[Paper Review] The constant objective value property for combinatorial optimization problems

Ante Ćustić, Bettina Klinz|arXiv (Cornell University)|May 19, 2014

Optimization and Packing Problems19 references3 citations

TL;DR

This paper characterizes combinatorial optimization problems with the constant objective value property (COVP), where all feasible solutions yield identical objective values. It establishes that axial and planar d-dimensional assignment problems have COVP if and only if their cost arrays are sum-decomposable; this result does not extend to general d-dimensional assignment problems, and similar characterizations are provided for minimum spanning tree, shortest path, and minimum weight maximum cardinality matching problems.

ABSTRACT

Given a combinatorial optimization problem, we aim at characterizing the set of all instances for which every feasible solution has the same objective value. Our central result deals with multi-dimensional assignment problems. We show that for the axial and for the planar $d$-dimensional assignment problem instances with constant objective value property are characterized by sum-decomposable arrays. We provide a counterexample to show that the result does not carry over to general $d$-dimensional assignment problems. Our result for the axial $d$-dimensional assignment problem can be shown to carry over to the axial $d$-dimensional transportation problem. Moreover, we obtain characterizations when the constant objective value property holds for the minimum spanning tree problem, the shortest path problem and the minimum weight maximum cardinality matching problem.

Motivation & Objective

• To characterize the set of combinatorial optimization instances in which every feasible solution yields the same objective value (COVP).
• To investigate the connection between COVP and admissible transformations, particularly in the context of sum objective functions.
• To extend known results on COVP from the linear assignment problem to higher-dimensional assignment and transportation problems.
• To determine whether sum-decomposability fully characterizes COVP instances across all multidimensional assignment problems.
• To provide COVP characterizations for classical problems such as minimum spanning tree, shortest path, and minimum weight maximum cardinality matching.

Proposed method

• Define the COVP as the condition that all feasible solutions have identical objective values for a given cost vector.
• Use the correspondence between COVP and admissible transformations to reframe the problem in terms of cost vector decompositions.
• Introduce sum-decomposable arrays as generalizations of sum matrices to higher dimensions, where cost entries are expressible as sums of d-dimensional vectors.
• Prove that axial and planar d-dimensional assignment problems have COVP if and only if their cost arrays are sum-decomposable.
• Construct a counterexample to show that sum-decomposability does not characterize COVP in general d-dimensional assignment problems.
• Leverage d-dimensional assignment results to derive COVP characterizations for the axial d-dimensional transportation problem.

Experimental results

Research questions

• RQ1 For which classes of d-dimensional assignment problems does the COVP hold if and only if the cost array is sum-decomposable?
• RQ2 Does the COVP characterization via sum-decomposable arrays extend to all d-dimensional assignment problems, or are there exceptions?
• RQ3 What conditions on edge weights ensure that all paths from source to sink in a graph have identical total weight (COVP for shortest path)?
• RQ4 Under what conditions is every perfect matching in a complete graph of even order guaranteed to have the same total weight (COVP for matching)?
• RQ5 Can the COVP in the minimum spanning tree problem be characterized by a structural property of the edge weight matrix?

Key findings

• The axial and planar d-dimensional assignment problems have the COVP if and only if their cost arrays are sum-decomposable.
• A counterexample demonstrates that the COVP does not hold for general d-dimensional assignment problems even when the cost array is sum-decomposable.
• The COVP for the axial d-dimensional transportation problem is characterized by the same sum-decomposable array condition as the axial d-dimensional assignment problem.
• For the shortest path problem in a directed graph, the COVP holds if and only if edge weights can be expressed as differences of vertex potentials: w(i,j) = a_j - a_i.
• In the undirected shortest path problem, the COVP holds if and only if edge weights are representable as differences of vertex potentials.
• For the minimum weight maximum cardinality matching problem on a complete graph: if n is odd, all edge weights must be equal; if n is even, edge weights must be expressible as w(i,j) = a_i + a_j for some real vector A.