# 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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### Triangle detection hardness in regular graphs

Consider a tripartite graph over $n^{1-\epsilon}$ vertices each in sets $I, J, K$. Suppose we impose a constraint that every vertex has degree $n^\epsilon/c$ for some constant $\epsilon > 0$ and ...
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### Given a partition and an element, find the subset that includes this element

I am interested in the following simple problem: Let $X$ be a set and $X_1\cup X_2\cup\cdots\cup X_k$ be a finite partition of $X$. Given $x\in X$, find the subset $X_i$ for which $x\in X_i$. I am ...
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### Algorithm to find $n$ player nash equilibrium

This is the question asked 10 years before. Most of the algorithm and software mentioned are out of date. I am wondering is there new approchs for this problem in the last 10 years? The game dealing ...
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### Efficient tools for checking SMT formulas with two quantifiers ($\exists\forall$)

I would like to check a sort of SMT formulas with two quantifiers where universal variables range over finite/bounded integer domains. An example formula is \exists x \forall y ((y \ge 1 \land y \le ...
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### Can we efficiently convert from NFA to smallest equivalent DFA?

Definitions For any automaton $X$, let $L(X)$ denote the language recognized by $X$. For any language $L$, let $sc(L)$ denote the number of states in the smallest DFA $X$ such that $L = L(X)$. ...
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### Minimum spanning tree, but with an unusual objective function

This is a problem that came up in my study of rumour networks. I was wondering if anyone had thoughts or references on this problem. If we have a rooted tree $T = (V,E)$ with root $r$, I first label ...
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### The Dyck Language Correction Problem

Given a word $w$ over $\{ [, ] \}$, the alphabet of the two square brackets, the Dyck correction problem is to find the shortest sequence of edit operations that would make $w$ a Dyck word, i.e., a ...
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### Fastest approximate triangle counting algorithms in dense graphs

One may compute the number of triangles in a graph by matrix multiplication in time $O(n^\omega)$. There is also a very simple algorithm that runs in time $O(n^3/(\epsilon^2 T))$ (where $T$ is the ...
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### Is there an approximation algorithm for the three-person stable roommates problem? [closed]

While there's an algorithm for solving the stable roommates problem, I understand that the three-people-per-room version of that problem, sometimes called the threesome roommates problem, is NP-...
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### Is this problem in P? Given a bipartite graph, find a minimum cardinality set of edges which intersect every vertex cover

This problem came up in my study of digraphs: Given a connected bipartite graph $G = (A \cup B, E)$, a vertex cover is a set $S$ of vertices such that every edge has some endpoint in $S$. Note that $A$...
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### Is the exponent in the rectangular matrix multiplication convex?

My question is regarding the paper "Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor". In the paper, the authors show an algorithm for multiplying a ...
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### Is $GCT$ necessarily a negative result program?

$GCT$ is a candidate program to separate permanent and determinant through symmetries. If indeed permanent and determinant can be handled in similar complexity class would $GCT$ be a program which can ...
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### Is the Triangle Finding decision problem in $coNTIME(\tilde{O}(n^2))$?

The Triangle Finding decision problem asks whether there exists a triangle in a graph $G$ containing $n$ vertices. A triangle is a triple of vertices $(a, b, c)$ such that $a$ is adjacent to $b$, $b$ ...
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### NP-hardness for one-dimensional facility location problem with entrance fee for each customer [closed]

We have $n$ customers, $(x_1, \dots, x_n)$, sorted on the read line. For convenience, we also use $x_i$ to denote its coordinate on the line. We need to locate $m$ facilities on the real line. We note ...
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### Can we always find a graph with a given algebraic connectivity?

This is crossposted from math stackexchange. This is my first time posting here, so let me know if I'm doing something wrong. I would like to experiment with various spectral properties of graphs, ...
1 vote
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### Dynamic permutation cycle data

Let $\pi \in S_n$ be a permutation of $\{1, \ldots, n\}$. Does there exist a simple data structure that admits the following operations in polylogarithmic time? sameCycle($\pi,x,y$): determines ...
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### An extension to the student/college admissions Gale-Shapley algorithm

The Gale-Shapley algorithm is an established algorithm that finds an optimal one-to-one match between two groups, each individual of which has a preference for the individuals in the other group (the &...
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### Computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$

This question is about computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$. The computation model we are considering is the unit-cost RAM machine with linear ...
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### Finding nodes with enough unique ancestors

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 ...
1 vote
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### Separating DAGs using separators consisting of lists of nodes and all ancestors

Suppose we are given a DAG, $G = (V, E)$ where $n = |V|$. We consider the sets $J_1, J_2, \dots, J_n$ to be lists of vertices where list $J_i$ consists of vertex $v_i \in V$ and all ancestors of $v_i$....
1 vote
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### Remove cycles from a stochastic comparison matrix, while doing the least amount of editing

Let $\mathcal P_n$ be the collection of all matrices $M \in [0, 1]^{n \times n}$ such that $M_{ij} + M_{ji} = 1$ for all $i, j \in [n]$. Such matrices are called comparison matrices. A comparison ...
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### Is there a competitive algorithm for this online scheduling problem to minimize the truncated gaps?

Time is discrete. There are $n$ time-slots and a single job that can be scheduled on one machine of budget $B$. If the job is scheduled at time-slot $t$, then it will consume $c(t)$ units of the ...
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### Has parameterized complexity led to better algorithms?

I know that for the vertex cover problem, if we know that the parameter $k$ (which is the number of vertices in the solution) is small, then we can expect to solve it feasibly in practice. So far, ...
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### Complexity of Encoding a Matroid Flow Problem in a Matrix

Context: Take a directed graph $G$ with a specified subset of source vertices $S$ and target vertices $T$. We say a subset $I\subseteq T$ of size $r$ is independent if there exist $r$ distinct ...
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### In an $m$ by $n$ Boolean matrix, can you find a square block whose four corners are ones in $O(m \cdot n)$ time?

Decision Problem Input: An $m$ by $n$ Boolean matrix $M$. Decision Question: Does there exist a square block within $M$ such that upper-left corner entry == upper-right corner entry == lower-left ...
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### Algorithm to calculate coordinated radio frequencies

I'm stuck on this algorithm problem for a project I'm working on. I can find no relevant documentation, and would very much appreciate any help that can be offered on this! I will try to explain it ...
Let's say we are using a binary tree to represent a set of elements, with operations $\mathsf{insert}(x)$ and $\mathsf{delete}(x)$. We will assume that the operations are used such that a deleted ...
### Generate cut $(A,B)$ in edge-colored graph $(V,E_1 \cup E_2)$ such that there are more red than white crossings, i.e $|E_1(A,B)| > |E_2(A,B)|$
Let $G=(V,E)$ be graph. Recall that a cut of $G$ is (or can uniquely be identified with) a pair $(A,B)$ of nonempty subsets of $V$ which partition it. Given a cut $(A,B)$, let \$E(A,B) := \{(a,b) \in ...