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12 votes
0 answers
530 views

Finding the diameter of an unknown tree: Is BFS optimal?

I'm interested on the following nice problem that is somewhat standard in CS, but I was surprised on the lack of references on the optimal algorithm to this problem. Ana and Banana plays the ...
Curious's user avatar
  • 63
12 votes
0 answers
1k views

Shortest path in Cayley graphs

The standard way to find the shortest path between 2 vertices, $v_1$ and $v_2$, of an undirected graph is BFS (breadth first search) which takes time $O(|E|)$ and space $O(|V|)$ (where $E$ is the set ...
user22070's user avatar
  • 121
12 votes
0 answers
349 views

Matroids with prescribed independent sets

Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...
ilyaraz's user avatar
  • 1,791
8 votes
0 answers
152 views

Disjoint Rooted Paths with Specified Patterns

Let $S:=$ { $s_i : i \in [k]$ } and $T:=$ { $t_i : i \in [k]$ } be disjoint subsets of vertices of a graph $G$. Furthermore, let $A$ be a subset of $S_k$ (the symmetric group on $[k]$). A set of ...
Tony Huynh's user avatar
  • 32.1k
7 votes
0 answers
186 views

How quickly can we test if a graph is distance-regular?

A (simple, finite, connected) graph $G$ is distance regular if there exist integers $b_i,c_i,i=0,...,D$ such that for any two vertices $x,y$ in $G$ and distance $i=d(x,y)$, there are exactly $c_i$ ...
Brendan McKay's user avatar
6 votes
0 answers
65 views

Vertex cover in bipartite graphs with bounds on cost and size

Suppose we have a bipartite graph $G$ with non-negative integer vertex costs. We would like to find a vertex cover of cost at most $C$ and size (number of vertices) at most $S$, where $C$ and $S$ are ...
Edith Elkind's user avatar
6 votes
0 answers
620 views

Detect cycle in graph in logspace and linear time?

Let's consider graphs of bounded degree. I know that it's possible to detect cycles in a graph in linear time -- essentially do a depth-first search, depositing a trail along the path you're currently ...
grok's user avatar
  • 2,519
6 votes
1 answer
534 views

How to find the Eulerian circuit with the minimum accumulative angular distance within an Eulerian graph?

Note: I originally posed this question to Mathematics, but it was recommended that I try here as well. Context For context, this problem is part of my attempt to determine the path of least inertia ...
mindTree's user avatar
  • 161
6 votes
0 answers
69 views

Digraph weak connectivity in $O(V)$ space and $O(V+E)$ time

A digraph is called weakly connected if its underlying undirected graph is connected. You are given a digraph $G$ with $V$ vertices and $E$ edges as a read-only data structure consisting of lists of ...
Brendan McKay's user avatar
6 votes
0 answers
315 views

Algorithms for computing the Resilience of Graphs

The definition of resilience with a graph $G$ w.r.t to a monotone property $\mathcal{P}$ is well known. (Global resilience) Let $\mathcal{P}$ be an increasing monotone property. The global ...
Pavan Sangha's user avatar
6 votes
0 answers
172 views

Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial maps ...
Vidit Nanda's user avatar
  • 15.5k
5 votes
0 answers
244 views

When does a "stable" assignment of buyers into goods exist?

Consider a setting of $n$ buyers and $m$ goods. We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...
R B's user avatar
  • 618
5 votes
0 answers
581 views

When is polytope compatible with network flow?

A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...
David Harris's user avatar
  • 3,475
4 votes
0 answers
204 views

Enumeration of stable graphs of genus $g$

Let $G=(V,E)$ be a connected undirected finite graph, let us call $G$ stable if each vertex has degree at least $3$. Is there a computer algorithm to efficiently enumerate (repetition allowed) all ...
user avatar
4 votes
0 answers
207 views

Disjoint paths in temporal graphs

Given a graph $G=(V,E)$ and a pair of source-destination nodes $s$ and $t$. Time is divided in periods with the total number of periods denoted by $T$. Each edge $e$ is either operational or broken at ...
lchen's user avatar
  • 367
4 votes
2 answers
381 views

Max weighted matching where edge weight depends on the matching

Given a bipartite graph $G$, we seek a maximal weighted matching $E$. The particularity is below. Once an edge $e$ is chosen, the action of choosing $e$ adds a negative weight $w(e,e')$ to any other ...
lchen's user avatar
  • 367
4 votes
0 answers
94 views

Efficient algorithm to construct path augmented graphs with smallest diameter?

