Algorithms on graphs, excluding heuristics.

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Total number of spanning trees of a set of graphs with constraint

This is an extension of the question "Total number of spanning trees of a set of graphs". The original problem has been shown to be #P-complete. Now a new constraint is added to the problem. I have ...
6
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82 views

Maximum weight “fair” matching

I'm interested in a variant of the maximum weight matching in a graph, which I call "Maximum Fair Matching". Assume that the graph is full (i.e. $E=V\times V$), has even number of vertices, and that ...
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Implementation of Min Max Matching in Christofides algorithm for approximating TSP

How is the corresponding step of finding the minimum-cost perfect matching on the odd-degree nodes is supposed to be implemented? The induced graph is not bipartite and all the algorithms I know for ...
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determing the max flow with only edge capacities from n/w with additional vertex capacities?

Let ((V, E); s, t; c) be an extended flow network where not only edge capacities, but also vertex capacities are constrained, i. e., c : E ∪ V → R^ + 0 and a flow f : E → R^ + 0 must satisfy, in ...
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Finding a maximum bipartite matching in O((|A| + |B|)^1.5) [closed]

I aim to solve the following puzzler I recently read: A toymaker is faced with a group of $|A|$ buyers for their stock of $|B|$ distinct toys. Each buyer can buy up to 3 toys if available for buying. ...
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15 views

Extending clique-percolation to bipartite graphs

Clique percolation is well-defined for general graphs. In bigraphs, however, there are only 2-cliques, so the algorithm does not work. There a several extensions of the definition of a clique for ...
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63 views

max spanning tree with conditional weights

Consider the max spanning tree problem in which for any $e \in G$ there is a fixed $f(e)$. Suppose I have a graph with conditional values of the following form: $$ f(e) = \begin{cases} v_1 & ...
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1answer
105 views

checking isomorphism between K regular graph

Problem Input is a k regular graph of n vertices and I have to check whether this is isomorphic to another given k regular graph G. This is a restricted version of graph isomorphism in the sense that ...
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56 views

Assign undirected edges in a mixed graph to make graph cyclic/acyclic [migrated]

What is the complexity of the following problem? Given a mixed (some edges directed, some undirected) graph, assign a direction to all the undirected edges to make the graph cyclic. What about to ...
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1answer
80 views

State of the art algorithms for community detection in graphs

Is anyone aware of the must read papers to get knowledge of the most recent algorithms and method for community detection in graphs, especially those that represent social networks?
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35 views

Number of k-expressions of graph (clique Width)

The clique-width of a graph $G$ is the minimum number of labels needed to construct G by means of the following 4 operations. The Construction of a graph $G$ using the four operations is represented ...
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2answers
108 views

Finding the paths through a graph that reuse as many of the nodes as possible

I'm implementing an encryption algorithm which does a bunch xor operations to mix up the columns. Because I want to find the lower bound of the number of ...
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1answer
66 views

Graphs whose maximal clique intersection graph has bounded chromatic number

In general, The chromatic number of the clique intersection graph a graph can again be arbitrarily high. For example, take $n$ maximal cliques of size $n$ such that every pair of maximal cliques has ...
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1answer
53 views

stable marriage breaking ties, how to?

I have found the following algorithm proposed by Halldorsson which is a randomized algorithm: ...
2
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1answer
151 views

Modifying Hopcroft-Karp algorithm to get approximate bipartite matching

I am trying to find an algorithm to find an $\epsilon$-approximate maximum matching $M_{\epsilon}$ in a bipartite graph in $O(m/\epsilon)$. The partite groups are of equal size, they are $A$ and $B$. ...
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1answer
141 views

Total number of spanning trees of a set of graphs

Given an undirected graph G with $n$ nodes, we can compute its number of spanning trees in polynomial time using Kirchhoff's matrix-tree theorem. Now consider a more complicated setting, in which each ...
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2answers
124 views

Upper bound for number of independent sets

What is the tightest upper bound known for the number of independent sets in a graph?
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78 views

