NP-hard problems are a class of problems in computational complexity theory that are at least as hard as the hardest problems in NP (nondeterministic polynomial time). The key properties of NP-hard problems include: 1. **Definition**: A problem is considered NP-hard if every problem in NP can be reduced to it in polynomial time. This means that if you could solve an NP-hard problem quickly (in polynomial time), you could also solve all NP problems quickly.
MAX-3LIN-EQN is a computational problem that falls within the realm of optimization and computational complexity theory. It is a specific case of the broader MAX-CSP (Maximum Constraint Satisfaction Problem) problems. In MAX-3LIN-EQN, the goal is to find an assignment of values (often binary, i.e., 0 or 1) to a set of variables such that the number of satisfied equations is maximized. Each equation is linear and involves three variables.
MAX-3SAT is an optimization problem that is a specific case of the broader boolean satisfiability problem (SAT). In MAX-3SAT, given a boolean formula in conjunctive normal form (CNF), the goal is to determine the maximum number of clauses that can be satisfied by any assignment of truth values to the variables.
MAXEkSAT (Maximum Excluded K-Satisfiability) is a variant of the Boolean satisfiability problem (SAT) in which the goal is to identify the maximum number of clauses that can be made true by assigning truth values to a set of boolean variables, while ignoring a specified number of clauses. This is typically formulated as a decision problem or an optimization problem, where the objective is to maximize the number of satisfied clauses subject to some constraints.
NP-hardness is a classification used in computational complexity theory to describe certain types of problems. Specifically, a problem is said to be NP-hard if: 1. **It is at least as hard as the hardest problems in NP (nondeterministic polynomial time)**: This means that any problem in NP can be reduced to it in polynomial time. In more technical terms, if you can solve an NP-hard problem efficiently (in polynomial time), you can also solve any NP problem efficiently.
The Quadratic Assignment Problem (QAP) is a classic problem in combinatorial optimization. It can be defined as follows: Imagine you have two sets: 1. A set of **n** facilities (like warehouses, factories, etc.). 2. A set of **n** locations (like sites or areas where the facilities can be placed).
The Quadratic Bottleneck Assignment Problem (QBAP) is an optimization problem that seeks to assign a set of agents to a set of tasks in such a way that the maximum cost associated with any assignment is minimized. It can be considered a generalization of the classic assignment problem, which focuses on minimizing the total cost of assignments without regard to the maximum individual costs. ### Problem Definition - **Agents**: A set of \( n \) agents (or workers).
A Rectilinear Steiner Tree (RST) is a concept used in network design and VLSI (Very Large Scale Integration) design to find the shortest network that interconnects a given set of points using only horizontal and vertical segments. The tree allows for additional points, called Steiner points, to be introduced to minimize the overall path length of the tree.
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