# Questions tagged [ds.algorithms]

Questions regarding well-defined instructions for completing a task, and relevant analysis in terms of time/memory/etc.

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### Deciding if all matrix multiplication entries have at least two witnesses

Consider two square matrices $A(x,y)$ and $B(y,z)$ of dimensions $N×N$ containing boolean entries. Consider the output product matrix $C(x,z)$ where $C=AB$ (not boolean matrix multiplication but the ...
1 vote
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### Efficient enumeration of connected functional digraphs (up to isomorphism)

Together with the research intern I am supervising, we are currently writing some software that requires us to enumerate all connected functional digraphs of $n$ vertices up to isomorphism (also known ...
1k views

### What is the problem in "closest pair problem" if all points share the same x-coordinate

The closest pair of points problem deals with the task to find a pair of points with the global minimum distance. There is a problem, when all points share the same x-coordinate, or at least a large ...
1k views

### Subgraph isomorphism with a tree

If we have a large (directed) graph $G$ and a smaller rooted tree $H$, what is the best known complexity for finding subgraphs of $G$ isomorphic to $H$? I am aware of results for subtree isomorphism ...
407 views

### What is the Complexity Status of Arbitarily Weighted Planar Max Cut?

If you search on the internet for the Complexity Status of Arbitarily Weighted Planar Max Cut you seem to get conflicting answers. On one hand, there are references that Barahona solved this problem ...
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### A problem in understanding an algorithm

I read a paper from John Hershberger with this title: "Minimizing the sum of diameters efficiently". That paper proposed a simple algorithm that finds a bipartition of points $S$ in the ...
4k views

### Find all pairs of values that are close under Hamming distance

I have a few million 32-bit values. For each value, I want to find all other values within a hamming distance of 5. In the naive approach, this requires $O(N^2)$ comparisons, which I want to avoid. ...
4k views

### Finding k shortest Paths with Eppstein's Algorithm

I'm trying to figure out how the Path Graph $P(G)$ according to Eppstein's Algorithm in this paper works and how I can reconstruct the $k$ shortest paths from $s$ to $t$ with the corresponding heap ...
1 vote
139 views

### Nontrivial Algorithms for Coloring (Parameterized by Pathwidth)

Let $k$ be a positive integer. In the $k$-coloring problem, we are given a graph $G$ on $n$ nodes, and want to determine if there is a way to assign a color to each vertex of $G$ such that no two ...
38 views

### Reducing computing the partition function to computing the number of min-cardinality (s, t) cut

Consider a partition function for a graph as follows: \begin{equation} \mathrm{Z}_\mathrm{G}(\beta) = \sum_{z \in \{-1, 1\}^{n}} \beta^{\underset{(i, j) \in E, i < j}{\sum} w_{i,j} ~z_i z_j}, \end{...
150 views

### Another variation of $k$-means problem in the plane

According to wikipedia, consider $k$-means problem in the plane : k-means clustering aims to partition the $n$ observations into $k (≤ n)$ sets $S = \{S_1, S_2, \dots, S_k\}$ so as to minimize the ...
2k views

### Graph families which have polynomial time algorithms for computing the chromatic number

Post updated on 31st of August: I added a summary of the current answers below the original question. Thanks for all the interesting answers! Of course, everyone can continue posting any new findings. ...
9k views

### What is the maximum number of stable marriages for an instance of the Stable Marriage Problem?

Stable Marriage Problem: http://en.wikipedia.org/wiki/Stable_marriage_problem I am aware that for an instance of a SMP, many other stable marriages are possible apart from the one returned by the ...
131 views

### Parameterized algorithm when the parameter is not known in advance?

In the setting of parameterized algorithms, we are typically given the problem instance as well as the value of the parameter. However, it seems like in applications the value of the parameter should ...
3k views

### minimizing size of regular expression

Suppose we have a regular language specified by a regex, for example, (ab|ac)* and we wish to find an equivalent regex with the minimal number of symbols, (a(b|c))*. Is there any efficient way to do ...
167 views

### Accessible entry for computational complexity theory through concrete problems

I am planning to start studying computational complexity theory. As the field is technical for a fresh undergrad alumni like me, I thought a good approach is to tackle it through areas I am more ...
1k views

### "Directed" problems that are easier than their "undirected" variant.

I was presenting a lecture on pancake sorting, and mentioned that: Sorting by reversals is NP-hard "signed" sorting by reversals is in P. Which got me thinking. There is a sense in which "signed" ...
68 views

### Interesting Variation on Subset Sum Problem

Does anyone have any ideas for this algorithms problem? Given an array $A$ with 40 integers ($-10^9 < A_i < 10^9$), how many ways are there to reach a target sum $X$. Normally, I would use ...
98 views

### Optimal solution for partitioning convex polygon into small pieces

Given a convex polygon $P$ (possibly) with holes. We want to partition $P$ into a minimum number of connected interior-disjoint small pieces $Q_1,...Q_s$. The definition of small can either be that ...
700 views

1 vote
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### 2-Center problem with forbidden pairs

Is there a nearly linear-time 2-approximation (or $O(1)$-approximation) algorithm for the following problem? 2-Center with Forbidden Pairs input: Bipartite graph $G=(V,E)$ where each vertex $v$ is a ...
72 views

### Is a grid graph a vertex-minor of a complete graph? [closed]

Consider a graph $G$. A graph $H$ is the vertex-minor of the graph $G$ if $H$ can be obtained from $G$ using vertex deletions and local complementations. For more information, look at Definition 2.1 ...
151 views

### Do irrational numbers contain an infinite number of (or all possible) patterns of sequences? [closed]

I guess the question is "does an 'infinite' number of patterns imply 'every' number of patterns?" For instance, if you could quickly calculate the decimal sequence of π, could you not (in ...
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### Fastest algorithm to compute maximum number of boxes that can fit inside each other

Given $n$ rectangles with widths $w_1,w_2,...,w_n$ and heights $h_1, ..., h_n$. A rectangle $i$ fits inside $j$ if and only if $h_i<h_j$ and $w_i<w_j$. We are interested in the maximum $k$ such ...
525 views

### OR-circuit complexity of a dense linear operator

Consider the following simple monotone circuit model: each gate is just a binary OR. What is the complexity of a function $f(x)=Ax$ where $A$ is a Boolean $n \times n$ matrix with $O(n)$ 0's? Can it ...
94 views

### optimization on graph edges selection

I have the below problem. I wonder if there exists a similar known class of problems (e.g., in optimization, graph theory) which I can relate my problem to, and find a similar solution there. I am ...
2k views

### Reducing #SAT to #MONOTONE-2SAT

The problem #MONOTONE-2SAT is known to be #P-complete. This means that #SAT can be reduced to it. My question is: given a #SAT instance $F$, which is the transformation that converts $F$ to its ...
4k views

### For which problems in P is it easier to verify the result than to find it?

For (search versions) of NP-complete problems, verifying a solution is clearly easier than finding it, since the verification can be done in polynomial time, while finding a witness takes (probably) ...
Given an undirected graph $G$, an acyclic orientation of $G$ is choice of orientation for each edge of $G$ (turning each edge into an arc) such that the resulting directed graph has no directed cycles....