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Questions tagged [algorithms]

An algorithm is a sequence of well-defined steps that defines an abstract solution to a problem. Use this tag when your issue is related to design and analysis of algorithms.

29
votes
0answers
321 views

Finding an $st$-path in a planar graph which is adjacent to the fewest number of faces

I am curious whether the following problems has been studied before, but wasn't able to find any papers about it: Given a planar graph $G$, and two vertices $s$ and $t$, find an $s$-$t$ path $P$ ...
20
votes
0answers
418 views

Approximate minimum-weighted tree decomposition on complete graphs

Say I have a weighted undirected complete graph $G = (V, E)$. Each edge $e = (u, v, w)$ is assigned with a positive weight $w$. I want to calculate the minimum-weighted $(d, h)$-tree-decomposition. By ...
19
votes
0answers
429 views

Largest set of cocircular points

Given $n$ points with integer coordinates in the plane, determine the maximum number of points that lie on the same circle (on its circumference, not its interior). This can be done in $O(n^3)$ ...
15
votes
0answers
286 views

Is finding a weight-balanced tree NP-hard?

In the following, we consider binary trees where only the leaves have weights. Let $T$ be a binary tree and $W(T)$ be the sum of the weights of its leaves. Let $T.l$ and $T.r$ be the left child and ...
12
votes
0answers
347 views

Choosing a subset of binary variables to maximize the sum of the highest $K$

Consider the following problem: Input: integers $n > m > k$; $n$ numbers $0 \leq p_1, \ldots, p_n \leq 1$; $n$ numbers $r_1, \ldots, r_n$ where ($r_i \geq 0$). Let $X_1,\dots,X_n$ be $n$ ...
12
votes
0answers
455 views

Test whether two languages are equal, when give in algebraic form

This sub-problem is motivated by Algorithm to test whether a language is regular. Suppose we have two languages $L_1,L_2$ that are expressed in "algebraic" form, as formalized below. I want to ...
11
votes
0answers
259 views

How fast can we compute the size of maximum matching in an unweighted bipartite graph?

Is there a way to compute the size of a maximum matching in an unweighted bipartite graph more efficiently (e.g. faster) than computing a maximum matching? It is a long shot but it is often an ...
11
votes
0answers
225 views

Change in the distances in a graph after removal of a node

Given an undirected unweighted graph $G=(V,E)$ and a node $s \in V$, we are looking for a vector $\operatorname{diff}[]$, such that, $$\operatorname{diff}[v] = \sum_{u \in V \setminus \{v\}}{(d^{G \...
11
votes
0answers
561 views

Fast algorithm for max-convolution with concave functions?

I'm interested in a discrete max-convolution problem, which is to compute $$r(c) = \max_{x | x \ge 0, \sum_k x_k = c} \left[ \sum_{k=1} f_k(x_k) \right] $$ for all values $c=0, \ldots, C$, where $x=(...
10
votes
0answers
412 views

Optimal meeting point in directed graph

Let $G(V, E)$ be a edge-weighted directed connected graph and $v_1, \dots, v_n \in V$ be some vertices. Let $d(a, b)$ denote the length of the shortest path from $a$ to $b$, for $a,b \in V$. I need ...
10
votes
0answers
277 views

Shift-resolve parsing - questions

I've recently came across a paper describing the parsing technique mentioned in the title. Unfortunately, the terminology used in said paper is somewhat beyond my comprehension, so I've been ...
10
votes
0answers
990 views

Alternatives to SVD for rank factorization

I have rank-deficient matrix $M \in \mathbb{R}^{n\times m}$ with $\text{rank}(M) = k$ and I want to find a rank factorization $M = PQ$ with $P \in \mathbb{R}^{n \times k}$ and $Q \in \mathbb{R}^{k \...
8
votes
0answers
112 views

Covering a complete graph with n copies of an arbitrary graph: NP-complete?

Given a complete graph $G$, an arbitrary graph $H$, and a positive integer $n$, are there subgraphs $A_1,\dots,A_n$ of $G$ (not necessarily disjoint) such that their union is $G$, and each of them ...
8
votes
0answers
128 views

Can you multiply complex 2x2 matrices in fewer than 21 real multiplies?

It is well known that 2x2 matrices can be multiplied using just 7 (instead of the obvious 8) multiplications in the ground field (Strassen-Winograd, etc.). It is also well known that complex numbers ...
8
votes
0answers
118 views

What is the best algorithm to compute ALL homomorphisms between two rooted labeled trees?

