Given an edge-weighted undirected connected graph G = (V, E, ρ , 𝒳, 𝒴) , where ρ : E →R^+∪{ 0 } is an edge-weight function, 𝒳⊂ V is a subset of clients, and 𝒴⊂ V is a subset of candidates, and a positive integer k < |𝒴| , the k-Supplier Problem (k SP) asks for an optimal subset of 𝒴 of cardinality at most k to minimize the radius from 𝒳 to the subset. In this paper, we focus on the case of 𝒳∩𝒴∅ , 𝒳, 𝒴 , and consider the scenario where the shortest path distances d(· , · ) in G satisfy a parameterized triangle inequality between 𝒳 and 𝒴 , i.e., d(x, y) + d(y, z) ≥α· d(x, z), ∀ x, y, z ∈{ u, v, w }, x y, y z, z x , where 1 ≤α≤ 2 is a parameter, for any three distinct vertices, v, u ∈𝒳 and w ∈𝒴 . We present a two-stage dual approximation algorithm ALG for the kSP with parameter triangle inequality between 𝒳 and 𝒴 . If it stops at the end of Stage 1 then it achieves a 2/α -approximation, and if it stops at the end of Stage 2 then it achieves a (2/α ^2 + 1/α) -approximation. ALG runs in a polynomial time and the above two parameterized performance factors of it are both strictly monotonic decreasing with respect to the value of parameter α . For the kSP instances with parameterized triangle inequality having 1 < α≤ 2 , it is implied by 1 ≤2/α < 2 and 1 ≤2/α ^2 + 1/α < 3 that ALG has better approximation ratios than the previously best polynomial-time 3-approximation algorithm of Hochbaum and Shmoys (J. ACM. 33: 533–550, 1986). Furthermore, regardless of the parameterized aspect, ALG achieves a 2-approximation if it stops at the end of Stage 1 and a 3-approximation if it stops at the end of Stage 2, for the general kSP.

We study the problem of covering the maximum number of vertices in a graph by a collection of vertex-disjoint stars, each with a number of satellites in a given interval [k, ℓ], where 1 ≤ k < ℓ and ℓ can be infinity. This is referred to as sequential [k, ℓ]-Star Packing problem. It is solvable in polynomial time when k = 1, but becomes strongly NP-hard when k ≥ 2. In this paper, we propose either the first or an improved approximation algorithm for the following four sequential settings: 1) a k+1/2-approximation algorithm when k ≥ 3 and ℓ = ∞, improving the previous best ratio of (k+1)^2/2k+1; 2) a 4/3-approximation algorithm when k = 2 and ℓ = ∞, improving the previous best ratio of 3/2; 3) the first (1 + ℓ/ℓ+1)-approximation algorithm when 2 = k < ℓ; and 4) the first (1 + max{k-1/2, (k+1) ℓ/3 (ℓ+1)})-approximation algorithm when 3 ≤ k < ℓ. Besides the main algorithmic techniques being local search coupled with amortized analysis, we observe augmenting configurations to bridge two distant neighborhoods for a local improvement operation. Additionally, the problem has been shown APX-hard when k ≥ 3; we prove its APX-hardness for the last remaining case where k = 2.

This paper studies the Connected k-Facility Location Problem (ConkFLP) on trees. Let T = ( V, E, w, rho, f ) be an undirected tree, where V = {v(1), v(2), . . . , v(n)} is the n-vertices set and E = {e(1), e(2), . . . , e(n- 1)} is the (n - 1)-edges set. A facility set F subset of V and a client set C subset of V are given. Each client vi is an element of C has one weight rho(i) >= 0 denoting the demand amount of v(i), and each facility v(i) is an element of F has a weight f(i) >= 0 denoting the opening cost at v(i), and each edge e(j), for j is an element of {1, 2, ... , n - 1}, is associated with a weight w(j) >= 0 denoting the connection cost of it. When some facilities S subset of F are opened, the overall cost involved in ConkFLP includes three parts: the cost of opening facilities S, K times the cost of Steiner tree interconnecting all the opened facilities where K is a fixed parameter, and the total connection cost of assigning each client to the closest facility in S. The goal of ConkFLP is to open at most k < n facilities to minimize the overall cost, for a given input parameter K >= 1. This paper focuses on the case of ConkFLP on trees where F = C = V , and as a result presents a polynomial-time exact dynamic programming algorithm and a computational experiment to illustrate it. Furthermore, a simple way is shown to adapt the algorithm to the general case of F subset of V and C subset of V. Finally, we apply the algorithm to a ConkFLP instance in a regional tree-like water transportation network.