I am interested in special graph constructions that have the smallest diameter. We have a path graph $P_n$ ($N$ is even). We add new set of edges $C$ between path nodes such that set $C$ forms a ...
Mohammad Al-Turkistany's user avatar
4 votes
0 answers
175 views

What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...
Thomas Kalinowski's user avatar
4 votes
0 answers
220 views

Navigation in a graph

The problem Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$. Assumptions about the graph: You may ...
real's user avatar
  • 323
4 votes
0 answers
209 views

Rough structure of the double coset space/Graph bijections up to automorphisms

I am dealing with bijective maps $\pi:\Gamma_1\to \Gamma_2$ between two graphs with the same number of vertices $N=O(10)$. The graphs have a significant automorphism group (these are disconnected ...
Slava Rychkov's user avatar
3 votes
0 answers
369 views

Perfect matching decomposition algorithm for bipartite regular graphs

It is a well-known result that a bipartite graph can be decomposed into edge-disjoint perfect matchings if and only if it is regular. Now here comes the question. Given a bipartite regular graph, is ...
CCC's user avatar
  • 51
3 votes
0 answers
280 views

Max flow with minimum number of edges

A max-flow problem may have multiple solutions. Among these max-flows, I seek the one with the minimum number of positive flow edges (by positive flow edges I mean the edges carrying positive flow). ...
lchen's user avatar
  • 367
3 votes
0 answers
254 views

Algorithm for regular graphs of tessellation {p,q}

We consider a particular class of tessellations $\{p,q\}$ on a Poincaré disk. There are few examples where a regular graph for a particular tessellation has been obtained. It is done by identifying ...
L.K.'s user avatar
  • 81
3 votes
0 answers
88 views

Infection on a complete graph

Suppose we have a complete graph on $2n$ vertices with one "infected" vertex. At each time step, we form a matching of the vertices. Then the vertices paired with infected vertices will also ...
PoissonSummation's user avatar
3 votes
0 answers
56 views

Karp hardness of two cycles which lengths differ by one

Our problem is as follows: NEARLY-EQUAL-CYCLE-PAIR Input: An undirected graph $G(V,E)$ Output: YES if there exists $2$ (simple) cycles in $G$ which lengths differ by $1$, otherwise NO Is it $NP$-...
T. D. Nguyen's user avatar
3 votes
0 answers
40 views

Constructing a graph with a given number of edges and a given triangle distribution

Give a number of edges $|E|$, number of vertices $|V|$ and a $|V|\times 1$ vector of integers $T=[t_1, \cdots, t_{|V|}]$, I wish to construct an undirected graph with $|V|$ vertices, $|E|$ edges such ...
Student88's user avatar
  • 503
3 votes
0 answers
111 views

Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this? ...
domotorp's user avatar
  • 18.9k
3 votes
0 answers
57 views

Algorithm to construct metric space endomorphism with controlled square

Given a finite metric space $(M,d)$ with parameters $K \geq 1$ and $\epsilon > 0$, I'd like to algorithmically check for the existence of a non-identity map $\phi:M \to M$ which happens to be $K$-...
Vidit Nanda's user avatar
  • 15.5k
3 votes
0 answers
350 views

Beating Kadane's Algorithm

I am seeking some reference on already existing work for the following problem. Given an $n$-dimensional square matrix $A=DP$ where $D$ is a diagonal and $P$ is a permutation matrix (think of Gaussian ...
Predrag Punosevac's user avatar
3 votes
0 answers
280 views

Additional Constraint Baum Welch for HMMs

I'm trying to derive a special form of the Baum Welch algorithm where there is an additional constraint that the sum of emission probabilities over all states sums to one for each output symbol. ...
John McGonagle's user avatar
3 votes
0 answers
620 views

Graph recognition software

ISGCI lists a lot of graph classes, many of which are recognizable in polynomial time. Is anyone here aware of actual implementations of these algorithms?
Anthony Labarre's user avatar
3 votes
1 answer
3k views

Fast algorithm for counting the number of acyclic paths on a directed graph

In short, I need a fast algorithm to count how many acyclic paths are there in a simple directed graph. By simple graph I mean one without self loops or multiple edges. A path can start from any node ...
Szabolcs Horvát's user avatar
2 votes
0 answers
135 views

Minimum cost k-edge connected subgraph

The problem of finding a k-edge connected spanning subgraph with the minimum number of edges is $ \mathcal{NP} $-hard in general. Is it the case for positive weighted graphs with "fractional ...
Bence's user avatar
  • 21
2 votes
0 answers
54 views

Do there (or might there) exist computable invariants for Aut(G)-invariant subgraphs of a graph G?