Lets consider two node-labeled rooted trees Q and D. According to wikipedia definition ( https://en.wikipedia.org/wiki/Tree_homomorphism ) a mapping m from the nodes of Q to the nodes of D is a tree ...
8
votes
0answers
174 views

Optimizing order of graph reduction to minimize memory usage

Having extracted the data-flow in some rather large programs as directed, acyclic graphs, I'd now like to optimize the order of evaluation to minimze the maximum amount of memory used. That is, given ...
7
votes
0answers
172 views

Algorithms for curve construction

I am interested in algorithms that construct continuous curves between two points in such a way that minimizes an energy functional of the curve. What sort of algorithms are most used for such tasks? ...
7
votes
0answers
130 views

When does greediness guarantee optimality?

I was wondering if there is any theoretical results characterizing under what condition does greedy algorithm actually finds the optimal solution. Here is a motivating example. Suppose you are trying ...
7
votes
0answers
269 views

How to efficiently divide a grid with some predefined sub-rectangles?

Given an $M$x$N$ grid with some predefined sub-rectangles. How to divide the grid into a set of sub-rectangles, which contains the predefined sub-rectangles, so that its the number of elements is ...
7
votes
0answers
172 views

Computing the “at least k friends in common” graph

Suppose we have the graph of a social network with symmetric connections (e.g. Facebook or LinkedIn). Suppose we would like to find all pairs of people who have at least k friends in common, in order ...
7
votes
0answers
137 views

Practical algorithms for the disjoint paths problem

Given an undirected graph $G$ and two pairs of vertices $(s_1, t_1), (s_2, t_2)$, the disjoint paths problem (DPP) asks for two vertex-disjoint paths, one from $s_1$ to $t_1$ and the other from $...
7
votes
0answers
1k views

What machine learning method for diabetes prediction SW?

I'm thinking of an application for diabetics, that, given previous values of blood glucose and insulin dosage, predicts the glucose level for the next few hours. I know a few things about neural ...
7
votes
0answers
137 views

Overlap Maximization problem

Here's the problem: I have a collection of collections, $C$, where each $c\in C$ is a collection of sets $X\subset U$. Denote $c_i$ as the i-th $X$ in $c$. Informally, I want to map all the sets in ...
7
votes
0answers
310 views

Worst-case sparse graphs for Hopcroft-Karp Algorithm

Of large sparse biparite graphs (say degree 4) with N verticies, roughly speaking, which of them cause the worst case running time of the Hopcroft-Karp algorithm? What is their general structure and ...
7
votes
0answers
1k views

Area of the union of rectangles anchored on the x-axis

I am trying to solve the following computational geometry problem. Let $S$ be a set of $n$ axis-parallel rectangles in the plane, so that the bottom edge of each rectangle in $S$ lies on the $x$-...
7
votes
0answers
715 views

Weighted Maximum 3-DIMENSIONAL-MATCHING with restricted weights (Approx Algo)

If the weights of the weighted 3-DIMENSIONAL-MATCHING problem are restricted to let's say, 1 and 2, is there a possibility to reduce this case to the unweighted 3-DIMENSIONAL-MATCHING problem? (...
6
votes
0answers
150 views

Testing algorithm for a modified sieve of Eratosthenes

Context: I am looking at a modified version of the sieve of Eratosthenes. I started by generalising Eratosthenes' sieve, like so: Choose some starting "root", $n_0\in\mathbb{N}$, a sieve limit (the ...
6
votes
0answers
336 views

Trying to find a human-usable method to figure out optimal round 1 openings for this game

I'm trying to figure out optimal round 1 openings for this game: http://generals.io/. For the purposes of this question, I've simplified some of the rules and mechanics of the game, and I assume that ...
6
votes
0answers
87 views

$3$-dimensional convex hull using only a desired number of planes

I would like to find the convex polytope with the smallest volume that envelops (contains) all the points of a given 3D point cloud and that can be constructed from only $k$ planes. This is similar to ...
6
votes
0answers
193 views

Is greedy minimax permutation rejecting sorting optimal?

I sketch an impractical, theoretical comparison sort. Initialize a list of all $n!$ permutations of size $n$. For each possible pair of indices $i, j$, count how many permutations would get rejected ...
6
votes
0answers
47 views

Linear functions of matrix exponential

Given a matrix $A$ and a vector $v$, I'm aware there are efficient algorithms for computing $e^Av$, where efficient means significantly faster than computing $e^A$ and multiplying by $v$. For a ...
6
votes
0answers
205 views

What's the complexity of solving a packing LP?

Linear Programming is in polynomial time weakly (when numbers are encoded in unary). AFAIK it remains open if it is possible to solve LP in polynomial time strongly (when numbers are encoded in ...
6
votes
0answers
54 views

Is it possible to reduce functional equations to SAT?

The problem of finding a solution for functional equations can be defined as: Let A0, A1, A2... An, B0, B1, B2... Bn, X be terms of the lambda calculus, all terms known, except for X, unknown. ...
6
votes
0answers
457 views

Parallel algorithm to find if a set of nodes is on an elememtry cycle in a directed/undirected graph

I'm looking to find / develop a simple parallel algorithm that does this: Input: vs: list of root vertices max_length: max cycle length max_dist: max distance to root Variants one variant of ...
6
votes
0answers
194 views

How are basic feasible solutions in linear programming related to vertices in its corresponding polytope?