Let G = (V, E, w, rho, X) be a weighted undirected connected graph, where V is the set of vertices, E is the set of edges, X subset of V is a subset of terminals, w(e) > 0, for all e is an element of E denotes the weight associated with edge e, and rho(v) > 0, for all v is an element of V denotes the weight associated with vertex v. Let T be a Steiner tree in G to interconnect all terminals in X. For any two terminals, t', t '' is an element of X, we consider the weighted tree distance on T from t' to t '', defined as the weight of t '' times the classic tree distance on T from t' to t ''. The longest weighted tree distance on T between terminals is named the weighted diameter of T. The Minimum Diameter Vertex-Weighted Steiner Tree Problem (MDWSTP) asks for a Steiner tree in G of the minimum weighted diameter to interconnect all terminals in X. In this paper, we introduce two classes of parameterized graphs (PG), < X, mu >-PG and (X, lambda)-PG, in terms of the parameterized upper bound on the ratio of two vertex weights, and a weaker version of the parameterized triangle inequality, respectively, and present approximation algorithms of a parameterized factor for the MDWSTP in them. For the MDWSTP in an edge-weighted < X, mu >-PG, we present an approximation algorithm of a parameterized factor mu+1/2. For the MDWSTP in a vertex-weighted (X, lambda)-PG, we first present a simple approximation algorithm of a parameterized factor lambda, where lambda is tight when lambda >= 2, and further develop another approximation algorithm of a slightly improved factor.

In the development of modern economic construction, the intelligent technological means of clothing design has become the core topic discussed by manufacturing enterprises. Although the main trend and related problems of the design mode are still fuzzy, with the continuous improvement of science and technology level, the clothing design with network technological means as the core has become the focus of the manufacturing industry, which can not only simulate the perception of life system, A dual feedback mechanism with clothing sensing technology as the core can also be constructed. Therefore, on the basis of understanding the research status of Internet-based intelligent safety clothing design technology, this paper mainly studies the application effect of intelligent safety clothing design method in the new era according to the structure and application of intelligent safety clothing design, in order to provide technical basis for the innovation and development of clothing manufacturing industry in the new era.

This paper considers a novel quality-of-service (QoS) routing problem from a source to a destination, named the Cheapest Deadline Path Problem (CDPP), which arises from the real-world scenario. Let \(D = (V, A, c, d, s, t)\) be a double-weighted strongly connected digraph, where each arc \(a \in A\) is associated with a cost, \(c(a) \in \mathbf {Z}^+\), and a delay, \(d(a) \in \mathbf {Z}^+\), and s and t are the indices of the designated source and destination, respectively, and let \(\mathcal {B} = \{ \mathbb {B}_1, \mathbb {B}_2, \ldots , \mathbb {B}_n \}\) be a set of positive constants, where \(\mathbb {B}_{i}, 1 \le i \le n\) represents the upper bound on delay at \(v_i \in V\). The objective of CDPP is to find a \(v_s\)-to-\(v_t\) path of the minimum cost in D such that the \(v_s\)-to-\(v_k\) delay along the path is at most \(\mathbb {B}_{k}\), for each vertex, \(v_k\), appearing in the path. This paper presents a fully polynomial time approximation scheme (FPTAS) for CDPP in \(D = (V, A, c, d, s, t)\) using a graph traverse based dynamic programming algorithm as a sub-procedure.

