4
votes
1answer
52 views

Complexity of calculating independence number of a hypergraph

Let $G$ be a "hypergraph", a collection of vertices $V=\{v_1,v_2,\ldots,v_n\}$ and a collection of "hyperedges" $E=\{e_1,e_2,\ldots,e_m\}$, where $e_i\subseteq V$ and unlike normal edges, an edge may ...
0
votes
1answer
43 views

Relation between digraph and NP-Complete problem

Can there be any relations regarding the number of nodes available in a digraph so that to qualify it as NP-Complete problem. If we consider this problem for instance: Input: A digraph $G=(V,E)$ and ...
2
votes
1answer
92 views

Proving NP-completeness of a graph coloring problem

Given a graph $G=(V,E)$ and a set of colors $k<V$. Find a assignment of colors to vertices that minimizes the number of adjacent vertices in conflict. (Two adjacent vertices are in conflict if they ...
2
votes
1answer
97 views

Why is determining the size of a maximum independent set or a clique in P?

I read that determining the size of the maximum independent set (and also a clique of maximum size) is in P. The versions that find the actual solution are known to be NP-hard. With respect to ...
2
votes
1answer
57 views

Reducing 3SAT to Triangle Cover Graph

The Triangle Cover Graph problem is this: Given a graph $G = (V,E)$ and an integer $k$, does there exist a set of at most $k$ vertices of $G$ such that every triangle contained in $G$ also ...
2
votes
0answers
97 views

Intractable properties of Two-factor in connected bridgeless cubic graphs

Petersen's Theorem states that every cubic, bridgeless graph $G(V, E)$ contains a 2-factor $F$ (and therefore a perfect matching $E-F$). Alternatively, 2-factor is a set of vertex disjoint cycles that ...
2
votes
1answer
114 views

Counting and finding all perfect/maximum matchings in general graphs

Recently i've been dealing with a problem that led me to the following questions: Is there a good algorithm to enumerate all maximum/perfect matchings in a general graph? Is there a good algorithm ...
2
votes
1answer
45 views

Reduction from max-cut to min-cut

Algorithms for the finding of an MST in a graph can be applied for both maximum and minimum spanning trees. It is well known, however, that the finding of a max-cut in a graph is an NP-hard problem ...
7
votes
4answers
140 views

Problems that are NP but polynomial on graphs of bounded treewidth

I heard here that the Hamiltonian cycle problem is polynomial on graphs of bounded treewidth. I am interested in examples/references to different problems which is essentially hard but having ...
9
votes
2answers
123 views

Hard computational problem on special class of bipartite graphs

I am interested in the properties of a class of bipartite graphs $G(X \cup Y, E)$ where all nodes in $X$ are 3-regular, all nodes in $Y$ are 2-regular, and $|X|=|2Y/3|$. First, Is this a well known ...
1
vote
1answer
24 views

Complexity class of Determining Hamiltonian cycle

I Know that determining Hamiltonian cycle in a graph is NP complete. For the sake of my clarification, I just want to know that whether the problem remains NP complete with following restrictions ? ...
8
votes
1answer
181 views

$O(n^{k-1}$) algorithm for K-clique problem

Clique problem is a well known $NP$-complete problem where the size of the required clique is part of the input. However, k-clique problem has a trivial polynomial time algorithm ($O(n^k)$ when $k$ is ...
7
votes
4answers
96 views

Negative weight cycle vs maximum weight cycle

I'm having trouble understanding why it's easy to detect negative-weight cycles (Bellman Ford) but hard to find the maximum weight cycle in an undirected graph. If we negate the weight of each edge, ...
6
votes
2answers
166 views

Software for testing graph homomorphism

I have graphs $G_k$ and $H_k$ with $|\mathcal{V}(G_k)|=|\mathcal{V}(H_k)|^{2k}=n^{2k}$ with $k\in\Bbb N$ that pass sanity checks such as no-homomorphism lemma. Are there free and easy to use tools to ...
2
votes
2answers
102 views

Direct reduction from Near-Clique to Clique

An undirected graph is a Near-Clique if adding one more edge would make it a clique. Formally, a graph $G=(V,E)$ contains a near-clique of size $k$ if there exists $S\subseteq V$ and $u,v\in S$ ...
6
votes
2answers
91 views

Courcelle's Theorem: Looking for papers

I am looking for an easy and introductory paper on the proof of Courcelle's Theorem. I am also interested in its connection to parameterized complexity regarding the treewidth. I am only a beginner ...
2
votes
1answer
53 views

Is there a graph product that is multiplicative in independence number?

I know that Stable set cannot be approximated to constant factor. I saw a simple proof using OR product sometime back. I am unable to recall it. If anyone here knows what I am talking about could help ...
2
votes
1answer
65 views

What will be minimum no of operation to make whole matrix zero if one is allowed to multiply a row or column by zero?

Suppose we are given an M×N matrix, with some elements are zero, some non-zero. We know the co-ordinates of non-zero elements. Now, if I am allowed to multiply a whole row or a whole column by zero ...
1
vote
0answers
52 views

Counting modified perfect matchings

Consider a bipartite graph with vertex set partitioned into $X=\{u_1,u_2,u_3\}$ and $Y=\{v_1,v_2,v_3\}$. Consider the graph has the following edges: $\{u_1,v_1\}$, $\{u_2,v_2\}$, $\{u_2,v_3\}$, ...
5
votes
1answer
121 views

Complexity of finding a spanning tree that minimizes the maximum interference

Given $n$ nodes in the plane, connect the nodes by a spanning tree. For each node $v$ we construct a disk centered at $v$ with radius equal to the distance to $v$’s furthest neighbor in the spanning ...
4
votes
1answer
49 views

Max cut in cubic graphs

The following question is related to the max cut problem in cubic graphs. In this survey paper Theorem 6.5 states A maximal cut of a cubic graph can be computed in polynomial time Browsing ...
7
votes
1answer
214 views

Find which vertices to delete from graph to get smallest largest component

Given a graph $G = (V, E)$, find $k$ vertices $\{v^*_1,\dots,v^*_k\}$, which removal would result in a graph with smallest largest component. I assume for large $n = |V|$ and large $k$ the problem ...
3
votes
1answer
152 views

Coordinated Attack Problem On The Arbitrary Graph

Let consider a general version of Two Generals' Problem, when there are $n$ generals located on the arbitrary graph and they should agree on exactly the same value whether to attack or not to attack. ...
-3
votes
3answers
106 views

Does reachability belong to P?

Reachability is defined as follows: a digraph $G = (V, E)$ and two vertices $v,w \in V$. Is there a directed path from $v$ to $w$ in $G$? Is it possible to write a polynomial time algorithm for it? ...
4
votes
1answer
351 views

Longest path in grid like graph

This was a question at SO, and I think it's very interesting, I thought about it, but I could not provide any efficient algorithm neither showing the NP-Hardness: Find the length of the longest ...
1
vote
1answer
62 views

Is there a program to solve a metric TSP for 80 edges at optimum?

i'm going to use the Christofides heuristic algorithm in order to solve a TSP for about 80 edges. Eventually i should have a solution, that is within the factor 1.5 of the optimum. But when i'm ...
1
vote
1answer
409 views

Show that Vertex-Cover is NP-complete, using Stable-Set

My task is to give proof, the Vertex-Cover problem is NP-complete, assuming it's already shown that the Stable-Set problem is NP-complete, too. My approach: i know, Stable-Set is NP-complete, and all ...
1
vote
3answers
130 views

k-path problem - P, NP or NPC?

I need to determine which complexity class this problem belongs to: Given a graph $G(V, E)$, two vertices $u$ and $v$ and a natural number $k$, does a path of length $k$ exist between thesee two ...
2
votes
0answers
48 views

End-Of-The-Line Augmented Problem of PPAD

Famous PPAD class of problems is formally defined by specifying one of its complete problems, known as End-Of-The-Line: End-Of-The-Line Problem: $G$ is a (possibly exponentially large) directed ...
3
votes
1answer
132 views

How to reduce INDEPENDENT SET to INDEPENDENT SET SIZE?

Suppose you are given a polynomial-time algorithm for the following problem related to INDEPENDENT SET: INDEPENDENT SET VALUE Input: An undirected graph G. Output:The size of the largest ...
0
votes
2answers
333 views

Can the edges of a graph be assigned directions such that all nodes in a given subset have in- or outdegree 0, and every other node indegree > 0?

In a directed graph, the indegree of a node is the number of incoming edges and the outdegree is the number of outgoing edges. Show that the following problem is NP-complete. Given an undirected graph ...
0
votes
3answers
295 views

How to check whether a graph is connected in polynomial time?

I have to solve the following problem: Consider the problem Connected: Input: An unweighted, undirected graph $G$. Output: True if and only if $G$ is connected. Show that Connected ...
2
votes
1answer
44 views

complexity of finding the hampath of length $k$ in a graph with $n$ vertexes where $k < n$

A simple question: What would be the complexity of finding whether a hampath of length $k$ exists in a graph with $n$ vertexes where $k < n$? Would this be in NP-complete or just NP?
5
votes
0answers
58 views

Decomposition of graphs that uses centers

Do you know of any kind of decomposition of graphs that involves centers, especially in the context of parametrized complexity? If so, please provide some reference. If not, do you see any reason ...
2
votes
1answer
185 views

How to analyze the Steiner tree problem?

