scholarly journals Permutation Reconstruction from Differences

10.37236/4086 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Marzio De Biasi
Keyword(s):  
Combinatorial Problems ◽  
Intermediate Step ◽  
Np Completeness ◽  
Np Complete ◽  
The Absolute ◽  
Decision Version

We prove that the problem of reconstructing a permutation $\pi_1,\dotsc,\pi_n$ of the integers $[1\dotso n]$ given the absolute differences $|\pi_{i+1}-\pi_i|$, $i = 1,\dotsc,n-1$ is $\sf{NP}$-complete. As an intermediate step we first prove the $\sf{NP}$-completeness of the decision version of a new puzzle game that we call Crazy Frog Puzzle. The permutation reconstruction from differences is one of the simplest combinatorial problems that have been proved to be computationally intractable.An addendum was added to this paper on the 9th of December 2015.

NP-completeness of combinatorial problems with unary notation for integers

1980 ◽  
Vol 3 (3) ◽  
pp. 397-400
Author(s):  
Martti Penttonen
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Combinatorial Problems ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete

Most NP-complete problems remain NP-complete even though the notation for integers is changed to unary. The knapsack problem is an exception, it becomes provably polynomial time recognizable. However, we present a modified knapsack problem that remains NP-complete also in unary notation.

scholarly journals FOLDING EQUILATERAL PLANE GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 75-92 ◽  
Author(s):  
ZACHARY ABEL ◽  
ERIK D. DEMAINE ◽  
MARTIN L. DEMAINE ◽  
SARAH EISENSTAT ◽  
JAYSON LYNCH ◽  
...  
Keyword(s):  
Plane Graph ◽  
Analogous Problem ◽  
Central Vertex ◽  
Polyhedral Complex ◽  
Plane Graphs ◽  
Single Vertex ◽  
Np Completeness ◽  
Np Complete ◽  
Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

COMPLEXITY AND APPROXIMABILITY OF DOUBLE DIGEST

2005 ◽  
Vol 03 (02) ◽  
pp. 207-223
Author(s):  
MARK CIELIEBAK ◽  
STEPHAN EIDENBENZ ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Real Life ◽  
Approximation Ratio ◽  
Additional Restriction ◽  
Np Hard ◽  
Model Errors ◽  
Np Completeness ◽  
The Third ◽  
Np Complete ◽  
Feasible Solutions

We revisit the DOUBLE DIGEST problem, which occurs in sequencing of large DNA strings and consists of reconstructing the relative positions of cut sites from two different enzymes. We first show that DOUBLE DIGEST is strongly NP-complete, improving upon previous results that only showed weak NP-completeness. Even the (experimentally more meaningful) variation in which we disallow coincident cut sites turns out to be strongly NP-complete. In the second part, we model errors in data as they occur in real-life experiments: we propose several optimization variations of DOUBLE DIGEST that model partial cleavage errors. We then show that most of these variations are hard to approximate. In the third part, we investigate variations with the additional restriction that coincident cut sites are disallowed, and we show that it is NP-hard to even find feasible solutions in this case, thus making it impossible to guarantee any approximation ratio at all.

NONLINEAR COMPUTABILITY BASED ON CHAOS

2000 ◽  
Vol 10 (02) ◽  
pp. 415-429 ◽  
Author(s):  
GABRIELE MANGANARO ◽  
JOSE PINEDA DE GYVEZ
Keyword(s):  
Hamiltonian Path ◽  
Combinatorial Problem ◽  
Combinatorial Problems ◽  
Information Coding ◽  
Computing Power ◽  
Complete Problems ◽  
Np Complete ◽  
Systematic Solution ◽  
Chaotic Simulated Annealing ◽  
Computing Models

Two new computing models based on information coding and chaotic dynamical systems are presented. The novelty of these models lies on the blending of chaos theory and information coding to solve complex combinatorial problems. A unique feature of our computing models is that despite the nonpredictability property of chaos, it is possible to solve any combinatorial problem in a systematic way, and with only one dynamical system. This is in sharp contrast to methods based on heuristics employing an array of chaotic cells. To prove the computing power and versatility of our models, we address the systematic solution of classical NP-complete problems such as the three colorability and the directed Hamiltonian path in addition to a new chaotic simulated annealing scheme.