The distribution on the solution space induced by randomized rounding

Consider the Goemans-Williamson algorithm for the MAX-CUT problem. It is known, that if $maxcut(G) \geq 1-\epsilon$, then the algorithm returns a cut $S$ of fractional size at least ...
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1answer
86 views

Expected length of longest construction path in Barabási–Albert Model

The Barabási-Albert Model is used for constructing scale-free networks using the preferential attachment technique. The essence, as I understand it, is that nodes are incrementally added to a graph by ...
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4answers
235 views

Finding outer face in plane graph (embedded planar graph)

I have a planar graph, for which I have computed a combinatorial embedding and coordinates in the plane. So all my arcs are now oriented in the plane, respective to their end vertices. Computing a ...
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2answers
207 views

Enumerating Planar Graphs of Bounded Treewidth

I am looking for references for the following problem: given integers $n$ and $k$, enumerate all non-isomorphic planar graphs on $n$ vertices and treewidth $\leq k$. I'm interested both in theoretical ...
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69 views

Definition of Clique width of graph

The clique width of graph $G$ is defined as minimum number of labels required to construct $G$ by using four operations. I would like to know why the name clique width is given to this definition. ...
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1answer
80 views

How to Quantify Entropy in a Data Set

I'm currently creating a program in Java to analysis the pathological cases of Quicksort. Namely, the transition of complexity from O(n^2) to O(nlogn) as a data set gets less ordered. Since Quicksort ...
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Fully dynamic algorithms for strong components of a directed graph

I'm faced with the problem of maintaining the strong components of a directed graph under insertions/deletions of edges and vertices. As noted in one of the answers to a closely-related question ...
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1answer
70 views

Graph (Forest) representation that supports edge deletion and efficient traversal

I am trying to write a data structure that given a general tree (or forest) will support the following operations: Edge deletion Connected(u,v) queries This problem is addressed in section two of ...
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1answer
95 views

Connecting partial paths to form a hamiltonian cycle [closed]

For an undirected graph that consists of partial paths such that each vertex is a part of one of those paths and that there are edges between all the paths, is there an efficient algorithm to connect ...
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1answer
151 views

Efficient Reduction from Min Cut to st-Min Cut

I am aware that many known algorithms for min cut problem is not by reducing the problem to $st$-min cut. But the question of efficient reduction from min cut to $st$-min cut is still interesting to ...
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2answers
463 views

How can I find the second cheapest spanning tree?

The classic Mininum Spanning Tree (MST) algorithms can be modified to find the Maximum Spanning Tree instead. Can an algorithm such as Kruskal's be modified to return a spanning tree that is strictly ...
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1answer
134 views

Finding cliques in weighted graph

We have given a weighted graph $G=\{V,E\}$, where $V=\{v_1, v_2,...,v_n\}$, and for all $i,j$, the weight of edges $w(v_i, v_j)\in (0,W)$. And we have also given a weight threshold s $w$ (where ...
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2answers
215 views

Find the cheapest cycle through two points on a rectangular grid

I have a task, but I have no idea how to solve it. We have to find a cycle going through $(1,1)$ and $(n,n)$ (X axis on the picture is indexed right to left) such that the sum of values in cells is ...
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2answers
339 views

Complexity of finding even cuts for a graph

Given a graph $G=(V,E)$, what is known about the classical computational complexity of finding a non-trivial cut which partitions the vertices into two sets $V_a$ and $V_b$ such that every vertex in ...
6
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85 views

General Results for Complicated Constraint Satisfaction Problem

Consider the following problem: on a finite two-dimensional grid (say the grid points are vertices of a graph), I need to color the vertices in such a way to satisfy the existence and nonexistence of ...
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2answers
91 views

Hamiltonian cycle on a subset of 2D points, constrained by maximum total length

We are given a list of 2d coordinates, each coordinate representing a node in a graph, and a scalar D, which is a constraint on total length of the cycle. The task is to find a Hamiltonian cycle on a ...
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Lower bounds on single-source shortest paths in directed graphs

Are there any non-trivial lower bounds on the complexity of single-source shortest paths (SSSP) in a directed graph, where all edges have non-negative edge weights? Can we rule out the possibility of ...
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1answer
85 views

Minimum vertex cover for bipartite graphs

I know that it is possible to calculate the minimum vertex cover of a bipartite graph, However, i want that the minimum vertex cover which contains vertices from only one partite set, which will be ...
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3answers
313 views

What separates easy global problems from hard global problems on graphs of bounded treewidth?