In Section 2.3.3 "Polytopes and LP" of the book "Combinatorial Optimization: Algorithms and Complexity" by Christos H. Papadimitriou, Theorem 2.4 establishes the relation between bfs's (basic feasible ...
6
votes
0answers
1k views

Stopping condition for goal-directed bidirectional search for shortest path

So I have a graph and need to find shortest path between two points in it. I need1 to do it it using bidirectional search. The bidirectional search should be goal-directed, i.e. A*. So let $l(u,v)$ ...
6
votes
0answers
409 views

Space filling between random 2D lines

Note that I had asked this question in GIS forum, although it has gotten many up-votes, still has not received any answer. Hope you can break the silence, some collaboration :) Consider a region (...
6
votes
0answers
640 views

Minimum vertex-weight directed spanning tree where the weight function depends on the tree

Given a directed graph $G=(V,E)$ and a node $r\in V$, I need to grow a tree $T$ rooted at $r$ that has a minimum weight and spans all reachable nodes in $G$. The weight function assigns a non-...
6
votes
0answers
1k views

A variation in Ford-Fulkerson algorithm

Suppose that we redefine the residual network to disallow edges into $s$. Argue that the procedure FORD-FULKERSON still correctly computes a maximum flow. I was thinking that when we augment a path ...
6
votes
0answers
142 views

What is the proof for the lemma “For every iteration of the Gomory-Hu algorithm, there is a representant pair for each edge”?

For a given undirected graph $G$, a Gomory-Hu tree is a graph which has the same nodes as $G$, but its edges represent the minimal cut between each pair of nodes in $G$. The Gomory-Hu algorithm finds ...
5
votes
0answers
40 views

Specialized Algorithm for Set-Cover with $k=3$

I know that the Set-cover problem with $n$ elements and a universe of size $N$ is NP complete. Also, the problem is has parameterized complexity regarding the number of sets $k$ that should cover the ...
5
votes
0answers
65 views

Specific quadratic 0-1 knapsack problem solvable in linear time?

I am interested in a simple variant of the quadratic knapsack problem. Let $\{w_1, \ldots, w_n\} \in \{0,1\}$ be $n$ weights and $\{v_1, \ldots, v_n\} \in \mathbb{R}$ be $n$ values. Furthermore, ...
5
votes
0answers
112 views

Partitioning through block moves to the end

Suppose we have a binary string $s$. We wish to partition this string in a series of $0$s followed by $1$s (alternatively: we wish to sort), using only one operation: moving three consecutive elements ...
5
votes
0answers
115 views

Count Wildcard Parenthesizations of a String

Let $\Sigma = \{ (, ), ? \}$ be an alphabet. For a given string $s \in \Sigma^*$, we denote by $f(s)$ the number of ways to replace each symbol $?$ either with $($ or with $)$ such that $s$ is ...
5
votes
0answers
58 views

Network Reconstruction from Flow Function

Suppose that $T$ is a set of vertices in an unknown network. We have oracle $F(X,Y)$ that returns maximum flow value between $X, Y \subseteq T$ in the unknown network. Can we reconstruct the unknown ...
5
votes
0answers
111 views

What is the intuition behind the Geometric Burrows-Wheeler Transform?

What is the intuition behind the Geometric Burrows-Wheeler Transform? And how can I use a GBWT with blocking factor $d$ to match a given pattern $P$ of length $|P| = m$ with $m \ge d$ $m < d$ ...
5
votes
0answers
336 views

How to apply ant colony optimization to the TSP but repeating nodes and edges

I'm learning the Ant Colony Optimization Algorithm and I would like to apply it to a variation of the TSP problem (find the path that start from a node, crosses all nodes and finish in the initial ...
5
votes
0answers
103 views

How to treat numerical errors in determinants of singular matrices when using LU decomposition

I want to calculate the determinant of a matrix. Currently I'm using LU decomposition. To check my algorithm I wrote a unit test with random matrices. In one part I set one row to be equal to ...
5
votes
0answers
123 views

Evaluating sums of subsets

Let $X=\{x_1,...,x_N\}$ be a set of real numbers. We consider $M$ sums over its subsets, e.g. $N=9$, $M=3$ $$ \begin{align*} s_1&=x_1+x_2+x_3+x_9,\\ s_2&=x_1+x_2+x_4+x_9,\\ s_3&=x_1+x_2+...
5
votes
0answers
470 views

Sorting in place & stable in linear time

Given an array with only 0 & 1. Can we have an algorithm which has all the following desirable characteristics- The algorithm runs in $O(n)$ time. The algorithm is stable. The algorithm sorts ...