Let \(G = (V, E, w, \rho , \mathcal {T})\) be a weighted connected graph, where V is the vertex set, E is the edge set, \(\mathcal {T} \subseteq V\) is a terminal subset, \(w: E \rightarrow \mathbb {R}^{+}\) is an edge-weight function and \(\rho : V \rightarrow \mathbb {R}^{+}\) is a vertex-weight function. The weighted diameter of a Steiner tree T in G spanning \(\mathcal {T}\) is referred to as the longest weighted tree distance on T between terminals. The objective of the Minimum Diameter Vertex-Weighted Steiner Tree Problem (MDWSTP) is to construct a Steiner tree in G spanning \(\mathcal {T}\) to minimize the weighted diameter.

Given an n-dimensional integer vector $$\mathbf {v} = (v_1, v_2, \ldots , v_n)$$ with $$v_1 \le v_2 \le \cdots \le v_n$$ , a pinwheel schedule for $$\mathbf {v}$$ is an infinite symbol sequence $$S_1 S_2 S_3 \cdots$$ , which satisfies that $$S_j \in \{ 1, 2, \ldots , n \}, \forall j \in \mathbb {Z}^{+}$$ and every $$i \in \{ 1, 2, \ldots , n \}$$ occurs at least once in every $$v_i$$ consecutive symbols $$S_{j + 1} S_{j + 2} \cdots S_{j + v_i}, \forall j \in \mathbb {Z}^{+} \cup \{ 0 \}$$ . If $$\mathbf {v}$$ has a pinwheel schedule then $$\mathbf {v}$$ is called (pinwheel) schedulable. The density of $$\mathbf {v}$$ is defined as $$d(\mathbf {v}) = \sum _{i = 1}^{n} \frac{1}{v_i}$$ . Chan and Chin (Algorithmica 9(5):425–462, 1993) made a conjecture that every vector $$\mathbf {v}$$ with $$d(\mathbf {v}) \le \frac{5}{6}$$ is schedulable. In this paper, we check the conjecture from the perspective of low-dimensional vectors, including 3-, 4- and 5-dimensional ones. We first find some simple but important properties of schedulable vectors, and then propose two comparing rules according to these properties. Also, we define a vector-space tree to represent all k-dimensional vectors, for any given integer $$k \ge 2$$ . Under the framework of the vector-space tree, we use the comparing rules to develop a Branch-and-Cut Approach to examining the schedulability of all k-dimensional vectors. As a result, we prove that the maximum density guarantee for the pinwheel schedulability of low-dimensional vectors is $$\frac{5}{6}$$ , which partially supports Chan and Chin’s conjecture.

The asymmetric p-center problem (ApCP) was proved by Chuzhoy et al. (STOC’04) to be NP-hard to approximate within a factor of $$\log ^*n - \Theta (1)$$ unless $$\mathrm {NP} \subseteq \mathrm {DTIME}(n^{\log \log n})$$. This paper studies ApCP and the vertex-weighted asymmetric p-center problem (WApCP). First, we propose four classes of parameterized complete digraphs, $$\alpha $$-CD, $$(\alpha , \beta )$$-CD, $$\langle \alpha , \gamma \rangle $$-CD and $$(\alpha , \beta , \gamma )$$-CD, from the angle of the parameterized upper bound on the ratio of two asymmetric edge weights between vertices as well as on the ratio of two vertex weights, and the parameterized triangle inequality, respectively. Using the greedy approach, we achieve a $$(1 + \alpha )$$- and $$\beta \cdot (1 + \alpha )$$-approximation algorithm for the ApCP in $$\alpha $$-CD’s and $$(\alpha , \beta )$$-CD’s, respectively, as well as a $$(1 + \alpha \gamma )$$- and $$\beta \cdot (1 + \alpha \gamma )$$-approximation algorithm for the WApCP in $$\langle \alpha , \gamma \rangle $$-CD’s and $$(\alpha , \beta , \gamma )$$-CD’s, respectively.