Plenty of hard graph problems are solvable in polynomial time on graphs of bounded treewidth. Indeed, textbooks typically use e.g. independet set as an example, which is a local problem. Roughly, a ...
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1answer
156 views

Max flow: either saturate an edge or avoids

Is there a way to create a max flow graph such that it satisfies the condition that a flow either saturates an edge or completely avoids it. It can't have half its flow through one edge and half ...
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1answer
97 views

Finding an exactly weighted st-path in a digraph [closed]

I have a weighted digraph graph $G = (V,E)$ where the weights are positive and negative integers. The graph $G$ is not necessarily acyclic. The question is: given 2 nodes $v_1$ and $v_2$, is there a ...
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Indexing structure for all-pairs min-cuts in a huge DAG

I have a huge DAG - e.g., the dependency graph of all packages in a linux distribution. Suppose I'd like to make a user-friendly tool that makes it very easy to understand how to break the transitive ...
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1answer
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280 views

Finding similar vectors in subquadratic time

Let $d:\{0,1\}^k\times \{0,1\}^k \to \mathbb{R}$ be a function which we refer to as the similarity function. Examples of similarity function are cosine distance, $l_2$ norm, Hamming distance, Jaccard ...
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2answers
322 views

MAX 1 in 2 SAT Algorithm

The maximum satisfiability problem (Max-Sat) is the problem of finding the maximum number of clauses that can be satisified in a Boolean satisfiability instance. The exactly 1 in 2 Sat problem asks, ...
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2answers
220 views

Algorithms for online clique detection

Are there any algorithms which let you detect cliques when adding/deleting edges based on previously detected cliques? What would be the time/memory complexity of this approach?
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85 views

Highway dimension

I'm interested in understanding some recent theoretical results on pathfinding. Specifically this paper: http://research.microsoft.com/apps/pubs/default.aspx?id=201061 I understand from the paper ...
8
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1answer
224 views

Fast algorithm for weighted bipartite matching problem

I have a set of $n$ agents and a set of $n$ tasks, and I need to assign each agent to exactly one task such that a cost is minimised. Some agents are incompatible with some tasks. I have an ...
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69 views

Pruning a graph by removing vertices not part of any minimal Steiner tree

Given a sizable graph $G = (V, E)$, with $|V| \approx 1400$ vertices and $|E| \approx 1600$ edges, I have an optimization problem to find a connected subgraph of $G$ of size $k$ such that a given ...
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1answer
85 views

Node-weighted steiner problem with few terminals

Consider the node-weighted steiner problem: Input: a graph $G=(V,E)$, a set $T\subseteq V$ of terminals, a weight function $w: V\setminus T \to \mathbb{R}_+$. Output: a minimum weight ...
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104 views

Complexity of an algorithm for deciding 3-colorability of graph by the chromatic polynomial modulo $x-3$

As explained on MO computing the chromatic polynomial $P(G,x)$ modulo $x-3$ is enough for deciding 3-colorability. For non adjacent vertices $u$ and $v$, $G+uv$ is the graph with the edge $uv$ added ...
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1answer
255 views

DAG reachability with O(n log n) space and O(log n)-time queries?

For a directed acyclic graph ${\langle}V,E{\rangle}$, is there a data structure that allows for reachability queries without requiring quadratic space or linear time? Ideally I seek an algorithm ...
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1answer
369 views

The randomized query complexity of the conjoined trees problem

An important 2003 paper by Childs et al. introduced the "conjoined trees problem": a problem admitting an exponential quantum speedup that's unlike just about any other such problem that we know of. ...