Structured Randomized Rounding and Coloring

Non-independent randomized rounding and coloring

Completing Quasigroups or Latin Squares: A Structured Graph Coloring Problem

We introduce a graph coloring challenge benchmark based on the problem of completing Latin squares. We show how the hardness of the instances can be finely controlled by varying the fraction of precolored squares. We compare three complete (exact) solution strategies on this benchmark: (1) a Constraint Satisfaction (CSP) based approach, (2) a hybrid Linear Programming / CSP approach, and (3) a ...

Deconstructing Intractability: A Case Study for Interval Constrained Coloring

The NP-hard Interval Constrained Coloring problem appears in the interpretation of experimental data in biochemistry dealing with protein fragments. Given a set of m integer intervals in the range 1 to n and a set of m associated multisets of colors (specifying for each interval the colors to be used for its elements), one asks whether there is a “consistent” coloring for all integer points fro...

Path Coloring on the Mesh

In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al [1] and Raghavan and Upfal [22] as a model for routing in all-optical networks. It is also related to questi...

Unbiased Rounding of Rational Matrices

Rounding a real-valued matrix to an integer one such that the rounding errors in all rows and columns are less than one is a classical problem. It has been applied to hypergraph coloring, in scheduling and in statistics. Here, it often is also desirable to round each entry randomly such that the probability of rounding it up equals its fractional part. This is known as unbiased rounding in stat...