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# Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

286 questions with no upvoted or accepted answers
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### Paper regarding the complexity of the longest path problem on weighted directed graphs of bounded treewidth

I would like to cite a paper/report/etc that solves the following problem polynomially in $n$: Given a weighted directed graph $G=(V,E)$, $|V|=n$, of bounded treewidth $k \in \mathbb{N}$ and a source-...
1k views

### Is there a better than brute-force solution to the shortest simple path problem?

Given as input graph which can possibly contain negative weight cycles, we can still ask for the weight of the shortest simple path between two vertices (i.e., a path that does not visit any vertex ...
55 views

### Examples of "Sandpile" TSP Instances

This question is closely related to this MO question. I would like to know, whether any (planar Euclidean) TSP instances are known, that exhibit avalanche effects similar to those ecountered in ...
297 views

### zero-sum path problem on a digraph

Consider a digraph $G=(V,A)$ with each arc weighted by either $+1$ or $-1$. A path is called a zero-sum path, if and only if all the arcs in its first half have weight $+1$, and all the arcs in the ...
610 views

### Path finding on graph with state dependent edge costs

I'm looking for a version of path planning that is able to find paths in a graph where edge costs depend on the state of the moving entity. In such cases, it is required to also consider trade-offs, i....
86 views

### Max common sub forest on $k$ graphs

Not sure how to phrase this really, but here goes. Suppose you are given $k$ simple graphs, each having exactly $m$ edges. The edges in each graph are labeled from 1 to $m$. The problem is to find ...
38 views

### Nearest Common Ancestor on DFS Tree (with Addition of Leaves in DFS Order) on Pointer Machines

What is the complexity status for the Nearest Common Ancestor Problem on Trees in which the leaves are attached to the tree in DFS order ? i.e. Suppose one is visiting a tree T in DFS, and at any ...
591 views

### Shortest non-crossing geometric paths

I have a plane graph $G$ and a set of $k$ vertex pairs $\{s_1,t_1\}, \dots, \{s_k, t_k\}$. The goal is to find $k$ non-crossing paths connecting the pairs of terminals $s_i$ with $t_i$ in the graph so ... 106 views

### Claw finding using quantum walk: superposition for Szegedy's framework

Within Claw Finding Algorithms Using Quantum Walk there is the subroutine $claw_{detect}$ described. As in above paper: Let $J_f(N, l)$ and $J_G(M, m)$ be Johnson graphs. Let $F$ and $G$ be vertices ...
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### centralized deterministic Spanner construction with low degree and low stretch

Does there exist a centralized deterministic spanner construction with low degree and low stretch both independent of the graph diameter (no log D factor), but can be dependent on the number of nodes. ...
183 views

### Hardness of approximately counting independent sets with a PRAS, rather than FPRAS

It is known that approximately counting the independent sets of a graph is hard, even if randomness can be used, and even if we restrict ourselves to bounded degree graphs with degree bound at least 6....
145 views

Can someone kindly help clarify as to exactly what is the generally accepted definition of the "sparsest cut" problem for a graph? (Isn't the set which achieves the Cheeger constant for a ...
147 views

### How much faster is solving Clique in properly colored graph?

Given a graph $G$ and a proper vertex coloring $C$ with the minimum numbers of colors, how much faster can a maximum clique be found than when just $G$ is given? Additional information doesn't make ...
102 views

### What mathematical models can analyze and optimize such message passing system?

I look for a mathematical model that can accommodate, analyze and suggest optimizations for a system that can be humanly described as message passing black box programs to which where optimal message ...
151 views

### Realization of a bipolar orientation by a mixed graph

Given an undirected graph $G(V,E)$ and a bipolar orientation $s$ over $G$, consider the problem of identifying $s$ by finding the minimum number of edges such that when orienting them in a particular ...
148 views

### Weighted graph as average of many unweighted graphs

I have a measure $L(G)$ that is only valid for unweighted graphs because is based on discrete distributions and can't be extended to weighted graphs. My idea is that if I consider a weighted graph as ...
195 views

### What is the name for this special case of the Travelling Salesman involving dynamic edge costs?

This is a modeling / taxonomy question. Is there a name for this type of problem? I came up with the following graph problem for a pizza delivery truck, which starts with zero dollars and enough fuel ...
178 views

### Graph partition with objective over intra-partition weights

I have a problem in which I need to find an optimal graph cut that maximizes an objective over weights not on the cut. I have looked at the literature but have not been able to find any similar ...
200 views

### Eulerian Triangulations

Hi i am looking for algorithms to decide whether a planar pointset has a eulerian triangulation i.e. a triangulation that makes every vertex of even degree. I cam across this page http://cs.anu.edu....
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### Hungarian Search and min net-cost-length paths

Consider the Hungarian Search algorithm for max/min weighted bipartite matching. Let G be a bipartite graph weighted by $w:E\rightarrow\mathbb{R}$ and let $M$ be a matching in $G$, let $S$ be a subset ...
1k views

### Maximum number of triangles in a constrained delaunay triangulation

I'm looking for an upper bound for the number of triangles in a constrained planar delaunay triangulation. I know for d=2 delaunay triangulation, there are at most n+1 triangles where n is the number ...
225 views

### Quadratic Binary Optimization formulation of Steiner Tree problem

can someone point out to me a solution or give advice on how to formulate as efficiently as possible in terms of number of bits the minimum Steiner tree problem as a 0-1 quadratic optimization problem?...
94 views

### Has anybody studied the problem of finding maximal weighted rooted spanning DAGs?

Let G=(V,E) be a directed weighted graph (not necessarily a tournament) and s be a special node of G so that all nodes in G are reachable from s. The problem is to find a subgraph G'=(V,E') of G so ...
233 views

