P∈Pi P∋e fi,P ≤ ce ∀e fi,P ≥ 0 Dual 1 solves the max-sum multi-commodity flow problem: ce represents the capacity of an edge, and fi,P is the amount of flow directed from si to ti along the path P . The LP tries to maximize the total amount of commodity flow. Lemma 17.1.1 Multicut is always larger than the corresponding max-sum multi-commodity flow. Lemma 17.1.2 Multicut is at most O(logK) time...

In the present work an attempt is being made to reduce the Maximum Flow Network Interdiction Problem (MFNIP) in to the Subset Sum Problem so as to get some algorithms solvable in polynomial time. Previously developed algorithms are either applicable to some special cases of MFNIP or they do not have a constant performance guarantee. Our reduction has paved the way towards the development of ful...

Let G be a graph. A zero-sum flow of G is an assignment of non-zero real numbers to the edges such that the sum of the values of all edges incident with each vertex is zero. Let k be a natural number. A zero-sum k-flow is a flow with values from the set {±1, . . . ,±(k − 1)}. It has been conjectured that every r-regular graph, r ≥ 3, admits a zero-sum 5-flow. In this paper we give an affirmativ...

We consider the problem of finding a feasible single-commodity flow in a strongly connected network with fixed supplies and demands, provided that the sum of supplies equals the sum of demands and the minimum arc capacity is at least this sum. A fast algorithm for this problem improves the worst-case time bound of the Goldberg-Rao maximum flow method[3] by a constant factor. Erlebach and Hageru...

an injective map f : e(g) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling of a graph g(p, q) if the induced vertex function f*: v (g) → z − {0} defined by f*(v) = (sigma e∈ev) f (e) is one-one, where ev denotes the set of edges in g that are incident with a vetex v and f*(v (g)) is either of the form {±k1, ±k2, · · · , ±kp/2} or {±k1, ±k2, · · · , ±k(p−1)/2} u {k(p+1)/2} accordin...

I derive a procedure to generate sum rules for the trace anomalies a and a′. Linear combinations of ∆a ≡ aUV − aIR and ∆a′ ≡ aUV − aIR are expressed as multiple flow integrals of the two-, threeand four-point functions of the trace of the stress tensor. Eliminating ∆a′, universal flow invariants are obtained, in particular sum rules for ∆a. The formulas hold in the most general renormalizable q...

Inverse maximum flow (IMDF), is among the most important problems in the field ofdynamic network flow, which has been considered the Euclidean norms measure in previousresearches. However, recent studies have mainly focused on the inverse problems under theHamming distance measure due to their practical and important applications. In this paper,we studies a general approach for handling the inv...