[Paper Review] Ordering with precedence constraints and budget minimization

Akbar Rafiey, Jeff Kinne+2 more|arXiv (Cornell University)|Jul 17, 2015

Advanced biosensing and bioanalysis techniques23 references3 citations

TL;DR

This paper introduces the bipartite graph ordering problem, a novel scheduling model with precedence constraints and budget minimization, where vertices in set B represent purchasable inputs and vertices in S represent sellable outputs, with edges indicating dependency. The authors present an exact algorithm running in O*(2^n) time and polynomial-time solutions for specific graph classes such as bipartite permutation graphs, trivially perfect bipartite graphs, and co-bipartite graphs, advancing a dichotomy classification of tractable cases.

ABSTRACT

We introduce a variation of the scheduling with precedence constraints problem that has applications to molecular folding and production management. We are given a bipartite graph $H=(B,S)$. Vertices in $B$ are thought of as goods or services that must be \emph{bought} to produce items in $S$ that are to be \emph{sold}. An edge from $j\in S$ to $i\in B$ indicates that the production of $j$ requires the purchase of $i$. Each vertex in $B$ has a cost, and each vertex in $S$ results in some gain. The goal is to obtain an ordering of $B\cup S$ that respects the precedence constraints and maximizes the minimal net profit encountered as the vertices are processed. We call this optimal value the \emph{budget} or \emph{capital} investment required for the bipartite graph, and refer to our problem as \emph{the bipartite graph ordering problem}. The problem is equivalent to a version of an NP-complete molecular folding problem that has been studied recently [12]. Work on the molecular folding problem has focused on heuristic algorithms and exponential-time exact algorithms for the un-weighted problem where costs are $\pm 1$ and when restricted to graphs arising from RNA folding. The bipartite graph present work seeks exact algorithms for solving the bipartite ordering problem. We demonstrate an algorithm that computes the optimal ordering in time $O^*(2^n)$ when $n$ is the number of vertices in the input bipartite graph. Our main result is a general strategy that can be used to find an optimal ordering in polynomial time for bipartite graphs that satisfy certain properties. We apply the technique to a variety of graph classes, obtaining polynomial-time solutions to the bipartite graph ordering problem for bipartite permutation graphs, trivially perfect, co-bipartite graphs, and trees.

Motivation & Objective

• To model and solve a new variant of scheduling with precedence constraints focused on minimizing the required capital investment (budget) rather than weighted completion time.
• To formalize the problem as the bipartite graph ordering problem, where vertices in B are purchased inputs and vertices in S are sold outputs with dependency edges from S to B.
• To identify graph classes for which the problem can be solved in polynomial time, aiming for a complete dichotomy classification of tractable vs. NP-complete cases.
• To generalize and extend prior work on molecular folding and RNA structure transformation by framing it as a budget-optimized ordering problem.
• To develop a general algorithmic strategy based on prime sets and lexicographic ordering of closures to guide optimal scheduling under budget constraints.

Proposed method

• The problem is modeled as a directed bipartite graph H = (B, S), where edges from S to B represent that a good in S requires a purchase of an item in B.
• The budget at each step is defined as the cumulative net profit (gain minus cost) of processed vertices, and the goal is to maximize the minimum budget encountered during the ordering.
• A key algorithmic strategy uses prime sets and closure sets (Superset_I(J)) to define a lexicographic ordering of vertex groups, enabling optimal reordering without exhaustive search.
• The algorithm recursively processes the lexicographically first prime subset I satisfying closure and dependency constraints, ensuring budget preservation.
• A correctness proof is established by showing that any optimal ordering can be transformed to respect the algorithm’s ordering rules without increasing the required budget.
• The running time is shown to be O*(2^n) in the general case, but becomes polynomial for specific graph classes such as bipartite permutation graphs, trivially perfect bipartite graphs, and co-bipartite graphs.

Experimental results

Research questions

• RQ1 For which classes of bipartite graphs can the budget-minimization ordering problem be solved in polynomial time?
• RQ2 Can the proposed algorithmic strategy based on prime sets and closures be generalized to identify new tractable graph classes?
• RQ3 Is there a dichotomy between polynomial-time solvable and NP-complete instances of the bipartite graph ordering problem?
• RQ4 Can the algorithm be adapted to provide approximation guarantees or fixed-parameter tractable solutions for broader graph families?
• RQ5 Does the algorithm’s structure allow for a characterization of all graph classes where it runs in polynomial time?

Key findings

• The paper presents an exact algorithm that computes the optimal ordering in O*(2^n) time for general bipartite graphs, where n is the number of vertices.
• Polynomial-time algorithms are developed for three specific graph classes: bipartite permutation graphs, trivially perfect bipartite graphs, and co-bipartite graphs.
• The algorithm uses a closure-based lexicographic ordering of prime sets to ensure that the budget constraint is preserved during reordering.
• The correctness of the algorithm is proven via induction and contradiction, showing that any optimal ordering can be transformed to follow the algorithm’s rules without increasing the required budget.
• The authors establish that the algorithm correctly identifies the minimal capital investment (budget) required for any given bipartite graph under precedence constraints.
• The work lays the foundation for a potential dichotomy theorem that separates polynomial-time solvable instances from NP-complete ones.