We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense " instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimum k-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degree\Omega\Gamma n), although our algorithms solve most of these problems so long as the average degree is \Omega\Gamma n). Denseness for non-graph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense " polynomials of constant degree.
|
7711
|
Computers and Intractability: A Guide to the Theory of NP-Completeness
– Garey, Johnson
- 1979
|
|
1587
|
Computational Complexity
– Papadimitriou
- 1994
|
|
574
|
Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
– Goemans, Williamson
- 1995
|
|
503
|
Optimization, approximation, and complexity classes
– Papadimitriou, Yannakakis
- 1991
|
|
457
|
A new polynomial-time algorithm for linear programming
– Karmarkar
- 1984
|
|
383
|
Some optimal inapproximability results
– H˚astad
|
|
317
|
On the hardness of approximating minimization problems
– Lund, Yannakakis
- 1994
|
|
288
|
Probabilistic checking of proofs: a new characterization of np
– Arora, Safra
- 1998
|
|
271
|
Property testing and its connection to learning and approximation
– Goldreich, Goldwasser, et al.
- 1996
|
|
271
|
Probabilistic construction of deterministic algorithms: Approximating packing integer programs
– Raghavan
- 1988
|
|
239
|
Randomized rounding: a technique for provably good algorithms and algorithmic proofs
– Raghavan, Thompson
- 1987
|
|
216
|
Approximating the permanent
– JERRUM, SINCLAIR
- 1989
|
|
207
|
An approximate max-flow min-cut theorems for uniform multicommodity flow problem with applications to approximation algorithms
– Leighton, Rao
- 1988
|
|
140
|
Fast approximation algorithms for the knapsack and sum of subset problems
– Ibarra, Kim
- 1975
|
|
110
|
A random polynomial time algorithm for approximating the volume of convex bodies
– Dyer, Frieze, et al.
- 1989
|
|
103
|
Graph Bisection Algorithms with Good Average Case
– Bui, Chaudhuri, et al.
- 1987
|
|
101
|
On the approximation of maximum satisfiability
– Yannakakis
- 1994
|
|
93
|
An efficient approximation scheme for the onedimensional bin-packing problem
– Karp, Karmarkar
- 1982
|
|
82
|
The regularity lemma and approximation schemes for dense problems
– Frieze, Kannan
- 1996
|
|
78
|
Clique is hard to approximate within n 1\Gammaffl
– Hastad
- 1996
|
|
73
|
Monte-Carlo approximation algorithms for enumeration problems
– Karp, Luby, et al.
- 1989
|
|
64
|
Eigenvalues and graph bisection: An average-case analysis
– Boppana
- 1987
|
|
57
|
A new rounding procedure for the assignment problem with applications to dense graph arrangement problems
– Frieze, Kaplan
|
|
53
|
Approximate algorithms for the 0-1 knapsack problem
– Sahni
- 1975
|
|
44
|
Randomness-efficient oblivious sampling
– Bellare, Rompel
- 1994
|
|
42
|
Hamiltonian circuits in random graphs
– Pósa
- 1976
|
|
39
|
On choosing a dense subgraph
– Kortsarz, Peleg
- 1993
|
|
36
|
Simulated annealing for graph bisection
– Jerrum, Sorkin
- 1993
|
|
35
|
A complete classification of the approximability of maximization problems derived from boolean constraint satisfaction
– Khanna, Sudan, et al.
- 1997
|
|
34
|
de la Vega, MAX-CUT has a randomized approximation scheme
– Fernandez
- 1996
|
|
34
|
Finding k-cuts within twice the optimal
– Saran, Vazirani
- 1995
|
|
32
|
Computing near-optimal solutions to combinatorial optimization problems
– Shmoys
- 1995
|
|
27
|
Randomness in Interactive Proofs
– Bellare, Goldreich, et al.
- 1993
|
|
20
|
Bin packing can be solved within 1 + ffl in linear time
– Vega, Lueker
- 1981
|
|
13
|
The complexity of colouring problems on dense graphs
– Edwards
- 1986
|
|
12
|
Polynomial time randomized approximation schemes for tutte-grothendieck invariants: the dense case. Random Structures Algorithms
– Alon, Frieze, et al.
- 1995
|
|
6
|
A chernoff bound for random walks on expanders
– Gillman
- 1993
|
|
4
|
Probability inequalities for sums of bounded random variables
– Hoffding
- 1964
|
|
2
|
Approximating the value of 2-prover proof systems, with applications to max-2sat and max-dicut
– Feige, Goemans
- 1995
|
|
2
|
A 7=8 \Gamma ffl approximation for MAX-3SAT
– Karloff, Zwick
- 1997
|
|
2
|
On the greedy heuristic for satisfiability
– Koutsoupias, Papadimitriou
- 1992
|
|