I have a problem where I am supposed to analyze the Steiner tree problem by doing the following 3 steps. 1) Look up what the Steiner tree problem is. 2) Find a ...
3
votes
1answer
172 views

Prove finding a near clique is NP-complete

An undirected graph is a near clique if adding an additional edge would make it a clique. Formally, a graph $G = (V,E)$ contains a near clique of size $k$ where $k$ is a positive integer in $G$ if ...
2
votes
0answers
57 views

Complexity of computing the first bits of a minimal permuted adjacency matrix

Given any graph $G$ on $V(G)=\{1,\dots,n\}$ and its adjacency matrix $$A(G)=\left(\matrix{ A_{1,1} & A_{1,2} & \dots & A_{1,n}\\ A_{2,1} & A_{2,2} & \dots & A_{2,n}\\ ...
0
votes
0answers
44 views

The name of “finding the path of a graph that is a variant of hamiltonian path”

Suppose that there is some graph, with $n$ vertexes. We wish to find the hamiltonian path, but we make the graph being searched a little different. There is a person A that travels each (undirected) ...
3
votes
3answers
208 views

Find node that splits tree in half

Given a tree $T = (V , F)$, find an algorithm which finds $u \in V$, so in the graph $T = (V \setminus \{u\} , F)$ the size of each connected component is $\lceil |V| / 2 \rceil$ at most. What is ...
1
vote
2answers
64 views

Prove that $0$-$1$ $\mathsf{ Ineq}$ is $\mathsf{NL}$-complete

I need to prove that the following problem $0$-$1$ $\mathsf{ Ineq}$ is $\mathsf{NL}$-complete. Given a finite set of variables $V$, a finite set of inequalities of the form $x \le y$ (where $x, y \in ...
1
vote
1answer
54 views

Prove that 2-Colourability is in L from Undir-Reachability is in L

Let Undir-Reachability be the following problem: given an undirected graph G and two specified vertices s and t in G, is there a path from s to t in G? I need to prove that the 2-Colourability is in ...
2
votes
0answers
89 views

Travelling salesman problem with detours

I am interested if there exists a following version of the travelling salesman problem: INSTANCE: A finite set $C = \{1,2,\dots,k\}$ of cities, a positive integer distance $\delta(i,j)$ for each ...
2
votes
2answers
1k views

How to understand the reduction from 3-Coloring problem to general $k$-Coloring problem?

3-Coloring problem can be proved NP-Complete making use of the reduction from 3SAT Graph Coloring (from 3SAT). As a consequence, 4-Coloring problem is NP-Complete using the reduction from 3-Coloring: ...
1
vote
1answer
67 views

For what values of A and B is the gap-VC-[A,B] problem NP-HARD?

For which values $A,B$ is the problem $\mathsf{gap\mathord-VC}\mathord-[A,B]$ NP-hard? VC is the vertex cover problem. I am given three options: $B=\frac{3}{4},A=\frac{1}{2}$ or ...
2
votes
1answer
2k views

Reducing Clique to Independent Set

The Clique problem takes a graph $G = (V,E)$ and an integer $k$ and asks if $G$ contains a clique of size $k$. (A clique is a set of vertices such that every pair of vertices in the set is adjacent.) ...
0
votes
2answers
106 views

Cliques in an alternate graph representation

EDITS: corrected $r$ to add edges just like $f$ does in paragraph 8 per the first comment below. Also specified the Clique problem of interest in paragraph 2. Having received no response to this ...
8
votes
1answer
2k views

Is the k-clique problem NP-complete?

In this Wikipedia article about the Clique problem in graph theory it states in the beginning that the problem of finding a clique of size K, in a graph G is NP-complete: Cliques have also been ...
6
votes
1answer
507 views

Minimum s-t cut in weighted directed acyclic graphs with possibly negative weights

I ran into the following problem: Given a directed acyclic graph with real-valued edge weights, and two vertices s and t, compute the minimum s-t cut. For general graphs this is NP-hard, since one ...
2
votes
1answer
198 views

Modification of Hamilton Path

Although I know that the Hamilton Path problem is ${\sf NP}$-complete, I think the following variant can be solved in polynomial time: Given a planar graph with vertex set $V$, edge set $E$, start ...
3
votes
2answers
342 views

Is the clique problem NP-complete also on bipartite or planar graphs?

We know that the clique problem is NP-complete. Is the restriction of the problem to bipartite graphs or planar graphs still NP-complete?