maximum number of edges passing between $V_1$ and $V_2$ taken over all partitions of the set of vertices of the graph into $V_1$ and $V_2$. We conjecture that the construction below is also hard for '''MaxCut''' in the streaming model. |
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 ''even chords'': these are chords that are an even number of vertices apart on the cycle. It is easy to see that $G'$ is bipartite. | 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 ''even chords'': these are chords that are an even number of vertices apart on the cycle. It is easy to see that $G'$ is bipartite. | ||
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The total number of edges in both $G$ and $G'$ is $3n/2$. It is easy to see that | The total number of edges in both $G$ and $G'$ is $3n/2$. It is easy to see that | ||
Revision as of 02:46, 5 June 2014
| Suggested by | Robert Krauthgamer |
|---|---|
| Source | Bertinoro 2014 |
| Short link | https://sublinear.info/67 |
The construction presented below is a hard instance for bipartiteness testing in sparse graphs [GoldreichR-02]. In the MaxCut problem the goal is to approximate the size of the maximum cut in the graph. More formally, the goal is to approximate the maximum number of edges passing between $V_1$ and $V_2$ taken over all partitions of the set of vertices of the graph into $V_1$ and $V_2$. We conjecture that the construction below is also hard for MaxCut in the streaming model.
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 even chords: these are chords that are an even number of vertices apart on the cycle. It is easy to see that $G'$ is bipartite.
The total number of edges in both $G$ and $G'$ is $3n/2$. It is easy to see that
- $G'$ has a cut of size $3n/2$,
- 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?
