Optimizing over Consecutive 1's and Circular 1's Constraints

We consider packing and covering optimization problems over constraints in consecutive and circular 1's. Such problems arise in the context of shift scheduling, and in problems related to interval graphs. Previous approaches to this problem depended on solving several minimum cost network flow problems. We devise here substantially more efficient and strongly polynomial algorithms based on parametric shortest paths approaches. The objective function in the covering and packing problems is to either minimize or maximize the number of sets that satisfy the constraints. The various problems studied are classified according to whether the constraints are all consecutive 1's or if there are also circular 1's constraints, and according to whether the constraints are all of covering type; all of packing type, or mixed. The running time of our algorithm for a pure covering all consecutive 1's constraints problem on $n$ variables and $m$ constraints is $O(m+n)$. For the pure packing problem with consecutive 1's constraints we present an $O(m+n\log n)$ time algorithm. For the "mixed" case with both covering and packing consecutive 1's constraints we present an $O(mn)$ time algorithm. An $O(mn+n^2\log n)$-time algorithm is presented for the case where the constraints are circular (consecutive 1's constraint is also circular) of pure type---either all covering constraints or all packing constraints. Finally, we show an $O(n \min \{ mn,n^2\log n+m\log^2 n\})$ time algorithm for the most general problem of mixed covering and packing case where the constraints are circular. All our algorithms are strongly polynomial and improve on the nonstrongly polynomial parametric minimum cost network flow or the (strongly polynomial) linear programming known approaches.