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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25 views

How to deal with the time to minimize a function in a given interval?

I'm writing a paper in which I designed an algorithm running in $O(n^2m)\cdot T(f)$ to solve my problem, where $n,m$ is the size of input and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a function, and $T(...
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1answer
171 views

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 ...
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1answer
65 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 ...
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0answers
48 views

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 ...
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0answers
85 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 ...
6
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1answer
278 views

Complexity of optimal elimination for a planar tensor network

Edit Dec 15 it's not obvious this problem is tractable when further restricting to trees, see cs.SE question Suppose we need to sum out variables in a tensor network (a factor graph where each ...
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1answer
102 views

Algorithm for finding traffic equilibrium

I watched a youtube video about a certain interesting property of springs and road networks. It made me think: if we represent a network of roads as a graph where edges are roads described by a ...
10
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0answers
122 views

Fastest Known Algorithm to Count Acyclic Orientations in a Graph

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....
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1answer
176 views

Young Diagrams and distinguishing between two distributions

Introduction: The reference for everything is this paper. The Robinson–Schensted–Knuth (RSK) algorithm is a well-known combinatorial algorithm with diverse applications throughout mathematics, ...
6
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1answer
234 views

How can we compute the VC dimension of a finite class of sets?

Let $F$ be a class of subsets of a finite set $X$ of cardinality $n$. What is the complexity of computing the VC dimension of $F$? Can we do better than looping through every subset of $X$ and ...
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0answers
59 views

Minimize Cumulative Cost on Topological Sort

We are given a n-vertex DAG $G=(V,E)$ and also given a cost function $c: V \rightarrow \Bbb N$. Given a topological sort $S = v_1,v_2,...,v_n$, it has associated a sorting cost $S_c = \sum_{i=1}^{n} C(...
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44 views

Approximate solution for maximum coverage problem with choice constraint

Suppose a sequence of sets $S_1,S_2,...,S_i$ where each set contains sets of elements. That is, each set $S$ contains many sets $a_1,a_2,...,a_{|S|}$. We are given an integer $k$ and we assume that $\...
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1answer
167 views

Are there an algorithm that find Minimum spanning tree in $O(n^2\log\log^*n)$?

Given completed metric weighted graph $G=(V,E)$ that have $n$ vertices. Are there an algorithm that find MST of $G$ in $O(n^2)$? I read abstract of this paper that mentioned an algorithm with running ...
5
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0answers
83 views

Fine-Grained Hardness for Undirected Hamiltonicity

The fastest known algorithm for detecting Hamiltonian cycles in directed graphs on $n$ nodes runs in essentially $2^n\text{poly}(n)$ time. However, for undirected graphs on $n$ nodes, there is an ...
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0answers
103 views

Fastest Known Algorithm for $k$-Dimensional Matching and $k$-Exact Cover

Given a $k$-uniform hypergraph $G$ (i.e., each edge of $G$ contains precisely $k$ vertices) on $n$ vertices, the $k$-Exact Cover problem is the task of deciding if there exists $n/k$ edges in $G$ ...
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1answer
146 views

Divide and Conquer Algorithm for 1-Median Problem

Let $P_1$ and $P_2$ be two disjoint point sets in $\mathbb{R}^d$ and $n = \vert P_1\vert = \vert P_2\vert$ and $P = P_1\cup P_2$. Let $c^\star$ be the optimal 1-median for $P$ and $opt^\star$ is the ...
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0answers
50 views

Decision tree vs. pebble game lower bounds

This question concerns two types of lower bounds. In a pebbling lower bound, we are concerned with the complexity of constructing the output from the input. For example, if the only way we could ...
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0answers
22 views

Fast algorithms for convex-convex quadratic fractional programming

What are the fastest algorithm(s) (possibly approximation algorithms) for solving convex-convex quadratic fractional programming problems, i.e. optimization problems of the form $$ \begin{align*} \sup&...
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1answer
210 views

Is this a novel technique for determining whether or not two rotated rectangles collide?

I was trying to determine whether or not two rectangles rotated around their centers were colliding and randomly thought to try the following algorithm: Rotate both rectangles by the negative rotation ...
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2answers
282 views

Partition a graph into two clusters

Suppose given a complete weighted graph $G=(V,E)$, with positive weight. Are there an algorithm that partition $G$ into two clusters $C_1,C_2$ such that sum of heaviest edges in $C_1,C_2$ minimized? ...
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0answers
93 views

How typical are odd-H-minor free graphs?

