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About 5,960 papers for “NP-complete problems”

NP-complete problems

Veronika Stankovianska · Digital Repository (National Repository of Grey Literature) · 2009

Cited by 0

Resolution of NP-complete problems

Gavaldà Mestre, Ricard · QRU Quaderns de Recerca en Urbanisme · 1995

These are the course notes for my lectures on "Resolution of NP-complete problems". These lectures are a part of the Sixth Advanced Course on Artificial Intelligence, ACAI'95, which this year is devoted to AI Algorithmics, and will take place on June 29th-July 6th in Perros-Guirec (France). They survey on three basic paradigms for combinatorial optimization: heuristics, branch-and-cut, and approximation algorithms.

Cited by 0Open access

Solving NP-Complete Problems Efficiently

Frank Vega · Preprints.org · 2024

The P versus NP problem is a fundamental question in computer science. It asks whether problems whose solutions can be quickly verified can also be quickly solved. Here, "quickly" refers to computational time that grows proportionally to the size of the input (polynomial time). While the problem's roots trace back to a 1955 letter from John Nash, its formalization is attributed to Stephen Cook and Leonid Levin. Despite extensive research, a definitive answer remains elusive. Closely tied to this is the concept of NP-completeness. If a single NP-complete problem could be solved efficiently, it would imply that all problems in NP can be solved efficiently, proving that P equals NP. This work posits that ONE-IN-THREE 3SAT, a notoriously difficult NP-complete problem, can be solved efficiently, thereby establishing the equivalence of P and NP.

Cited by 1Open access

edge-deletion NP-complete problems

Mihalis Yannakakis · 1978

If ~ is a graph property, the general node(edge) deletion problem can be stated as follows: Find the minimum number of nodes(edges), whose deletion results in a subgraph satisfying property ~. In this paper we show that if ~ belongs to a rather broad class of properties (the class of properties that are hereditary on induced subgraphs) then the node-deletion problem is NP-complete, and the same is true for several restrictions of it. For the same class of properties, requi~ing the remaining graph to be connected does not change the NP-complete status of the problem; moreover for a certain subclass, finding any "reasonable " approximation is also NP-complete. Edge-deletion problems seem to be less amenable to such generalizations. We show however that for several common properties (e.g. planar, outer-planar, line-graph, transitive digraph) the edge-deletion problem is NPcomplete.

Cited by 0

Two NP-Complete Augmentation Problems

Tibor Jordán · 1997

We prove that the Simplicity Preserving Edge-Connectivity Augmentation Problem and the problem of Increasing Edge-Connectivity by Reinforcing Edges are NP-complete. 1 Introduction The k-edge-connectivity augmentation problem is to find a smallest set F of new edges which makes a given (di)graph G = (V; E) k-edge-connected, that is, for which G 0 = (V; E [ F ) is k-edge-connected and jF j is as small as possible. (Note, that the augmented graph and also the augmenting set F itself may contain parallel edges.) This problem -- and a number of its extensions -- can be solved in polynomial time. More details on this and other connectivity augmentation problems can be found in the survey paper [3] by A. Frank. There are several interesting versions of this problem, however, which have not been solved yet by means of efficient algorithms. As it is noted in [3], "it is an important open problem to find algorithms which do not add parallel edges". This kind of simplicity requirement leads...

Cited by 23

New NP-complete partition problems

Shahrokh Saeednia · IEEE Transactions on Information Theory · 2002

Three new partition problems are proposed. The fact that these problems are NP-complete is shown.

Cited by 2

Solving NP-complete Problems Efficiently

Frank Vega · IPI Letters · 2024

The P versus NP problem is a fundamental question in computer science. It asks whether problems whose solutions can be quickly verified can also be quickly solved. Here, "quickly" refers to computational time that grows proportionally to the size of the input (polynomial time). While the problem's roots trace back to a 1955 letter from John Nash, its formalization is attributed to Stephen Cook and Leonid Levin. Despite extensive research, a definitive answer remains elusive. Closely tied to this is the concept of NP-completeness. If a single NP-complete problem could be solved efficiently, it would imply that all problems in NP can be solved efficiently, proving that P equals NP. Garey and Johnson defined K-CLOSURE such that for any edge $(u, v)$ in the directed graph, either node $u$ is in the set $V'$ or node $v$ is not in $V'$. This implies that either both nodes are in $V'$ or both are not in $V'$. Our previous work in IPI Letters presented a polynomial-time algorithm for K-CLOSURE. While no errors have been identified in this work, many believe that Garey and Johnson's original definition was incorrect, and their citation of Queyranne was a misunderstanding. Many argue that the empty set serves as a simple counterexample to Garey and Johnson's definition of K-CLOSURE. This paper proposes that K-CLOSURE is actually an NP-complete problem, which would imply that P equals NP.

Cited by 0Open access

Solving NP-Complete Problems Efficiently

Frank Vega · Preprints.org · 2024

The P versus NP problem is a fundamental question in computer science. It asks whether problems whose solutions can be quickly verified can also be quickly solved. Here, "quickly" refers to computational time that grows proportionally to the size of the input (polynomial time). While the problem's roots trace back to a 1955 letter from John Nash, its formalization is attributed to Stephen Cook and Leonid Levin. Despite extensive research, a definitive answer remains elusive. Closely tied to this is the concept of NP-completeness. If a single NP-complete problem could be solved efficiently, it would imply that all problems in NP can be solved efficiently, proving that P equals NP. This work posits that MONOTONE ONE-IN-THREE 3SAT, a notoriously difficult NP-complete problem, can be solved efficiently, thereby establishing the equivalence of P and NP.

Cited by 0Open access

Some NP-complete Geometric Problems

M. R. Garey, et al. · 2008

We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NP-complete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NP-hard if the distance I~asure is the (unmodified) Euclidean metric. However, for reasons we discuss, there is some question as to whether these problems, or even the well-solved MINIMUM SPANNING TREE problem, are in NP when the distance measure is the Euclidean metric.

Cited by 0

On Feasibly Solving NP-complete Problems

Frank Vega · 2023

NAE-3SAT consists in knowing whether a Boolean formula ϕ in 3CNF has a truth assignment such that for each clause at least one literal is true and at least one literal is false. NAE-3SAT remains NP-complete when all clauses are monotone. We create a polynomial time reduction which converts the monotone version into a bounded number of linear constraints on real numbers. Since the linear optimization on real numbers can be solved in polynomial time, then we can decide this NP-complete problem in polynomial time. Certainly, the problem of solving linear constraints on real numbers is equivalent to solve the particular case when there is a linear optimization without any objective to maximize or minimize. If any NP-complete can be solved in polynomial time, then we obtain that P = NP. Moreover, our polynomial reduction is feasible since it can be done in linear time.

Cited by 1Open access