# A new rounding procedure for the assignment problem with applications to dense graph arrangement problems

@article{Arora1996ANR, title={A new rounding procedure for the assignment problem with applications to dense graph arrangement problems}, author={Sanjeev Arora and Alan M. Frieze and Haim Kaplan}, journal={Mathematical Programming}, year={1996}, volume={92}, pages={1-36} }

Abstract.We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satisfies any linear inequality, then with high probability, the new matching satisfies that linear inequality in an approximate sense. This extends the well-known LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.¶We use our rounding procedure to design an additive approximation…

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## References

SHOWING 1-10 OF 36 REFERENCES

### Polynomial time approximation schemes for dense instances of NP-hard problems

- Computer ScienceSTOC '95
- 1995

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…

### The regularity lemma and approximation schemes for dense problems

- MathematicsProceedings of 37th Conference on Foundations of Computer Science
- 1996

The central point here is that the Regularity Lemma provides an explanation of why these Max-SNP hard problems turn out to be easy in dense graphs.

### An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms

- Computer Science[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
- 1988

The main result is an algorithm for performing the task provided that the capacity of each cut exceeds the demand across the cut by a Theta (log n) factor.

### An efficient approximation scheme for the one-dimensional bin-packing problem

- Computer Science23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
- 1982

It is proved that the LP relaxation of bin packing, which was solved efficiently in practice by Gilmore and Gomory, has membership in P, despite the fact that it has an astronomically large number of variables.

### Constructing a perfect matching is in random NC

- Mathematics, Computer ScienceSTOC '85
- 1985

We show that the problem of constructing a perfect matching in a graph is in the complexity class Random NC; i.e., the problem is solvable in polylog time by a randomized parallel algorithm using a…

### Probabilistic construction of deterministic algorithms: Approximating packing integer programs

- Computer Science, Mathematics27th Annual Symposium on Foundations of Computer Science (sfcs 1986)
- 1986

### Polynomial time randomised approximation schemes for the Tutte polynomial of dense graphs

- Mathematics, Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

A general technique is developed that supplies fully polynomial randomised approximation schemes for approximating the valve of T(G; x,, y) for any dense graph G, that is, any graph on n vertices whose minimum degree is /spl Omega/(n).

### Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems

- Mathematics, Computer ScienceJACM
- 1975

An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum.

### Polynomial time approximation schemes for Euclidean TSP and other geometric problems

- Computer ScienceProceedings of 37th Conference on Foundations of Computer Science
- 1996

A polynomial time approximation scheme for Euclidean TSP in /spl Rfr//sup 2/ that finds a (1+/spl epsiv/)-approximation to the optimum traveling salesman tour in time n/sup 0(1//spl ePSiv/)/.

### An approximation algorithm for the generalized assignment problem

- BusinessMath. Program.
- 1993

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei…