An instance of the quadratic assignment problem (QAP) with cost matrix Q is said to be linearizable if there exists an instance of the linear assignment problem (LAP) with cost matrix C such that for each assignment, the QAP and LAP objective function values are identical. The QAP linearization problem can be solved in O(n4) … Read more

The minimum spanning tree problem with conflict constraints and its variations

We consider the minimum spanning tree problem with conflict constraints (MSTC). It is observed that computing an $\epsilon$-optimal solution to MSTC is NP-hard for any $\epsilon >0$. For a general conflict graph, computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded … Read more

Integer Network Synthesis Problem for Hop Constrained Flows

Hop constraint is associated with modern communication network flows. We consider the problem of designing an optimal undirected network with integer-valued edge-capacities that meets a given set of single-commodity, hop-constrained network flow value requirements. We present a strongly polynomial, combinatorial algorithm for the problem with value of hop-parameter equal to three when values of flow … Read more

Trioid: A generalization of matroid and the associated polytope

We consider a generalization of the well known greedy algorithm, called $m$-step greedy algorithm, where $m$ elements are examined in each iteration. When $m=1$ or $2$, the algorithm reduces to the standard greedy algorithm. For $m=3$ we provide a complete characterization of the independence system, called trioid, where the $m$-step greedy algorithm guarantees an optimal … Read more

On cost matrices with two and three distinct values of Hamiltonian paths and cycles

Polynomially testable characterization of cost matrices associated with a complete digraph on $n$ nodes such that all the Hamiltonian cycles (tours) have the same cost is well known. Tarasov~\cite{TARA81} obtained a characterization of cost matrices where tour costs take two distinct values. We provide a simple alternative characterization of such cost matrices that can be … Read more