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

*(English)*Zbl 1154.90602Summary: 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 P. Raghavan and C. D. Thompson [Combinatorica 7, 365–374 (1987; Zbl 0651.90052)], which is usually used to round fractional solutions of linear programs.

We use our rounding procedure to design an additive approximation algorithm to the Quadratic Assignment Problem. The approximation error of the algorithm is \(\varepsilon n^2\) and it runs in \(n^{O(\log n/\varepsilon^2)}\) time.

We also describe Polynomial Time Approximation Schemes (PTASs) for dense subcases of many well-known NP-hard arrangement problems, including minimum linear arrangement, minimum cut linear arrangement, maximum acylic subgraph, and betweenness.

We use our rounding procedure to design an additive approximation algorithm to the Quadratic Assignment Problem. The approximation error of the algorithm is \(\varepsilon n^2\) and it runs in \(n^{O(\log n/\varepsilon^2)}\) time.

We also describe Polynomial Time Approximation Schemes (PTASs) for dense subcases of many well-known NP-hard arrangement problems, including minimum linear arrangement, minimum cut linear arrangement, maximum acylic subgraph, and betweenness.