Questions related to NP-hardness and NP-completeness.

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Definition of Planar 3-SAT

What is the standard definition of Planar 3-SAT? I have seen a number of different definitions. What was the original paper that defined it and proved it to be NP-complete?
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1answer
91 views

Hardness of an extended maximum set packing problem

(Edited) The maximum set packing problem when the sets are all of equal size, say $k$, is known to be NP-hard for $k \ge 3$. The requirement in this problem is that the sets in the solution will be ...
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Maximal-clique to domination number

Suppose for a graph family, i have an algorithm to compute all the maximal cliques in polynomial time. Is it possible to compute the Dominating set in polynomial time? So far, i have tried to use the ...
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0answers
82 views

Preserving connectivity from a vertex by edges deletion [migrated]

Given a connected graph $G$ and a vertex $v$, is it polynomially solvable to find a maximal cardinality set of edges incident to $v$, which deletion (still) leaves vertex $v$ to be connected with all ...
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1answer
101 views

Complexity class of a subset problem similar to subset sum

What could be the complexity class of the following problem: Given a positive integer $K$, a positive integer $d$ (say $2$) and a set $S$ of all non-negative integers less than $K$ find a $S' ...
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2answers
987 views

Hamiltonian cycle on graphs without small cycles

While answering this question on cstheory, I (informally) proved on the fly the following theorem: Theorem: For any fixed $l \geq 3$ the Hamiltonian cycle probem remains NP-complete even if ...
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0answers
76 views

Hard extendability problems

In extendability problem, we are given part of the solution and we want to decide whether we can extend it to a complete solution. Some extendability problems are efficiently solvable while other ...
5
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0answers
135 views

NP-completeness of the Dominating set problem for planar graphs of maximum degree 3?

I am trying to learn about some techniques that are used for proving the NP-completeness of domination related problems. One of the problems that is known to be NP-complete is the domination number of ...
2
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1answer
220 views

Hardness of 3-dimensional matching variant

I am thinking about a variant of 3-dimensional matching. In normal 3d matching, we have three sets of vertices $X$, $Y$ and $Z$, and a set of edges $E \subseteq X \times Y \times Z$. We want to choose ...
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0answers
67 views

NP-hardness of a bilinear program?

Given $a_i, b_i, c_i, d_{ij}\in [0,1]$ for $i,j\in [n]$ and $i\neq j$ such that $\sum_{i\in [n]} a_i=1$ and $d_{ij}=d_{ji}$. I have the following bilinear program: max $\sum_{i=1}^n (x_i-a_i)y_i$ ...
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0answers
97 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 ...
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0answers
48 views

Help in NP-Hardness proof of a certain type of Class Cover problem

Class Cover Problem is nothing but finding an optimal cover of certain class (Point Set) with a particular shape only i.e. finding minimum number of a certain shaped polygon (for example, rectangle) ...
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1answer
91 views

Set cover with budget on sets

I am wondering if this is a studied variant of the Set Cover problem. We are given a universe $X$, a collection of sets $S = \{S_1, ..., S_m\}$ and integers $c_i$. We want to cover all elements in ...
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1answer
233 views

NP-hardness proof: looking for some good restricted np-hard problems

To show the NP-hardness of a problem, one need to choose a known NP-hard problem and find a polynomial reduction from the known problem to his problem in hand. Theoretically, any NP-hard problem can ...
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2answers
220 views

For which values of $k$ is the $k$-disjoint paths problem in $\mathcal{P}$?

The $k$-Vertex-Disjoint Paths Problem ($k$-$\text{DPP}$) is defined as follows: Input: A graph $G=(V,E)$ and $k$ pairs of vertices $(s_1,t_1),\ldots,(s_k,t_k)$. Question: Does there exist ...
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3answers
285 views

Multidimensional knapsack STRONGLY NP-complete

Who can point me to a reference where it is actually shown that multidimensional knapsack is strongly NP-complete? I have found loads of papers where they claim it is, without citation; I have found ...
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0answers
56 views

Different hardness proofs w.r.t different classes

Consider a language $L$ which is hard for some class $C$ (e.g. PSPACE-hard). Trivially, $L$ is also $D$-hard for every class $D\subseteq C$ under the same type of reduction (e.g. NP-hard). Is there a ...
6
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1answer
157 views

Max-weight connected subgraph problem in planar graphs

The maximum-weight connected subgraph problem is as follows: Input: a graph $G=(V,E)$ and a weight $w_i$ (possibly negative) for each vertex $i \in V$. Output: a maximum-weight subset $S$ of ...
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1answer
84 views

NP Hardness of Metric Steiner Tree

It is known that the metric steiner tree problem is NP hard (Garry and Johnson [1977]). I wanted to know if there is a simpler way of proving the same. Specifically, I am trying to find a polynomial ...
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0answers
133 views

What relations are there between a problem hardness and the hardness of verifying a witness?

I had some hard times trying to formulate the question, so I'll start with some examples: Suppose you are given a Dominating Set instance, $<G,k>$. Now suppose I give you a set of vertices $D$ ...
2
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0answers
99 views

How does part 5 of the paper “computational complexity of Minesweeper” work?

Marzio De Biasi mentioned this paper in this answer, and that paper's claim in part 5 would appear to resolve my question. However, there seems to be a significant gap in the proof. Even though one ...
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1answer
304 views

Is it NP-hard to _play_ minesweeper perfectly?

This paper shows that it is NP-hard "to determine if there is some pattern of mines in the blank squares that give rise to the numbers seen." If there is a way to "lead a perfect player into" such ...
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0answers
71 views

another solution problem (ASP) of integer multicommodity flow: is it NP-complete?

I know that integer multicommodity flow is NP-complete. It was proven in "On the complexity of time table and multi-commodity flow problems" (http://dl.acm.org/citation.cfm?id=1382492) that any SAT ...
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1answer
64 views

Complexity of Haemers' minimum rank

In 1978 Willem H. Haemmers published "An upper bound on the Shannon capacity of a graph". Tims has a survey of more recent results his thesis. What is the computational complexity of computing ...
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8answers
644 views

Problems that are easy on unweighted graphs, but hard for weighted graphs

Many algorithmic graph problems can be solved in polynomial time both on unweighted and weighted graphs. Some examples are shortest path, min spanning tree, longest path (in directed acyclic graphs), ...
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0answers
295 views

$NP$-completeness of recognizing the difference of two permutations

Shor stated, in his comment to anonymous moose's answer to this question Can you identify the sum of two permutations in polynomial time?, that it is $NP$-complete to identify the difference of two ...
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3answers
164 views

Comparing graphs

I am looking for a kick in the right direction. What i am trying to do is to compute "similarity" between two graphs, where I define "similarity" as the number of shared paths. example: ...
9
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1answer
272 views

Complexity of finding the maximal number of pair-wise disjoint sets

Assume that I have $P$ sets with elements taken from $r$ possible ones. Each set is of size $n$ ($n<r$), where the sets can overlap. I want to determine whether the following two problems are ...
2
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2answers
73 views

complexity of CSPs with 2-variable constraints and multi-valued variables

(CSP = "constraint satisfaction problem") CSPs for which either the variables are restricted to less than two values or the constraints take less than two inputs are obviously trivial, and Schaefer's ...
13
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4answers
443 views

Graph problems which are NP-Complete on directed graphs but polynomial on undirected graphs

I'm looking for problems which are known to be NPC for directed graphs but has a polynomial algorithm for undirected graphs. I've seen the question regarding the other way around here “Directed” ...
28
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3answers
2k views

Techniques for showing that problem is in hardness “limbo”

Given a new problem in $\mathsf{NP}$ whose true complexity is somewhere between $\mathsf{P}$ and being NP-complete, there are two methods that I know of that might be used to prove that resolving this ...
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6answers
1k views

NP-hard problems on paths

everybody knows there exist many decision problems which are NP-hard on general graphs, but I'm interested in problems that are even NP-hard when the underlying graph is a path. So, can you help me to ...
13
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3answers
558 views

Transitive feedback arc set (TFAS): NP-complete?

Some time ago, I posted a reference request for graph problems where we want to find a 2-partition of the edges where both sets fulfill a property not related to their cardinality. I was trying to ...
2
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0answers
338 views

What's the hardest problem with a non-trivial exact algorithm?