Given an edge-weighted undirected graph G=(V,E,w) and a subset T⊆V of p terminals, a k-Steiner forest spanning all the terminals in T includes k branches, where every branch is a Steiner tree. The diameter of a k-Steiner forest is referred to as the maximum distance between two terminals of a branch. This paper studies the minimum diameter k-Steiner forest problem (MDkSFP) and establishes the relationship between MDkSFP and the absolute k-Steiner center problem (AkSCP). We first obtain a 2-factor dual approximation algorithm for AkSCP, and then achieve a 2-approximation algorithm for MDkSFP based on the 2-approximation to AkSCP. Furthermore, we develop an improved 2ρ-approximation algorithm for MDkSFP, where ρ<1 in general, by perturbing the sites of facilities and re-clustering the terminals.

Given an n-dimensional integer vector v = (v_1, v_2, … , v_n) with 2 ≤ v_1 ≤ v_2 ≤⋯≤ v_n , a pinwheel schedule for v is referred to as an infinite symbol sequence S_1 S_2 S_3 ⋯ , which satisfies that S_j ∈{ 1, 2, … , n }, ∀ j ∈ℤ and every i ∈{ 1, 2, … , n } occurs at least once in every v_i consecutive symbols S_j + 1 S_j + 2⋯ S_j + v_i, ∀ j ∈ℤ . If v has a pinwheel schedule then v is called (pinwheel) schedulable. The density of v is defined as d(v) = ∑ _i = 1^n1/v_i . Chan and Chin [4] made a conjecture that every vector v with d(v) ≤5/6 is schedulable. In this paper, we examine the conjecture from the point of view of low-dimensional vectors, including 3-, 4- and 5-dimensional ones. We first discover some simple but important properties of schedulable vectors, and then apply these properties to test whether or not a vector is schedulable. As a result, we prove that the maximum density guarantee for low-dimensional vectors is 5/6 , which partially support this conjecture.

Given a square matrix M = (u_ij)_n × n and an m-order matrix polynomial f_m(M) = ∑ _k=0^m a_k M^k = a_0 I + a_1M + a_2 M^2 + ⋯ + a_m M^m , if M is a dense matrix and is perturbed to become M' at a single entry, say u_pq , a straightforward re-calculation of f_m(M') would require O(n^ω·α (m)) arithmetic operations, where ω < 2.3728639 and α (m) depends on the strategy of computing M'^k, 1 ≤ k ≤ m appearing in f_m(M') , using the fastest square matrix multiplication algorithm by François Le Gall (ISSAC’14). In this paper, we assume that M is a dense matrix and that f_m(M) is known while no other additional information is available. From the perspective of the naive (a.k.a., standard row-by-column) matrix multiplication, we discuss the update of matrix polynomials. First, we present O(n)-, O(n^2) - and O(n^2) -operations update algorithms for 2-order, 3-order and 4-order matrix polynomials, respectively. Furthermore, we discuss the update of high-order matrix polynomials with a sparse coefficient vector and as a result, propose a combinatorial heuristic updating method based on directed Steiner tree in a directed acyclic graph.

This paper studies the asymmetric p-center problem (ApCP) and the vertex-weighted asymmetric p-center problem (WApCP) in complete digraphs (CD) satisfying the triangle inequality. First, we propose two classes of parameterized complete digraphs, α -CD and ⟨α , β⟩ -CD from the angle of the parameterized upper bound on the ratio of two asymmetric edge-weights between two different vertices and on the ratio of two vertex-weights, respectively. Using the greedy method, we design a (1 + α ) -approximation algorithm for the ApCP in α -CD’s and a (1 + αβ ) -approximation algorithm for the WApCP in ⟨α , β⟩ -CD’s, respectively.