On the Hardness of Devising Interval Routing Schemes

1997 ◽  
Vol 07 (01) ◽  
pp. 39-47 ◽  
Author(s):  
Michele Flammini
Keyword(s):  
Time Complexity ◽  
Compact Routing ◽  
Routing Schemes ◽  
Np Completeness ◽  
Routing Scheme ◽  
Np Complete ◽  
Interval Routing ◽  
General Networks

The k-Interval Routing Scheme (k-IRS) is a compact routing scheme on general networks. It has been studied extensively and recently been implemented on the latest generation of the INMOS transputer router chips. In this paper we investigate the time complexity of devising a minimal space k-IRS and we prove that the problem of deciding whether there exists a 2-IRS for any network G is NP-complete. This is the first hardness result for k-IRS where k is constant and the graph underlying the network is unweighted. Moreover, the NP-completeness holds also for linear and strict 2-IRS.

scholarly journals On subset sum problem in branch groups

2020 ◽  
Vol Volume 12, issue 1 ◽  
Author(s):  
Andrey Nikolaev ◽  
Alexander Ushakov
Keyword(s):  
Final Version ◽  
Subset Sum Problem ◽  
Grigorchuk Group ◽  
Np Completeness ◽  
Subset Sum ◽  
Np Complete ◽  
Np Hardness

We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly regular branch groups, which implies NP-completeness if the group is, in addition, contracting. Comment: v3: final version for journal of Groups, Complexity, Cryptology. arXiv admin note: text overlap with arXiv:1703.07406

scholarly journals Near-Colorings: Non-Colorable Graphs and NP-Completeness

10.37236/3509 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. Montassier ◽  
P. Ochem
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Fixed Integer ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7. 

A New Algorithm to Solve the Maximal Connected Subgraph Problem Based on Parallel Molecular Computing

2016 ◽  
Vol 13 (10) ◽  
pp. 7692-7695
Author(s):  
Nan Guo ◽  
Jun Pu ◽  
Zhaocai Wang ◽  
Dongmei Huang ◽  
Lei Li ◽  
...  
Keyword(s):  
Dna Computing ◽  
Combinatorial Problems ◽  
Search Problem ◽  
Connected Subgraph ◽  
Complete Problems ◽  
Portfolio Problem ◽  
Path Search ◽  
Np Complete ◽  
Connected Subgraphs ◽  
Connected Vertex

DNA computing is widely used in complex NP-complete problems, such as the optimal portfolio problem, the optimum path search problem. DNA computing, having the characteristics of high parallelism, huge storage capacity and low energy loss, is very suitable for solving complex combinatorial problems. The maximal connected subgraph problem aims to find a connected vertex subset with maximal number of vertices in a simple undirected graph. Using biological computing technology, we give a new DNA algorithm to solve the maximal connected subgraphs problem with O(n) time complexity, which can significantly reduce the complexity of computing compared with the previous algorithms.

scholarly journals NP-Completeness in the gossip monoid

2018 ◽  
Vol 28 (04) ◽  
pp. 653-672 ◽  
Author(s):  
Peter Fenner ◽  
Marianne Johnson ◽  
Mark Kambites
Keyword(s):  
Algebraic Model ◽  
Decision Problems ◽  
Equivalence Relations ◽  
Membership Problem ◽  
Np Completeness ◽  
Important Decision ◽  
Pure Mathematics ◽  
Finite Set ◽  
Np Complete ◽  
Shed Light

Gossip monoids form an algebraic model of networks with exclusive, transient connections in which nodes, when they form a connection, exchange all known information. They also arise naturally in pure mathematics, as the monoids generated by the set of all equivalence relations on a given finite set under relational composition. We prove that a number of important decision problems for these monoids (including the membership problem, and hence the problem of deciding whether a given state of knowledge can arise in a network of the kind under consideration) are NP-complete. As well as being of interest in their own right, these results shed light on the apparent difficulty of establishing the cardinalities of the gossip monoids: a problem which has attracted some attention in the last few years.

scholarly journals 3-Colorability of Pseudo-Triangulations

2015 ◽  
Vol 25 (04) ◽  
pp. 283-298
Author(s):  
Oswin Aichholzer ◽  
Franz Aurenhammer ◽  
Thomas Hackl ◽  
Clemens Huemer ◽  
Alexander Pilz ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Plane Graphs ◽  
Np Completeness ◽  
Complete Problem ◽  
Polynomial Time Algorithms ◽  
Complexity Results ◽  
The Family ◽  
General Plane ◽  
Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

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