An approximation algorithm for finding a d-regular spanning connected subgraph of maximum weight in a complete graph with random weights of edges

We consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal. If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated right-hand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve. Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified. The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.

Graph-based approaches for simulating pedestrian dynamics in building models

An improved approximation algorithm for the traveling salesman problem with relaxed triangle inequality

Consider a complete graph Kn with edge weights drawn independently from a uniform distribution U(0, 1). The weight of the shortest (minimum‐weight) path P1 between two given vertices is known to be , asymptotically. Define a second‐shortest path P2 to be the shortest path edge‐disjoint from P1, and consider more generally the shortest path Pk edge‐disjoint from all earlier paths. We show that the cost Xk of Pk converges in probability to uniformly for all k ≤ n − 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapest k edge‐disjoint paths, that is, a minimum‐cost k‐flow. We also obtain the expectation of Xk conditioned on the existence of Pk.

In this paper, a bottom-up salient object detection method is proposed by modeling image as a random graph. The proposed method starts with portioning input image into superpixels and extracting color and spatial features for each superpixel. Then, a complete graph is constructed by employing superpixels as nodes. A high edge weight is assigned into a pair of superpixels if they have high similarity. Next, a random walk prior on nodes is assumed to generate the probability distribution on edges. On the other hand, a complete directed graph is created that each edge weight represents the probability for transmitting random walker from current node to next node. By considering a threshold and eliminating edges with higher probability than the threshold, a random graph is created to model input image. The inbound degree vector of a random graph is computed to determine the most salient nodes (regions). Finally, a propagation technique is used to form saliency map. Experimental results on two challenging datasets: MSRA10K and SED2 demonstrate the efficiency of the proposed unsupervised RG method in comparison with the state-of-the-art unsupervised methods.

We study paths of time-length $t$ of a continuous-time random walk on $\mathbb{Z}^{2}$ subject to self-interaction that depends on the geometry of the walk range and a collection of random, uniformly positive and finite edge weights. The interaction enters through a Gibbs weight at inverse temperature $\beta$; the “energy” is the total sum of the edge weights for edges on the outer boundary of the range. For edge weights sampled from a translation-invariant, ergodic law, we prove that the range boundary condensates around an asymptotic shape in the limit $t\to\infty$ followed by $\beta\to\infty$. The limit shape is a minimizer (unique, modulo translates) of the sum of the principal harmonic frequency of the domain and the perimeter with respect to the first-passage percolation norm derived from (the law of) the edge weights. A dense subset of all norms in $\mathbb{R}^{2}$, and thus a large variety of shapes, arise from the class of weight distributions to which our proofs apply.

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.

We consider the capacitated multi-source multicast tree routing problem (CMMTR) in an undirected graph G = (V,E) with a vertex set V, an edge set E and an edge weight w(e) ≥ 0, e ∈ E. We are given a source set S ⊆ V with a weight g(e) ≥ 0, e ∈ S, a terminal set M ⊆ V-S with a demand function q : M → R+, and a real number κ > 0, where g(s) means the cost for opening a vertex s ∈ S as a source in a multicast tree. Then the CMMTR asks to find a subset S′ ⊆ S, a partition {Z1,Z2,₀,Zl} of M, and a set of subtrees T1,T2,₀,Tl of G such that, for each i, Σt∈Ziq(t) ≤ κ and Ti spans Zi ∪ {s} for some s ∈ S′. The objective is to minimize the sum of the opening cost of S′ and the constructing cost of {Ti}, i.e., Σs∈S′g(s) + Σi=1lw(Ti), where w(Ti) denotes the sum of weights of all edges in Ti. In this paper, we propose a (2ρUFL + ρST)-approximation algorithm to the CMMTR, where ρUFL and ρST are any approximation ratios achievable for the uncapacitated facility location and the Steiner tree problems, respectively. When all terminals have unit demands, we give a ((3/2)ρUFL + (4/3)ρST -approximation algorithm.

