## #StackBounty: #ds.algorithms #graph-theory #graph-algorithms #directed-acyclic-graph Finding nodes with enough unique ancestors

### Bounty: 50

Given a DAG $$G = (V, E)$$, let $$T subseteq V$$ be a set of nodes of $$V$$ that is computed via the following process. Assuming the nodes of $$G$$ are sorted in topological order, $$v_1, dots, v_n$$. We process the nodes in this topological order. Suppose we are processing vertex $$v_i$$ in the order. Let $$r(v_i)$$ be the set of ancestors of $$v_i$$. We add $$v_i$$ and all ancestors of $$v_i$$ into $$T$$ if and only if the number of ancestors of $$v_i$$ (including $$v_i$$ itself) that are not in $$T$$ is at least $$frac{|r(v_i)| + 1}{f}$$ where $$f$$ is some constant $$f > 1$$ provided as part of the input. If a vertex is added to $$T$$, then all its ancestors are also added into $$T$$. Any algorithm that computes $$T$$ must satisfy the following invariant: any vertex $$v_i$$ added into $$T$$ must have at least $$frac{|r(v_i)| + 1}{f}$$ of its ancestors (including itself) not in $$T$$ when it was added, and any vertex $$v_i$$ not added into $$T$$ must have $$< frac{|r(v_i)| + 1}{f}$$ of its ancestors (including itself) not in $$T$$.

Problem: Provide an algorithm that runs in $$tilde{O}(m + n)$$ time (meaning ignoring polylogarithmic factors) that provides a valid set $$T$$ given an input graph $$G = (V, E)$$ and the processing order of the vertices is the toposort order. Approximation (meaning constant error on the $$frac{r(v_i)+1}{f}$$ condition) and Las Vegas algorithms are welcome also. You may preprocess the graph but the preprocessing time must also be $$tilde{O}(m + n)$$.

Trivial Solution in More Time We can do this trivially in $$O(nm)$$ time. For each $$v_i$$, we find via BFS its set of ancestors and check whether each one is in $$T$$. From this we can compute the exact fraction that is not in $$T$$ and either put $$v_i$$ and $$r(v_i)$$ in $$T$$ or not.

Example Problem: Suppose you have the following trivial DAG (below) and $$f= 2$$.

a -> b -> c

Then, $$T = {a, b}$$. $$a$$ is processed first and $$T = emptyset$$ at first so $$a$$ is added into $$T$$. Then, $$b$$ is processed after $$a$$ and is added to $$T$$ since $$1/2$$ of the vertices in $${a, b}$$ are not in $$T$$. Finally, $$c$$ is processed and not added to $$T$$ since it has at most $$1/3$$ of the vertices in $${a, b, c}$$ are not in $$T$$. Note that in this example, a trivial $$O(n)$$ algorithm for any line is to keep a counter $$c$$ of the index of the last element added to $$T$$ (since adding the element at $$c$$ also means adding all its ancestors) and add the $$i$$-th element of the line to $$T$$ if $$frac{i-c}{i} geq frac{1}{f}$$. However, this problem is much less trivial for general graphs.

Has anyone seen a problem similar to this in the literature or algorithms which solve similar problems?

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## #StackBounty: #cc.complexity-theory #ds.algorithms #graph-theory #linear-algebra #max-flow-min-cut Complexity of Encoding a Matroid Flo…

### Bounty: 50

Context:
Take a directed graph $$G$$ with a specified subset of source vertices $$S$$ and target vertices $$T$$.
We say a subset $$Isubseteq T$$ of size $$r$$ is independent if there exist $$r$$ distinct vertices in $$S$$ which can be connected to distinct vertices of $$I$$ via a collection of $$r$$ vertex-disjoint paths in $$G$$. In other words, there is a flow of size $$r$$ from $$S$$ to $$I$$.

It turns out that a graph together with independent subsets of vertices defined in this way forms a structure known as a gammoid, which is itself a special case of a structure known as a matroid.
Many algorithms that work with matroids take as input a linear representation of the matroid. In the context of this problem, this representation is a matrix $$M$$ whose column vectors $$vec{v}u$$ are indexed by vertices $$u$$ in $$G$$, with the property that a subset $$I$$ of vertices is independent if and only if the corresponding list of column vectors $${vec{v}_u}$${uin I}\$ is linearly independent.

Several sources (see here and here for example) observe that is well-known that given $$G$$, $$S$$, and $$T$$, one can build a matrix representation of the associated gammoid in randomized polynomial time. However, I have not been able to find any reference to the precise running time of this algorithm.
This motivates the following question.

Question: Given a graph $$G$$ with source set $$S$$ and target set $$T$$, how quickly can we build a matrix representation of the associated gammoid? Randomness is allowed.

I’m mainly interested in the dependence on the the number of vertices $$n$$ in the graph, but it would be good to know the dependence on the number of source vertices $$|S| = k$$ as well.

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## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!

## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!

## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!

## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!

## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!

## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!

## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!

## #StackBounty: #cc.complexity-theory #graph-theory #sat #counting-complexity #planar-graphs Disproving \$oplus\$ETH by reducing \$oplus k…

### Bounty: 100

In this question and its answer, they discuss about reducing CNF-SAT with $$n$$ variables and $$m$$ clauses to a (problem on) planar graph $$G=(V,E)$$ with $$|V|$$ as small as possible. It is said that the best known reduction has $$|V| = m^2$$, and that if a better reduction with $$|V| in o(m^2)$$ is found, that would refute ETH.

There is a reduction from $$oplus k$$-SAT with $$n$$ variables and $$m$$ clauses to $$oplus$$VERTEX COVER where the output graph $$G=(V,E)$$ is planar and has $$|V| = 51(k+1)nm$$. Such reduction clearly meets the $$|V| in o(m^2)$$ requirement when $$k$$ is a constant and $$m$$ is superlinear in $$n$$.

Question
Can the same line of reasoning made within the linked question be applied here in order to refute $$oplus$$ETH, or am I missing some important detail?

Get this bounty!!!