Upper bounds for the total rainbow connection of graphs

A two-agent scheduling problem on parallel machines is considered. Our objective is to minimize the makespan for agent A, subject to an upper bound on the makespan for agent B. When the number of machines, denoted by \(m\), is chosen arbitrarily, we provide an \(O(n)\) algorithm with performance ratio \(2-\frac{1}{m}\), i.e., the makespan for agent A given by the algorithm is no more than \(2-\frac{1}{m}\) times the optimal value, while the makespan for agent B is no more than \(2-\frac{1}{m}\) times the threshold value. This ratio is proved to be tight. Moreover, when \(m=2\), we present an \(O(nlogn)\) algorithm with performance ratio \(\frac{1+\sqrt{17}}{4}\approx 1.28\) which is smaller than \(\frac{3}{2}\). The ratio is weakly tight.     

Total completion time minimization in online hierarchical scheduling of unit-size jobs

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least \(1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}\), where \(x=\sqrt{2m+4}\). We then present a greedy algorithm with tight competitive ratio of \(1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}\). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when \(m=2\), we propose an optimal online algorithm with competitive ratio of \(16\) \(/\) \(13\), which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., \(n\rightarrow \infty \).     

Online tradeoff scheduling on a single machine to minimize makespan and maximum lateness

In this paper, we consider the following single machine online tradeoff scheduling problem. A set of n independent jobs arrive online over time. Each job \(J_{j}\) has a release date \(r_{j}\), a processing time \(p_{j}\) and a delivery time \(q_{j}\). The characteristics of a job are unknown until it arrives. The goal is to find a schedule that minimizes the makespan \(C_{\max } = \max _{1 \le j \le n} C_{j}\) and the maximum lateness \(L_{\max } = \max _{1 \le j \le n} L_{j}\), where \(L_{j} = C_{j} + q_{j}\). For the problem, we present a nondominated \(( \rho , 1 + \displaystyle \frac{1}{\rho } )\)-competitive online algorithm for each \(\rho \) with \( 1 \le \rho \le \displaystyle \frac{\sqrt{5} + 1}{2}\).     

On judicious partitions of graphs

Let \(k, m\) be positive integers, let \(G\) be a graph with \(m\) edges, and let \(h(m)=\sqrt{2m+\frac{1}{4}}-\frac{1}{2}\). Bollobás and Scott asked whether \(G\) admits a \(k\)-partition \(V_{1}, V_{2}, \ldots , V_{k}\) such that \(\max _{1\le i\le k} \{e(V_{i})\}\le \frac{m}{k^2}+\frac{k-1}{2k^2}h(m)\) and \(e(V_1, \ldots , V_k)\ge {k-1\over k} m +{k-1\over 2k}h(m) -\frac{(k-2)^{2}}{8k}\). In this paper, we present a positive answer to this problem on the graphs with large number of edges and small number of vertices with degrees being multiples of \(k\). Particularly, if \(d\) is not a multiple of \(k\) and \(G\) is \(d\)-regular with \(m\ge {9\over 128}k^4(k-2)^2\), then \(G\) admits a \(k\)-partition as desired. We also improve an earlier result by showing that \(G\) admits a partition \(V_{1}, V_{2}, \ldots , V_{k}\) such that \(e(V_{1},V_{2},\ldots ,V_{k})\ge \frac{k-1}{k}m+\frac{k-1}{2k}h(m)-\frac{(k-2)^{2}}{2(k-1)}\) and \(\max _{1\le i\le k}\{e(V_{i})\}\le \frac{m}{k^{2}}+\frac{k-1}{2k^{2}}h(m)\).     

Semi-online scheduling with known maximum job size on two uniform machines

In this paper, we investigate the semi-online scheduling problem with known maximum job size on two uniform machines with the speed ratio s≥1. The objective is to minimize the makespan. Two algorithms are presented, where the first is optimal for \(1\leq s\leq\sqrt{2}\), and the second is optimal for 1.559≤s≤2 and \(s\ge \frac{3+\sqrt{17}}{2}\). In addition, the improvement on lower bounds is made for \(2.     

Approximation for maximizing monotone non-decreasing set functions with a greedy method

We study the problem of maximizing a monotone non-decreasing function \(f\) subject to a matroid constraint. Fisher, Nemhauser and Wolsey have shown that, if \(f\) is submodular, the greedy algorithm will find a solution with value at least \(\frac{1}{2}\) of the optimal value under a general matroid constraint and at least \(1-\frac{1}{e}\) of the optimal value under a uniform matroid \((\mathcal {M} = (X,\mathcal {I})\), \(\mathcal {I} = \{ S \subseteq X: |S| \le k\}\)) constraint. In this paper, we show that the greedy algorithm can find a solution with value at least \(\frac{1}{1+\mu }\) of the optimum value for a general monotone non-decreasing function with a general matroid constraint, where \(\mu = \alpha \), if \(0 \le \alpha \le 1\); \(\mu = \frac{\alpha ^K(1-\alpha ^K)}{K(1-\alpha )}\) if \(\alpha > 1\); here \(\alpha \) is a constant representing the “elemental curvature” of \(f\), and \(K\) is the cardinality of the largest maximal independent sets. We also show that the greedy algorithm can achieve a \(1 - (\frac{\alpha + \cdots + \alpha ^{k-1}}{1+\alpha + \cdots + \alpha ^{k-1}})^k\) approximation under a uniform matroid constraint. Under this unified \(\alpha \)-classification, submodular functions arise as the special case \(0 \le \alpha \le 1\).     

