In the undirected Edge-Disjoint Paths problem with Congestion (EDPwC), we are given an undirected graph with V nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the Edge-Disjoint Paths (EDP) pr...

We consider the energy complexity of leader election problem in single-hop radio network model, where each device v has a unique identifier ID ( ) ∈{ 1, 2, ⋖ , N } . Energy is scarce resource for small battery-powered devices. For such devices, most often spent on communication, not computation. To approximate actual cost, an algorithm defined as maximum over all devices number time slots trans...

Let X ⊂ R d be a set of n points in general position (i.e. no d + 1 points of X lie on the same hyperplane.) For a positive 0 < ε < 1, we define A finite set S ⊂ R d is called a (weak) ε-net of X if S ∩ F = ∅ for any F ∈ F ε. We note that S is called a strong ε-net if, in addition, it also satisfies S ⊂ X. Since we are only going to discuss weak ε-nets, we will simply call these ε-nets. Our goa...

We study 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph G = (V,E), an alphabet set Σ and, for each edge {u, v} ∈ E, a constraint Cuv ⊆ Σ× Σ, the goal is to find an assignment σ : V → Σ that satisfies as many constraints as possible, where a constraint Cuv is said to be satisfied by σ if (σ(u), σ(v)) ∈ Cuv. While the approximability of 2...

We prove that, assuming the Unique Games Conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSP) where each constraint has constant arity is approximation resistant. In other words, we show that if ρ is the expected fraction of constraints satisfied by a random ordering, then obtaining a ρ′ approximation, for any ρ′ > ρ is UG-hard. For the simplest order...

We consider the Cauchy problem for Klein–Gordon equation in $${\mathbb{R}}^{d},d\ge 1,$$ with random initial data. introduce a family of measures $${\mu }_{0}^{\varepsilon },\varepsilon >0$$ depending on small parameter ε. The }$$ are assumed to be locally homogeneous or slowly varying under spatial shifts order o(ε−1) and inhomogeneous ε−1. Moreover, correlation functions decrease uniformly ε ...

and Applied Analysis 3 then there exists ε0 > 0 such that for all 0 < ε ≤ ε0 the mapping Θ : Ω ε → R3 is an injective mapping and the three vectors gεi x ε : ∂εiΘ x ε ∂α ∂/∂xα, ∂ ε 3 ∂/∂x ε 3 are linear independent for each x ∈ Ωε. The injectivity of the mapping Θ : Ωε → R3 ensures that the physical problem described below is meaningful. The three vectors gεi x ε form the covariant basis at the...

‎in this paper, using the idea of convexificators, we study boundedness and nonemptiness of lagrange multipliers satisfying the first order necessary conditions. we consider a class of nons- mooth fractional programming problems with equality, inequality constraints and an arbitrary set constraint. within this context, define generalized mangasarian-fromovitz constraint qualification and show t...