Advances in Combinatorial Optimization

Linear Programming Formulations of the Traveling Salesman and Other Hard Combinatorial Optimization Problems

Combinational optimization (CO) is a topic in applied mathematics, decision science and computer science that consists of finding the best solution from a non-exhaustive search. CO is related to disciplines such as computational complexity theory and algorithm theory, and has important applications in fields such as operations research/management science, artificial intelligence, machine learning, and software engineering.

Advances in Combinatorial Optimization presents a generalized framework for formulating hard combinatorial optimization problems (COPs) as polynomial sized linear programs. Though developed based on the 'traveling salesman problem' (TSP), the framework allows for the formulating of many of the well-known NP-Complete COPs directly (without the need to reduce them to other COPs) as linear programs, and demonstrates the same for three other problems (e.g. the 'vertex coloring problem' (VCP)). This work also represents a proof of the equality of the complexity classes "P" (polynomial time) and "NP" (nondeterministic polynomial time), and makes a contribution to the theory and application of 'extended formulations' (EFs).

On a whole, Advances in Combinatorial Optimization offers new modeling and solution perspectives which will be useful to professionals, graduate students and researchers who are either involved in routing, scheduling and sequencing decision-making in particular, or in dealing with the theory of computing in general.

Errata(s)
Errata (267 KB)

Contents:

• Introduction
• Basic IP Model Using the TSP
• Basic LP Model Using the TSP
• Generic LP Modeling for COPs
• Non-Symmetry of the Basic (TSP) Model
• Non-Applicability of Extended Formulations Theory
• Illustrations for Other NP-Complete COPs

Readership: Professionals, graduate students and researchers who are either involved in routing, scheduling and sequencing decision-making in particular, or in dealing with the theory of computing in general.
Key Features:

• The book offers a new proof of the equality of the complexity classes "P" and "NP"
• Although our approach is developed using the framework of the TSP, it has natural analogs for the other problems in the NP-Complete class thus providing a unified framework for modeling many combinatorial optimization problems (COPs)
• The book makes a contribution to the theory and application of Extended Formulations (EFs) refining the notion of EFs by separating the case in which that notion is degenerate from the case in which the notion of EF is well defined/meaningful. It separates the case in which the addition of redundant constraints and variables (for the purpose of establishing EF relations) matters from the case in which the addition of redundant constraints and variables does not matter