Paired-domination number of claw-free odd-regular graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph, denoted by \(\gamma _{pr}(G)\). Let G be a connected \(\{K_{1,3}, K_{4}-e\}\)-free cubic graph of order n. We show that \(\gamma _{pr}(G)\le \frac{10n+6}{27}\) if G is \(C_{4}\)-free and that \(\gamma _{pr}(G)\le \frac{n}{3}+\frac{n+6}{9(\lceil \frac{3}{4}(g_o+1)\rceil +1)}\) if G is \(\{C_{4}, C_{6}, C_{10}, \ldots , C_{2g_o}\}\)-free for an odd integer \(g_o\ge 3\); the extremal graphs are characterized; we also show that if G is a 2 -connected, \(\gamma _{pr}(G) = \frac{n}{3} \). Furthermore, if G is a connected \((2k+1)\)-regular \(\{K_{1,3}, K_4-e\}\)-free graph of order n, then \(\gamma _{pr}(G)\le \frac{n}{k+1} \), with equality if and only if \(G=L(F)\), where \(F\cong K_{1, 2k+2}\), or k is even and \(F\cong K_{k+1,k+2}\).     

Nordhaus–Gaddum type result for the matching number of a graph

For a graph G, \(\alpha '(G)\) is the matching number of G. Let \(k\ge 2\) be an integer, \(K_{n}\) be the complete graph of order n. Assume that \(G_{1}, G_{2}, \ldots , G_{k}\) is a k-decomposition of \(K_{n}\). In this paper, we show that (1)

$$\begin{aligned} \left\lfloor \frac{n}{2}\right\rfloor \le \sum _{i=1}^{k} \alpha '(G_{i})\le k\left\lfloor \frac{n}{2}\right\rfloor . \end{aligned}$$

(2) If each \(G_{i}\) is non-empty for \(i = 1, \ldots , k\), then for \(n\ge 6k\),

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n+k-1}{2}\right\rfloor . \end{aligned}$$

(3) If \(G_{i}\) has no isolated vertices for \(i = 1, \ldots , k\), then for \(n\ge 8k\),

$$\begin{aligned} \sum _{i=1}^{k} \alpha '(G_{i})\ge \left\lfloor \frac{n}{2}\right\rfloor +k. \end{aligned}$$

The bounds in (1), (2) and (3) are sharp. (4) When \(k= 2\), we characterize all the extremal graphs which attain the lower bounds in (1), (2) and (3), respectively.

     

Further results on the reciprocal degree distance of graphs

The reciprocal degree distance of a simple connected graph \(G=(V_G, E_G)\) is defined as \(\bar{R}(G)=\sum _{u,v \in V_G}(\delta _G(u)+\delta _G(v))\frac{1}{d_G(u,v)}\), where \(\delta _G(u)\) is the vertex degree of \(u\), and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). The reciprocal degree distance is an additive weight version of the Harary index, which is defined as \(H(G)=\sum _{u,v \in V_G}\frac{1}{d_G(u,v)}\). In this paper, the extremal \(\bar{R}\)-values on several types of important graphs are considered. The graph with the maximum \(\bar{R}\)-value among all the simple connected graphs of diameter \(d\) is determined. Among the connected bipartite graphs of order \(n\), the graph with a given matching number (resp. vertex connectivity) having the maximum \(\bar{R}\)-value is characterized. Finally, sharp upper bounds on \(\bar{R}\)-value among all simple connected outerplanar (resp. planar) graphs are determined.     

Semi-online scheduling with combined information on two identical machines in parallel

This paper is concerned with a semi-online scheduling problem with combined information on two identical parallel machines to minimize the makespan, where all the jobs have processing times in the interval \([1,\,t]\)  \((t\ge 1)\) and the jobs arrive in non-increasing order of their processing times. The objective is to minimize the makespan. For \(t\ge 1\), we obtain a lower bound \(\max _{N=1,2,3,\ldots }\left\{ \min \{\frac{4N+3}{4N+2}\,,\frac{Nt+N+1}{2N+1}\}\right\} \) and show that the competitive ratio of the \(LS\) algorithm achieves the lower bound.     