In a complete graphwith independent uniform(or exponential) edge weights, letbe the minimum‐weight spanning tree (MST), andthe MST after deleting the edges of all previous trees. We show that each tree's weightconverges in probability to a constant, with, and we conjecture that. The problem is distinct from Frieze and Johansson's minimum combined weightofkedge‐disjoint spanning trees; indeed,. With an edge of weightw“arriving” at time, Kruskal's algorithm defines forests, initially empty and eventually equal to, each edge added to the first possible. Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component ofconverges to some. We conjecture that the functionstend to time translations of a single function.

A channel scheduling problem for a given graph G(V,E) is to select a subset E′ of edges of E and assign channel to each one in E′ under the restriction that all edges of E′ are interference free. We introduce the notion of two sides approximation for the channel scheduling problem. A pair of parameters (f,g) controls the accuracy of approximation in this paper. A (f,g)-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\) and \(\sum_{e\in E-E'}W(e)\le g\sum_{e\in E- Opt^*}W(e)\), where Opt * is the set of edges assigned channels in an optimal solution, and W(e) is the weight of edge e. An f-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\). We show that a simple greedy algorithm can obtain an (O(1), O(1))-approximation for the single channel scheduling problem. In many cases, the greedy algorithms gives much more accurate result than the worst ratio. Furthermore, we develop an \(|E|^{O({1\over \epsilon})}\) time (1 − ε, O(1))-approximation algorithm for the single channel scheduling problem. We also show that a simple greedy algorithm can obtain an O(1)-approximation for the multi-channel scheduling problem which satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over \Omega(1)}\). We also develop a \(|E|^{O({d\over \epsilon})}\) time (1 − ε)-approximation algorithm the multi-channel scheduling problem, where d is the number of channels. This improves the existing approximation scheme for multi-channel scheduling problem with \(|E|^{O({d\over \epsilon^2})}\) time by Cheng et al. We also develop a polynomial time constant factor greedy approximation algorithm for the multi-channel scheduling that allows variate radii of interference among those nodes.

Utility maximization data scheduling in drone-assisted vehicular networks

We consider the practical NP-complete problem of finding a minimum weight spanning tree with both edge weights and inner nodes weights. We present two polynomial time algorithms with approximation factors of 2.35 · ln n and 2Hn, respectively, where n is the number of nodes in the graph and Hn is the n-th Harmonic number. This nearly matches the lower bound of (l-\in )Hn, for any \in \ge 0. We also give an approximation algorithm with approximation factor \Delta - 1, where \Delta is the maximum degree of the graph. For metric spaces, we give a 3.105-approximation algorithm and show that an approximation factor of 1.463 is impossible unless {NP \subseteq DTIME[n^{O(\log longn)} ]}.

In this work we consider the m-d-UMST problem. The m-d-UMST is to find m edge-disjoint spanning trees with the given diameter d of a minimal weight in given n-vertex weighted Undirected graph \(G = (V, E)\). We give an \(O(n^2)\)-time approximation algorithm solving this problem. Then we prove its asymptotic optimality in the case of complete undirected graphs. We also assume that weights of graph edges are independent random variables identically distributed from the class of biased exponential distribution.KeywordsGiven-diameter minimal spanning treeProbabilistic analysisAsymptotic optimalityBiased exponential distribution

Given an edge weighted undirected graph \(G=(V,E)\) with \(|V|=n\), and a function \(f:V\rightarrow \mathbb {N}\), we consider the problem of finding a connected f-factor in G. In particular, for each constant \(c \ge 2\), we consider the case when \(f(v)\ge \frac{n}{c}\), for all v in V. We characterize the set of graphs that have a connected f-factor for \(f(v) \ge \frac{n}{3}\), for every v in V, and this gives polynomial time algorithm for the decision version of the problem. Extending the techniques we solve the minimization version. On the class of instances where the edge weights in G form a metric and \(f(v) \ge \frac{n}{c}\), c is a fixed value greater than 3, we give a PTAS. For each \(c \ge 3\) and \(\epsilon > 0\), our algorithm takes as input a metric weighted undirected graph G and a function \(f:V\rightarrow \mathbb {N}\) such that \(f(v) \ge \frac{n}{c}\), for every v in V, and computes a \((1+\epsilon )\)-approximation to the minimum weighted connected f-factor in polynomial time.

Irregular message passing networks