### Map points from one plane into another

Given a point on a plane A, I want to be able to map to a corresponding point on plane B. I have a set of N corresponding pairs of reference points between the two planes, however, the overall mapping ...
240 views

### Initial paper of the Moore Neighborhood algorithm

I don't exactly know if this is the place to ask it, but I'm looking for the original paper of the Moore Neighborhood algorithm. I need to make a reference to it (or whoever came up with it). I can't ...
270 views

### Algorithm for choosing unique items from a collection of sets of items

I have a number of sets, each containing items with a numerical value and a string. I want to choose one item out of each set, so that: 1. the strings of the chosen items form a set (i.e. the strings ...
510 views

### Data set for Degree Constrained MST?

Degree Constrained Minimum Spanning Tree is an NP-hard problem. It differs from Minimum Spanning Tree in that, degree of every vertex should be $\leq$ some degree constrained. This is a well studied ... 1 vote
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### Max Flow Routing

Let G = (V,E,S,I,T) be a directed flow network with nodes V, edges E with unit capacity, source nodes S $\subseteq$ V, intermediate nodes I $\subseteq$ V, and target nodes T $\subseteq$ V. The problem ...
1 vote
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### What are the fastest known parameterized algorithms for Grid Tiling?

Let $k$ and $n$ denote positive integers. In the $k$-GridTiling problem, for every pair of indices $(i,j)\in \{1, \dots, k\}^2$ we get a subset $S_{ij}\subseteq \{1, \dots, n\}^2$ of pairs of the ...
1 vote
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### Finding a Hamiltonian cycle in a graph if we are guaranteed that there are not many of them in the graph

Problem: Given an undirected simple graph $G=(V,E)$ on $n$ vertices, such that there are not more than $c^n$ ($c<2$) Hamiltonian cycles in $G$, find a Hamiltonian Cycle in $G$ if there exists one. ...
1 vote
51 views

### Approximation algorithm for non-bipartite Euclidean matching

What is the current best (in terms of running time) (1+\epsilon)-approximation algorithm (both randomized and deterministic) for non-bipartite Euclidean (in higher dimension) matching? There are ...
1 vote
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### Minimum vertex-separators under edge addition

I am trying to prove the following claim. Let $T$ be a minimum $st$-separator in an undirected graph $G$, and let $x \in T$. Let $S\neq T$ be a minimal $st$-separator (i.e., not necessarily minimum), ...
1 vote
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### On the borderline between natural and artificial problems

While there is no formal definition of what constitutes a natural algorithmic problem, in most cases there is pretty good consensus whether a specific problem is natural or artificial. Natural usually ...
1 vote
26 views

### Can input-output matrices optimize bidirectional search?

Given a bidirectional search on a weigthed digraph, could a modified input-output matrix guess what nodes are more likely to belong to the shortest path and the search be done through these nodes ...
1 vote
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### Are there good analogues to Sparsest Cut/Balanced cut for vertex separators instead of edges cuts?

Most problems about cutting graphs into roughly equal parts such as Sparsest cut, Graph partition, Balanced Cut, etc are based on minimizing the size of an edge cut. Even if all of those problems are ...
1 vote
106 views

### 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 ...
1 vote
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### Is there an FPT or XP algorithm known for this version of $k$-edge disjoint paths problem?

The shortest $k$-edge disjoint paths problem is defined as follows: Input: An undirected graph $G=(V,E)$ and $k$ pairs of vertices $(s_1,t_1),\ldots,(s_k,t_k)$. Question: Find (if exist) $k$-pairwise ...
1 vote
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### Cheapest Insertion is $2$-approximation for TSP

Consider the Cheapest Insertion Algorithm on a complete graph with $n$ vertices, where each edge $uv$ has a weight $w(uv)$, and the weights satisfy the triangle inequality $w(xz)\leq w(xy)+w(yz)$ for ...
1 vote
108 views

### A reduction from the maximum $k$-closure problem to the clique problem

Fix a partially ordered set $(P, \le)$ with $N$ elements and real weights $w(p)$ for each $p \in P$. A subset $S \subset P$ is called closed if for any $x, y$ with $y \in S$ and $x \le y$ we also ...
1 vote
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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 ...
1 vote
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### granularity of bidirectional breadth-first search

I tried posting this on stack overflow and it got no takers, decided to cross post here: One thing that I've never seen discussed about bidirectional breadth-first-search (which I'll abbreviate as ...
1 vote
41 views

### Efficient algorithm for finding segregators in a directed acyclic graph

Given a directed acyclic graph $G=(V,E)$, we define a $(\alpha,\beta)$-segregator of $G$ to be a subset $S$ of $V$ of size $\alpha$ such that no vertex in $G\setminus S$ has more than $\beta$ ...
1 vote
82 views

### Are the intermediary sets in maximum cardinality search optimal in some way?

The maximum cardinality search (MCS) algorithm works as follows. Given a weighted graph $G = (V, E)$ where $w(u, v)$ denotes the weight of the edge $\{u, v\}$, we select a start node $a \in V$ and do ...
1 vote
52 views

### Minimum rank graph cut

Consider the following problem: Input: A graph $G=(V,E)$ and a matroid $M$ on $E$, given by an independence oracle. Task: Find a cut $C\subseteq E$ in the graph, such that the rank of $C$ in the ...
1 vote
Instance: an undirected graph $G=(V,E)$ with edge-weights $w:E\to{\mathbb{R}}$; a source $s\in V$ and a sink $t\in V$; a ground set $X=\{x_1, ..., x_k\}$, and for every $v\in V$ a corresponding ... 