Given an undirected connected graph G = (V, E, w) , where V is the set of n vertices, E is the set of m edges and each edge e ∈ E has a positive weight w(e) > 0 , a subset 𝒯⊆ V of p terminals and a subset ℰ⊆ E of candidate edges, the absolute 1-center problem (A1CP) asks for a point on some edge in ℰ to minimize the distance from it to 𝒯 . We prove that a vertex 1-center (V1C) is just an absolute 1-center (A1C) if the all-pairs shortest paths distance matrix from the vertices covered by the edges in ℰ to 𝒯 has a (global) saddle point. Furthermore, we define the local saddle point of an edge and conclude that the candidate edge having a local saddle point can be sifted. By combining the tool of sifting edges with the framework of Kariv and Hakimi’s algorithm, we design an O(m + p m^*+ n p log p) -time algorithm for A1CP, where m^* is the number of the remaining candidate edges. Applying our algorithm to the classic A1CP takes O(m + m^*n + n^2 log n) time when the distance matrix is known and O(m n + n^2 log n) time when the distance matrix is unknown, which are smaller than O(mn + n^2 log n) time and O(mn + n^3) time of Kariv and Hakimi’s algorithm, respectively.

Given an edge-weighted undirected graph G = (V, E, w) and a subset 𝒯⊆ V of p terminals, a k-Steiner forest spanning all the terminals in 𝒯 includes k branches, where every branch is a Steiner tree. The diameter of a k-Steiner forest is referred to as the maximum distance between two terminals of a branch. This paper studies the minimum diameter k-Steiner forest problem (MDkSFP) and establishes the relationship between MDkSFP and the absolute Steiner k-center problem (ASkCP). We first obtain a 2-approximation to ASkCP by a dual approximation algorithm and then achieve a 2-approximation to MDkSFP. Further, we achieve a (better) 2 ρ -approximation to MDkSFP, where ρ < 1 in general, by modifying the sites of centers and re-clustering all the terminals.

Given a vertex-weighted undirected connected graph \(G = (V, E, \ell , \rho )\), where each edge \(e \in E\) has a length \(\ell (e) > 0\) and each vertex \(v \in V\) has a weight \(\rho (v) > 0\), a subset \(T \subseteq V\) of vertices and a set S containing all the points on edges in a subset \(E' \subseteq E\) of edges, the generalized absolute 1-center problem (GA1CP), an extension of the classic vertex-weighted absolute 1-center problem (A1CP), asks to find a point from S such that the longest weighted shortest path distance in G from it to T is minimized. This paper presents a simple FPTAS for GA1CP by traversing the edges in \(E'\) using a positive real number as step size. The FPTAS takes \(O( |E| |V| + |V|^2 \log \log |V| + \frac{1}{\epsilon } |E'| |T| {\mathcal {R}})\) time, where \({\mathcal {R}}\) is an input parameter size of the problem instance, for any given \(\epsilon > 0\). For instances with a small input parameter size \({\mathcal {R}}\), applying the FPTAS with \(\epsilon = \Theta (1)\) to the classic vertex-weighted A1CP can produce a \((1 + \Theta (1))\)-approximation in at most O(|E| |V|) time when the distance matrix is known and \(O(|E| |V| + |V|^2 \log \log |V|)\) time when the distance matrix is unknown, which are smaller than Kariv and Hakimi’s \(O(|E| |V| \log |V|)\)-time algorithm and \(O(|E| |V| \log |V| + |V|^3)\)-time algorithm, respectively.