Exact algorithms for NP-complete problems are sometimes feasible, if the input is small enough. I’ve also came across some algorithms which are not practical even for very small inputs, and their ...
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1answer
120 views

Hardness of XSAT

The standard NP-hard SAT problem is the problem of Boolean satisfiability of conjunctions of clauses, where clauses are disjunctions of literals. I am interested in the problem of the Boolean ...
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2answers
526 views

Co-NP-completeness of minimal TSP tour?

This problem came out of my recent blog post, suppose you are given a TSP tour, is it co-NP-complete to determine if it is a minimal one? More precisely is the following problem NP-complete: ...
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1answer
172 views

Complexity of an edit distance problem

Given an array $A[1...n]$ of non-negative integers, we want to transform $A$ into $A'$ such that $|A[I] - A[I + 1]| \leq 1$ in the minimum number of operations. One operation consist of picking ...
4
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1answer
179 views

For what k is MAX-2-SAT-k NP-complete?

It is well-known that it is NP-complete to decide whether in a 2-CNF at least s clauses are satisfiable. It also follows from the reduction from 3-SAT-3 that we can suppose that every literal occurs ...
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1answer
220 views

Design of exact algorithms for non-local hard problems

In the connected dominating set problem (CDS) we are given an $n$-vertex undirected graph, and asked to find the smallest connected subset $S$ of vertices such that each vertex not in $S$ is adjacent ...
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1answer
256 views

Does there exist the idea of “RP-complete”, like NP-complete?

NP-hardness and NP-completeness play an important role in complexity theory. My question is, does there exist a language $L$ in RP to which any language $M$ in RP can be reduced in polynomial time? ...
5
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1answer
287 views

Edge and vertex fault tolerance in graphs

Suppose we are given two graphs $G$ and $H$, where $H$ is a subgraph of $G$. What is the maximum number $k$ such that if any $k$ edges are removed from $G$, $H$ still remains a subgraph of $G$? What ...
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3answers
814 views

Decide whether a matrix's kernel contains any non-zero vector all of whose entries are -1, 0, or 1

Given an $m$ by $n$ binary matrix $M$ (entries are $0$ or $1$), the problem is to determine if there exists two binary vectors $v_1 \ne v_2$ such that $Mv_1 = Mv_2$ (all operations performed over ...
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2answers
211 views

Is Almost-2-SAT NP-hard?

Is a CNF SAT problem NP hard when the total number (but not the width) of the 3-or-more-term clauses is bounded above by a constant? What about specifically when there's only one such clause?
5
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1answer
217 views

Graph partitioning by node deletion

Given an undirected graph and an integer $B$, we ask to find the minimum set of nodes whose removal partitions the generated graph into connected components, each having at most $B$ nodes. We can ...
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1answer
128 views

Computational complexity of finding a (1D deterministic) CA that achieves a desired spacetime history “patch” after $t$ iterations

Question description: Consider the problem of finding a minimum $n$-color $k$-state one-dimensional cellular automata (minimizing $k$ for some fixed value of $n$ or vice versa), with communication ...
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1answer
197 views

Maximize the expected number of “losers” - Is it NP-hard?

I am trying to find a reduction for a problem that seems NP-hard: Let me start from a toy example. Consider 3 elements, $a$, $b$, and $c$. You want to choose two pairs out of the three pairs and ...
2
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1answer
214 views

Is it NP-Complete to determine if a quadratic program (QP) has multiple solutions?

If this is known, can someone point me to a proof? Edit: A QP is essentially a LP with a quadratic objective. That is, it looks like: minimize $\frac{1}{2} x^T Q x + c^T x$ s.t. $Ax \leq b$ It's ...
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3answers
868 views

NP complete graph problems about structural properties

(This question is a bit of a "survey".) I'm currently working on a problem where I'm trying to partition the edges of a tournament into two sets, both of which are required to fulfill some structural ...
4
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0answers
136 views

The weakly NP-complete problems and their associated counting problem

Are there weakly NP-complete problems whose associated counting problem can be computed in pseudo-polynomial time? And if one were to be found (and assuming it is #P-complete), what would be the ...
7
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1answer
493 views

Linear diophantine equation in non-negative integers

There's only very little information I can find on the NP-complete problem of solving linear diophantine equation in non-negative integers. That is to say, is there a solution in non-negative ...