I am interested in algorithms for computing all subgraphs (not necessarily induced) of a graph $G$ up to $Aut(G)$ isomorphism. I had the idea of partitioning the edges of the graph like so $$\{F|E(G)\...
healynr's user avatar
  • 161
2 votes
0 answers
106 views

Decomposing a planar graph

Thomassen proved that the vertex set of every planar graph can be decomposed into two sets inducing a 1-degenerate graph and a 2-degenerate graph, respectively (C. Thomassen, Decomposing a planar ...
jack's user avatar
  • 3,153
2 votes
0 answers
79 views

Pagerank Markov chain reductions

In short: if a Markov chain models a (generalized) pagerank, is it always possible to remove any of its state and obtain a Markov chain that models a pagerank close to the initial one? Full details. ...
Matthieu Latapy's user avatar
2 votes
0 answers
69 views

Are two degree sequences compatible, for random simple graph generation?

Consider a set $V$ of $n$ vertices, and three degree sequences $a_i$, $b_i$ and $c_i$ such that $c_i = a_i+b_i$, $i=1..n$. Assume these degree sequences are graphical: there exist simple graphs (no ...
Matthieu Latapy's user avatar
2 votes
0 answers
388 views

A fast algorithm for deciding if a given undirected graph contains a C4 subgraph

I'm looking for an algorithm for deciding if a given undirected graph G contains C4 as a sub graph, not necessarily induced. I'm not interested in finding such a cycle, if it exists. I was told there ...
Itai Pelles's user avatar
2 votes
0 answers
33 views

Algorithm for lightest unnested planar vertex-disjoint cycle-cover

Question: given a finite set $\mathcal{P}$ of disjoint points in the Euclidean plane and the set $\mathcal{C}$ of all simple polygons whose corners are subsets of $\mathcal{P}$, what is the ...
Manfred Weis's user avatar
  • 13.2k
2 votes
0 answers
65 views

The best way for obtaining the canonical label of a matroid?

For a graph, we can canonical labelings with Nauty or other methods. But for a matroid, I do not find an easy way to labeling. Dillon Mayhewa, Gordon F. Royle introduced the hyperplane graph of a ...
Xie's user avatar
  • 51
2 votes
0 answers
48 views

Planarity of a subgraph

Given a complete symmetric graph $G(V,E)$ with real-valued edgeweights and assume that every induced $k_4$ that is induced by a quadruplet from $v$'s vertices has a unique perfect matching of maximal ...
Manfred Weis's user avatar
  • 13.2k
2 votes
0 answers
288 views

3-uniform hypergraphs and their circuit space

So, I'll break this post into two questions. Both concern 3-uniform hypergraphs. A 3-uniform hypergraph $H=(V,E)$ consists of a set of vertices $V$ and a set of edges $E$, where each edge $e\in E$ is ...
anthony mann's user avatar
2 votes
1 answer
329 views

Worst case performance of heuristic for the non-Eulerian windy postman problem

The windy postman problem seeks the cheapest tour in a complete undirected graph, that traverses each edge at least once; the cost of traversing an edge is positive and may depend on the direction, in ...
Manfred Weis's user avatar
  • 13.2k
2 votes
0 answers
162 views

Calculating Minimum Spanning Trees in Very Big Graphs

I need to determine Minimum Spanning Trees (MST) of very big complete graphs, whose edgeweights can be calculated from data that is associated with the vertices. In the planar euclidean case, for ...
Manfred Weis's user avatar
  • 13.2k
2 votes
0 answers
27 views

Complexity of weighted fractional edge coloring

Given an edge-weighted multigraph $G=(V,E)$ with a positive, rational weight function $(w(e): e \in E)$, the weighted fractional edge coloring problem (WFECP) is to compute ($\min 1^T x$ subject to $...
mo2019's user avatar
  • 151
2 votes
0 answers
520 views

Succinct circuits and NEXPTIME-complete problems

I am fascinated by a recent fact I was reading: Succinct Circuits are simple machines used to descibe graphs in exponentially less space, which leads to the downside that solving a problem on that ...
ACGT's user avatar
  • 41
2 votes
0 answers
76 views

Fastest Algorithm to calculate Graph pebbling number?

I am interested in Graph Pebbling, and in particular what are the fastest known algorithm is to find the pebbling number of a graph. Also, i am interested whether there are lower limits on the runtime ...
ACGT's user avatar
  • 41
2 votes
0 answers
228 views

Known Methods for "Mutexing" Antiparallel Arcs in Graphs

I recently faced the problem of calculating shortest paths in undirected graphs in the presence of negative edge weights; I could not find any applicable algorithms via online search. Transforming the ...
Manfred Weis's user avatar
  • 13.2k
2 votes
0 answers
110 views

Does there exist a linear-time algorithm to find a basis of the null space of the adjacency matrix of a tree?

I am working on a decomposition of trees based on the null space of the adjacency matrix of the tree. Most algorithm on trees are really fast. The decomposition could give some algorithms to find ...
Daniel Alejandro Jaume's user avatar
2 votes
0 answers
120 views

Blossoms and Colorings

There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...
Manfred Weis's user avatar
  • 13.2k