This paper studies the incremental single-source shortest paths (SSSP) problem in general digraphs with arbitrary positive arc weights. First, we examine several properties of single source shortest paths in general digraphs with arbitrary positive arc weights, and devise a nontrivial local search algorithm LSA to handle a single arc weight increase in such a digraph, which takes at most O(n.max{1, nlogn/m}) expected update time where n is the number of nodes and m is the number of arcs in the digraph. LSA also works on undirected graphs. Furthermore, this paper analyzes the expected update time of LSA dealing with edge weight increases or edge deletions in Erdos-Renyi (a.k.a., G(n, p)) random graphs. For weighted G(n, p) random graphs with arbitrary positive edge weights, LSA takes at most O(h(T-s)) expected update time to deal with a single edge weight increase as well as O(pn(2)h(T-s)) total update time, where h(Ts) is the height of input SSSP tree T-s. For G(n, p) random graphs, LSA takes 0 (Inn) expected update time to handle a single edge deletion as well as O(pn(2) Inn) total update time when 20lnn/n <= p < root 2Inn/n, and O(1) expected update time to handle a single edge deletion as well as 0 (pn2) total update time when p > root 2lnn/n. Specifically, LSA takes the least total update time of O (nInnh(T-s)) for weighted G(n, p) random graphs with p = c Inn/n, c > 1 as well as O (n(3/2) (Inn)(1/2)) for G(n, p) random graphs with p = c root lnn/n, c > root 2. (C) 2017 Elsevier B.V. All rights reserved.

This paper studies the restricted vertex 1-center problem (RV1CP) and restricted absolute 1-center problem (RA1CP) in general undirected graphs with each edge having two weights, cost and delay. First, we devise a simple FPTAS for RV1CP with O ( m n 3 ( 1 ϵ + log ⁡ log ⁡ n ) ) running time, based on FPTAS proposed by Lorenz and Raz (1999) [11] for computing end-to-end restricted shortest path (RSP). During the computation of the FPTAS for RV1CP, we derive a RSP distance matrix. Next, we discuss RA1CP in such graphs where the delay is a separable (e.g., linear) function of the cost on edge. We investigate an important property that the FPTAS for RV1CP can find a ( 1 + ϵ )-approximation of RA1CP when the RSP distance matrix has a saddle point . In addition, we show that it is harder to find an approximation of RA1CP when the matrix has no saddle point. This paper develops a scaling algorithm with at most O ( m n 3 K ( log ⁡ K η + log ⁡ log ⁡ n ) ) running time where K is a step-size parameter and η is a given positive number, to find a ( 1 + η )-approximation of RA1CP.

This paper studies the continuous connected 2-facility location problem (CC2FLP) in trees. Let T = (V, E, c, d, ℓ , μ ) be an undirected rooted tree, where each node v ∈ V has a weight d(v) ≥ 0 denoting the demand amount of v as well as a weight ℓ (v) ≥ 0 denoting the cost of opening a facility at v, and each edge e ∈ E has a weight c(e) ≥ 0 denoting the cost on e as well as is associated with a function μ (e,t) ≥ 0 denoting the cost of opening a facility at a point x(e, t) on e where t is a continuous variable on e. Given a subset 𝒟⊆ V of clients, and a subset ℱ⊆𝒫(T) of continuum points admitting facilities where 𝒫(T) is the set of all the points on edges of T, when two facilities are installed at a pair of continuum points x_1 and x_2 in ℱ , the total cost involved by CC2FLP includes three parts: the cost of opening two facilities at x_1 and x_2 , K times the cost of connecting x_1 and x_2 , and the cost of all the clients in 𝒟 connecting to some facility. The objective is to open two facilities at a pair of continuum points in ℱ to minimize the total cost, for a given input parameter K ≥ 1 . This paper considers the case of 𝒟 = V and ℱ = 𝒫(T) . We first study the discrete version of CC2FLP, named the discrete connected 2-facility location problem (DC2FLP), where two facilities are restricted to the nodes of T, and devise a quadratic time edge-splitting algorithm for DC2FLP. Furthermore, we prove CC2FLP is almost equivalent to DC2FLP in trees, and develop a quadratic time exact algorithm based on the edge-splitting algorithm.