Questions tagged [duality]

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What does the complementary slackness theorem say intuitively/in layman's terms?

In layman's terms: weak duality theorem states that all feasible solutions to the primal LP is less than or equal to all feasible solutions to the dual LP. strong duality theorem states that the ...
1 vote
1 answer
19 views

Dual of a graph where the faces are unclear

To get the dual of a planar graph, each of the faces becomes a vertex. And then, two vertices in the dual graph are connected if they share a common face. My question is about planar graphs that have ...
1 vote
1 answer
166 views

What's an example of a planar graph with two embeddings whose geometric duals are nonisomorphic?

How to prove that the dual of the dual of a connected planar graph $G$ is isomorphic to $G$? In the post linked above, the user "plop" gives a great response where they claim, in particular, ...
3 votes
2 answers
546 views

How to prove that the dual of the dual of a connected planar graph $G$ is isomorphic to $G$?

I find in many references the fact that if $G$ is a connected planar graph, then for any embedding, $G^{**} \cong G$. However, all those references either say that this fact is trivial, or give the ...
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0 answers
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How can I show a problem is in the intersection of np and co-np using duality and Farkas-lemma?

Currently, I have a hard time to find out the solution to this problem: Given a matrix $A \in Z^{m \times n}$, $b \in Z^m$, $c \in R^n$ and $\lambda \in R$. Is there $x \in R^n$ with $Ax \leq b$ and $...
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2 votes
1 answer
133 views

What is the intuition behind the way of reading off a dual optimal solution from simplex primal tabular in CLRS?

Section 29.4 "Duality" of CLRS (3rd Edition) describes the way of reading off an optimal dual solution from the last slack form of the primal as follows: Suppose that the last slack form of the ...
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2 votes
1 answer
708 views

Using LP to prove the max matching - min cover theorem

Konig's theorem says that, in a bipartite graph, the size of the maximum matching equals the size of the minimum vertex cover. This theorem has several proofs; I would like to know if the following ...
1 vote
1 answer
949 views

What is a min-max theorem in graph theory?

I'm currently studying a paper which uses extensively the term 'min-max theorems' in graph theory, and claims to present a tool allowing to generalize these ...
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4 votes
1 answer
154 views

What is a not-well-founded cotree?

I'm reading the paper "Dual of substitution is Redecoration". And I'm struggling with understanding the usage of the word "not-well-founded cotrees". what is a cotree compared to a tree ? I suspect ...
2 votes
1 answer
529 views

Comparing dual of a canonical primal program - Directly and by dual of the standard program

I have it as a homework question to compare dual programs in the following way: Take a canonical program and write its dual Take the same canonical program, write it as a standard program, take the ...
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4 votes
1 answer
187 views

Finding a minimal width strip which encloses a set of points in the plane

Problem: Consider a set of $n$ points in the plane, how could we find a strip of minimal vertical distance that contains all points? Definitions: A strip is defined by two parallel lines and the ...
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4 votes
0 answers
253 views

Intuitive self-contained proof of Farkas' Lemma

I've been studying the proof of Farkas' Lemma, and given my rather fuzzy memory of Linear Algebra, am having some trouble with it. One version of Farkas' lemma states: For any convex cone generated ...
4 votes
0 answers
136 views

Showing a linear program is infeasible or finding a feasible solution

I'm aware that for any given maximize/minimize LP problem, if its dual is unbounded then the primary is infeasible and vice versa. But what if there is no maximize/minimize objective function? For ...
2 votes
1 answer
366 views

What's the dual problem of stable matching?

So the dual problem of max-flow is min-cut. What's the dual problem of stable matching?
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10 votes
1 answer
5k views

Short and slick proof of the strong duality theorem for linear programming

Consider the linear programs \begin{array}{|ccc|} \hline Primal: & A\vec{x} \leq \vec{b} \hspace{.5cm} & \max \vec{c}^T\vec{x} \\ \hline \end{array} \begin{array}{|ccc|} \hline Dual: & \...
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