Algorithms for the Simple Equal Flow Problem

Abstract

The minimum cost flow problem is to determine a least cost shipment of a commodity through a network G = (N, A) in order to satisfy demands at certain nodes from available supplies at other nodes. In this paper, we study a variant of the minimum cost flow problem where we are given a set R \subseteq A of arcs and require that each arc in R must carry the same amount of flow. This problem, which we call the simple equal flow problem, arose while modeling a water resource system management in Sardinia, Italy. We consider the simple equal flow problem in a directed network with n nodes, m arcs, and where all arc capacities and node supplies are integer and bounded by U. We develop several algorithms for the simple equal flow problem---the network simplex algorithm, the parametric simplex algorithm, the combinatorial parametric algorithm, the binary search algorithm, and the capacity scaling algorithm. The binary search algorithm solves the simple equal flow problem in O(log(nU)) applications of any minimum cost flow algorithm. The capacity scaling algorithm solves it in O(m(m + n logn) log (nU)) time, which is almost the same time needed to solve the minimum cost flow problem by the capacity scaling algorithm. These algorithms can be easily modified to obtain an integer solution of the simple equal flow problem.minimum cost flow problem, network simplex algorithm, scaling algorithm, parametric programming, network flows

Integer equal flows

“…Ahuja et al [2] have shown that this problem is solvable in polynomial time when = 1, but they did not address the complexity for greater values of . Our results are for the minimum cost flow version of the problem, though they also hold for the maximum flow version via a standard transformation [1].…”

“…Ahuja et al [2] studied the minimum cost simple equal flow problem as a means of modeling a water resource system in Sardinia, Italy. They detailed several different methods of solving the problem, including a version of the network simplex algorithm and a parametric simplex method.…”

Behavioral synthesis with activating unused flip-flops for reducing glitch power in FPGA

“…However, different from the general characteristics of networks, network H in our formulation requires that the flows on certain edges are equal. The min-cost flow problem with equal integral flow constraints is a difficult problem (NPhard) [2]. To trade off solution quality with runtime, we can use heuristic algorithms, such as the one presented in [3], where the authors used a Lagrangian relaxation technique to speed up the min-cost equal-flow problem.…”

Optimality study of resource binding with multi-Vdds

“…However, except the general characteristics of networks, H S d requires that the flow on edges (v a , v) and (v, v b ) are equal. The min-cost flow problem with equal integral flow constraint is a difficult problem (NP-hard) [2]. First of all, after we form the flow network, it becomes straight forward to reduce the problem into an ILP problem so we can take advantage of existing ILP solvers.…”

“…We prove that when we achieve the min-cost flow under the equal-flow constraint, we achieve the optimal binding solution under dual-Vdds, i.e., maximizing the total number of low-Vdd operations and minimizing the total switching activity simultaneously. Although the min-cost flow problem with equal integral flow constraint is a difficult problem (NP-hard) [2], our formulation has several values: (i) It leads to an easy reduction to the integer linear programming formulation, so that we can use an exiting ILP solver to get an optimal solution; (2) It enables efficient approximate solution by other mathematical programming method, such as Lagrangian relaxation as shown in [3]; (3) We hope that our formulation provides a graph-theoretical framework for us and others to study the exact complexity of the problem (note that there is no conclusion yet whether the multi-Vdd resource binding problem with fixed voltage configuration is NP-hard or polynomial-time solvable); (4) Finally, using the optimal solution provided by the ILP solver, for the first time, we are able to measure the optimality gap of the previous heuristics to our solution.…”