Combinatorial Optimization

Schrijver, A. (2003). Combinatorial Optimization: Polyhedra and Efficiency. Springer.

Type

Book

Intellectual base

Review or monograph

Year

2003

Link

https://openlibrary.org/works/OL5096223W

Cited from

operations_research

Cited by

4 citations across 4 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Network Flow Models
  • The totally-unimodular structure of network-flow constraint matrices guarantees integer optimal solutions without explicit integer-programming treatment, making network-flow formulations distinctly more tractable than general ILPs, a property Schrijver (2003) develops within the broader theory of totally unimodular matrices and integer polyhedra.

    This sourceComprehensive treatment of network flow within combinatorial optimization, including totally-unimodular structure and polynomial-time algorithms.

Domain-specific

• Goldberg–Seymour Theorem
  • This is an integer-rounding phenomenon specific to the multigraph edge-coloring polytope and odd-set constraints.
• Independence system
  • … empty-set inclusion, hereditary axiom, maximal independent sets, rank or augmentation properties and qualifications that distinguish matroids, greedoids and arbitrary hypergraphs are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Matching
  • The matching polytope — the convex hull of characteristic vectors of matchings in \(G\) — has a complete linear-programming description for bipartite graphs (nonnegativity per edge plus the degree constraints per vertex), and that description is integral, the bipartite incidence matrix being totally unimodular; for general graphs those same constraints define only a fractional relaxation whose vertices are half-integral, and the exact description additionally requires Edmonds's odd-set inequalities, placing matching at the center of the polyhedral combinatorics literature

    This sourceThe standard reference for the polyhedral picture this sentence states: the bipartite matching polytope described exactly by nonnegativity and the degree constraints and integral there by total unimodularity, the same constraints yielding only a half-integral relaxation in a general graph, and Edmonds's odd-set inequalities as the additional family the exact description requires.

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