Can anything be said about how typical are odd-H-minor free graphs? (definition of odd-minor-free is in Section 2.2 of notes, page 20 of slides). For instance, for a random graph with $n$ vertices, $...
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0answers
18 views

Fast filter for concurrent version vectors (vector clocks, lamport clocks) from a set

I have a set of version vectors (or vector clocks) and need to implement a filter function that takes this set and returns a new set where all elements are either "equal", "happened ...
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1answer
96 views

Max-k-cut with negative edge weights

Let $G=(V,E,w)$ be a graph and for edge $e\in E$, there is associated weight $w_e$. The max-k-cut wants to partition V into k subsets $P_1,\cdots,P_k$ and maximize $\sum_{1\leq r<s\leq k}\sum_{i\in ...
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0answers
301 views

Problem in the paper "Stable Minimum Space Partitioning in Linear Time"

The paper Stable Minimum Space Partitioning in Linear Time describes an algorithm that stably sorts a binary array (an array whose elements can only have two distinct values) in $O(n)$ time complexity ...
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0answers
73 views

Parametrization of context-sensitive language in polynomial time

Let $\Sigma$ be a finite alphabet. Let $L\subset \Sigma^*$ be a context-sensitive language containing a word of every length. Can we always find $f:\Sigma^*\to L$ computable in polynomial time in ...
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1answer
55 views

Set cover where consecutive sets differ by at most one item [closed]

First I define my version of the set cover problem: We have a collection of sets such as $S_1, \dots, S_m$ where each $S_i$ is a subset of $M=\{1,\dots, m\}$. The goal is to find the minimum number of ...
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1answer
51 views

How to build the tree with the "most different" solutions of a clustering?

Illustrate the question with an example : we have a similarity matrix for 1000 people, and the similarity represents how much their hobbies are the same (it does not really matter how it's built). Let'...
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0answers
147 views

Theorem on non-decreasing probability of success of an algorithm

Question: What's a standard name/framework for the following, or some variant? Stated briefly for a probabilistic algorithm: define $p_n$ as the conditional probability of success of a fork of the ...
9
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3answers
243 views

Sublinear Time Regular Expression Search

Does there exist a data structure with the following properties. Given a string $s$, it performs some polynomial amount of precomputation to construct the data structure. After construction, it allows ...
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0answers
147 views

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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2answers
189 views

Finding the point that maximizes a linear function

Consider $N$ two-dimensional points of the form $(x_i, y_i)$ where all $x_i, y_i > 0$ are positive integers. We will be given a workload of queries $Q = \{c_1, \dots, c_k\}$ where for each $c_j \in ...
2
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1answer
121 views

Do there exist two equivalent objective functions one of which can be approximated but another one cannot?

I have two equivalent problems A and B, meaning that the optimal solution of one must be the optimal solution of another one. However, it seems that problem A can be approximated but B cannot. Below ...
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0answers
83 views

Flipping one bit to maximize BMM output

Consider a boolean matrix $A$ of size $N \times N$ and let $A^\top$ be its transpose. Let $C = AA^\top$ be the boolean matrix multiplication (BMM) result and let $c$ be the number of non-negative ...
1
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1answer
128 views

Finding output with unique witness in matrix multiplication

Consider two square matrices $A(x,y)$ and $B(y,z)$ of dimensions $N \times N$ containing boolean entries. Consider the output product matrix $C(x,z)$ where $C = AB$ (not boolean matrix multiplication ...
3
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1answer
343 views

How hard is this combinatorial optimisation problem?

Suppose we have multiple intervals $R_1,R_2,...,R_i$ of non-negative integers. These intervals may overlap and we use $R_h(\mathrm{median})$ to denote the median integer in the $h$-th interval $R_h$, ...
2
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1answer
162 views

Characterization of integral polyhedra

A rational polyhedron $P \subseteq \mathbb{R}^n$ is an integral polyhedron if it is the convex hull of its integer points. That is, if $P = conv(P \cap \mathbb{Z}^n)$. Equivalently, $P$ is integral if ...
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4answers
3k views

Algorithms Careers

I’ve been writing software for a living for a number of years now. I have graduate background in mathematics and I am wondering whether knowledge of higher algorithms is utilized anywhere in industry. ...
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0answers
203 views

How to find the second smallest cut in a graph?

For an undirected graph, how do we find the second smallest $s,t-$cut(s) for some $s,t\in V$? What's the time complexity of this computation? What if we only cared about finding a cut of size $p+1$, ...
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0answers
49 views

Generalizing PageRank for tripartite graphs

Problem I have the following directed tripartite graph $G(E\cup V\cup P, A)$, where there is a many-to-one symmetric relationship between the subsets V and E - $e\in E,v\in V,[e, v]\in A \iff [v, e]\...
0
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1answer
71 views

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 ...
3
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1answer
96 views

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 ...
2
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1answer
117 views

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 ...
4
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1answer
422 views

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 $L = L(X)$. ...
5
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0answers
164 views

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 ...
4
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2answers
173 views

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 ...
3
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1answer
169 views

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 ...
2
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0answers
73 views

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-...
5
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0answers
171 views

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 $A$...
12
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2answers
191 views

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 ...
4
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1answer
192 views

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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