Open Problems:67 - Revision history

173.72.39.74 at 21:24, 2 November 2020

2020-11-02T21:24:30Z

Krzysztof Onak at 04:50, 28 April 2017

2017-04-28T04:50:03Z

Sassadi at 16:14, 20 April 2017

2017-04-20T16:14:57Z

Krzysztof Onak: Krzysztof Onak moved page Waiting:Robert Krauthgamer to Open Problems:67

2014-06-13T23:02:31Z

Krzysztof Onak moved page Waiting:Robert Krauthgamer to Open Problems:67

Krzysztof Onak at 23:02, 13 June 2014

2014-06-13T23:02:16Z

Krzysztof Onak at 14:46, 12 June 2014

2014-06-12T14:46:26Z

Krzysztof Onak at 12:40, 5 June 2014

2014-06-05T12:40:55Z

Krzysztof Onak at 12:37, 5 June 2014

2014-06-05T12:37:23Z

132.76.50.6 at 05:58, 5 June 2014

2014-06-05T05:58:27Z

Krzysztof Onak at 02:46, 5 June 2014

2014-06-05T02:46:26Z

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 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 $G'$ 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.
+Let $G_1$ 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, $G_1$ is an expander and at least $0.01 n$ edges have to be removed to turn it into a bipartite graph. Let $G_2$ 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 $G_2$ is always bipartite.
-The total number of edges in both $G'$ and $G''$ is exactly $3n/2$. It is easy to see that
+The total number of edges in both $G_1$ and $G_2$ is exactly $3n/2$. It is easy to see that
-* $G''$ has a cut of size $3n/2$,
+* $G_2$ has a cut of size $3n/2$,
-* with high probability, $G'$ has no cut of size greater than $(3/2 - 0.01)n$.
+* with high probability, $G_1$ 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 $\Omega(n)$?
 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'''?

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 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'''?
-== Update ==
+== Updates ==
+The progress on the complexity of '''Max-Cut''' is described in updates on [[Open_Problems:45|Problem 45]].

← Older revision		Revision as of 16:14, 20 April 2017
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	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'''?
		+
		+	== Update ==
		+
		+	See {{ProblemLink\|45}}.

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 {{Header
 |source=bertinoro14
 |who=Robert Krauthgamer
 }}
 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.

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 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 $G'$ 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 always 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 ''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 $G'$ and $G''$ is exactly $3n/2$. It is easy to see that

← Older revision	Revision as of 23:02, 13 June 2014
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 }}
-The following construction is a hard instance for bipartiteness testing in sparse graphs {{cite|GoldreichR-02}}. We conjecture that it is also hard for '''MaxCut''' in the streaming model.
+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.
 The total number of edges in both $G$ and $G'$ is $3n/2$. It is easy to see that