Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

An elementary h-route flow, for an integer h ≥ 1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h ≤ 3: The size of a minimum h-route cut is at least f/h and at most O(log 3 k·f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h = 3 that has an approximation ratio of O(log 3 k). Previously, polylogarithmic approximation was known only for h-route cuts for h ≤ 2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem

Mathematical Models and Algorithms for Network Flow Problems Arising in Wireless Sensor Network Applications

We examine multiple variations on two classical network flow problems, the maximum flow and minimum-cost flow problems. These two problems are well-studied within the optimization community, and many models and algorithms have been presented for their solution. Due to the unique characteristics of the problems we consider, existing approaches cannot be directly applied. The problem variations we examine commonly arise in wireless sensor network (WSN) applications. A WSN consists of a set of sensors and collection sinks that gather and analyze environmental conditions. In addition to providing a taxonomy of relevant literature, we present mathematical programming models and algorithms for solving such problems. First, we consider a variation of the maximum flow problem having node-capacity restrictions. As an alternative to solving a single linear programming (LP) model, we present two alternative solution techniques. The first iteratively solves two smaller auxiliary LP models, and the second is a heuristic approach that avoids solving any LP. We also examine a variation of the maximum flow problem having semicontinuous restrictions that requires the flow, if positive, on any path to be greater than or equal to a minimum threshold. To avoid solving a mixed-integer programming (MIP) model, we present a branch-and-price algorithm that significantly improves the computational time required to solve the problem. Finally, we study two dynamic network flow problems that arise in wireless sensor networks under non-simultaneous flow assumptions. We first consider a dynamic maximum flow problem that requires an arc to transmit a minimum amount of flow each time it begins transmission. We present an MIP for solving this problem along with a heuristic algorithm for its solution. Additionally, we study a dynamic minimum-cost flow problem, in which an additional cost is incurred each time an arc begins transmission. In addition to an MIP, we present an exact algorithm that iteratively solves a relaxed version of the MIP until an optimal solution is found

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Single Source Multiroute Flows and Cuts on Uniform Capacity Networks

For an integer h ≥ 1, an elementary h-route flow is a flow along h edge disjoint paths between a source and a sink, each path carrying a unit of flow, and a single commodity h-route flow is a non-negative linear combination of elementary h-flows. An instance of a single source multicommodity flow problem for a graph G = (V, E) consists of a source vertex s ∈ V and k sinks t1,..., tk ∈ V; we denote it I = (s; t1,..., tk). In the single source multicommodity multiroute flow problem, we are given an instance I = (s; t1,..., tk) and an integer h ≥ 1, and the objective is to maximize the total amount of flow that is transferred from the source to the sinks so that the capacity constraints are obeyed and, moreover, the flow of each commodity is an h-route flow. We study the relation between classical and multiroute single source flows on networks with uniform capacities and we provide a tight bound. In particular, we prove the following result. Given an instance I = (s; t1,..., tk) such that each s − ti pair is hconnected, the maximum classical flow between s and the ti’s is at most 2(1−1/h)-times larger than the maximum h-route flow between s and the ti’s and this is the best possible bound for h ≥ 2. This, as we show, is in contrast to the situation of general multicommodity multiroute flows that are up to k(1−1/h)-times smaller than their classical counterparts. As

Single Source Multiroute Flows and Cuts on Uniform Capacity Networks

For an integer h ≥ 1, an elementary h-route flow is a flow along h edge disjoint paths between a source and a sink, each path carrying a unit of flow, and a single commodity h-route flow is a non-negative linear combination of elementary h-route flows. An instance of a single source multicommodity flow problem for a graph G = (V,E) consists of a source vertex s ∈ V and k sinks t1,...,tk ∈ V corresponding to k commodities; we denote it I = (s;t1,...,tk). In the single source multicommodity multiroute flow problem, we are given an instance I = (s;t1,...,tk) and an integer h ≥ 1, and the objective is to maximize the total amount of flow that is transferred from the source to the sinks so that the capacity constraints are obeyed and, moreover, the flow of each commodity is an h-route flow. We study the relation between classical and multiroute single source flows on undirected networks with uniform capacities and we provide a tight bound. In particular, we prove the following result. Given an instance I = (s;t1,...,tk) such that each s − ti pair is h-connected, the maximum classical flow between s and the ti’s is at most (2 − 2/h)-times larger than the maximum h-route flow between s and the ti’s and this is the best possibl

Single Source Multiroute Flows and Cuts on Uniform Capacity Networks

For an integer h ≥ 1, an elementary h-route flow is a flow along h edge disjoint paths between a source and a sink, each path carrying a unit of flow, and a single commodity h-route flow is a non-negative linear combination of elementary h-route flows. An instance of a single source multicommodity flow problem for a graph G = (V, E) consists of a source vertex s ∈ V and k sinks t1,..., tk ∈ V; we denote it I = (s; t1,..., tk). In the single source multicommodity multiroute flow problem, we are given an instance I = (s; t1,..., tk) and an integer h ≥ 1, and the objective is to maximize the total amount of flow that is transferred from the source to the sinks so that the capacity constraints are obeyed and, moreover, the flow of each commodity is an h-route flow. We study the relation between classical and multiroute single source flows on networks with uniform capacities and we provide a tigh