The 2-surviving rate of planar graphs without 5-cycles

Let \(G\) be a connected graph with \(n\ge 2\) vertices. Let \(k\ge 1\) be an integer. Suppose that a fire breaks out at a vertex \(v\) of \(G\). A firefighter starts to protect vertices. At each step, the firefighter protects \(k\)-vertices not yet on fire. At the end of each step, the fire spreads to all the unprotected vertices that have a neighbour on fire. Let \(\hbox {sn}_k(v)\) denote the maximum number of vertices in \(G\) that the firefighter can save when a fire breaks out at vertex \(v\). The \(k\)-surviving rate \(\rho _k(G)\) of \(G\) is defined to be \(\frac{1}{n^2}\sum _{v\in V(G)} {\hbox {sn}}_{k}(v)\), which is the average proportion of saved vertices. In this paper, we prove that if \(G\) is a planar graph with \(n\ge 2\) vertices and without 5-cycles, then \(\rho _2(G)>\frac{1}{363}\).     

Improved upper bound for the degenerate and star chromatic numbers of graphs

Let \(G=G(V,E)\) be a graph. A proper coloring of G is a function \(f:V\rightarrow N\) such that \(f(x)\ne f(y)\) for every edge \(xy\in E\). A proper coloring of a graph G such that for every \(k\ge 1\), the union of any k color classes induces a \((k-1)\)-degenerate subgraph is called a degenerate coloring; a proper coloring of a graph with no two-colored \(P_{4}\) is called a star coloring. If a coloring is both degenerate and star, then we call it a degenerate star coloring of graph. The corresponding chromatic number is denoted as \(\chi _{sd}(G)\). In this paper, we employ entropy compression method to obtain a new upper bound \(\chi _{sd}(G)\le \lceil \frac{19}{6}\Delta ^{\frac{3}{2}}+5\Delta \rceil \) for general graph G.     

An upper bound on the double Roman domination number

A double Roman dominating function (DRDF) on a graph \(G=(V,E)\) is a function \(f : V \rightarrow \{0, 1, 2, 3\}\) having the property that if \(f(v) = 0\), then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with \(f(w)=3\), and if \(f(v)=1\), then vertex v must have at least one neighbor w with \(f(w)\ge 2\). The weight of a DRDF f is the value \(f(V) = \sum _{u \in V}f(u)\). The double Roman domination number \(\gamma _{dR}(G)\) of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23–29, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality \(\gamma _{dR}(G)\le \frac{6n}{5}\) and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, \(\gamma _{dR}(G)\le \frac{8n}{7}\).     

Online MapReduce scheduling problem of minimizing the makespan

MapReduce system is a popular big data processing framework, and the performance of it is closely related to the efficiency of the centralized scheduler. In practice, the centralized scheduler often has little information in advance, which means each job may be known only after being released. In this paper, hence, we consider the online MapReduce scheduling problem of minimizing the makespan, where jobs are released over time. Both preemptive and non-preemptive version of the problem are considered. In addition, we assume that reduce tasks cannot be parallelized because they are often complex and hard to be decomposed. For the non-preemptive version, we prove the lower bound is \(\frac{m+m(\Psi (m)-\Psi (k))}{k+m(\Psi (m)-\Psi (k))}\), higher than the basic online machine scheduling problem, where k is the root of the equation \(k=\big \lfloor {\frac{m-k}{1+\Psi (m)-\Psi (k)}+1 }\big \rfloor \) and m is the quantity of machines. Then we devise an \((2-\frac{1}{m})\)-competitive online algorithm called MF-LPT (Map First-Longest Processing Time) based on the LPT. For the preemptive version, we present a 1-competitive algorithm for two machines.     

Semi-online scheduling on two uniform machines with the known largest size

This paper investigates semi-online scheduling on two uniform machines with the known largest size. Denote by s _j the speed of each machine, j=1,2. Assume 0<s ₁≤s ₂, and let s=s ₂/s ₁ be the speed ratio. First, for the speed ratio \(s\in [1,\sqrt{2}]\), we present an optimal semi-online algorithm \(\mathcal{LSMP}\) with the competitive ratio \(\mathrm{max}\{\frac {2(s+1)}{2s+1},s\}\). Second, we present a semi-online algorithm \(\mathcal{HSMP}\). And for \(s\in(\sqrt{2},1+\sqrt{3})\), the competitive ratio of \(\mathcal{HSMP}\) is strictly smaller than that of the online algorithm \(\mathcal{LS}\). Finally, for the speed ratio s≥s ^*≈3.715, we show that the known largest size cannot help us to design a semi-online algorithm with the competitive ratio strictly smaller than that of \(\mathcal{LS}\). Moreover, we show a lower bound for \(s\in(\sqrt{2},s^{*})\).     

An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities

We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \).     

Strong minimum energy hierarchical topology in wireless sensor networks

Given a set of \(n\) sensors, the strong minimum energy topology (SMET) problem in a wireless sensor network is to assign transmit powers to all sensors such that (i) the graph induced only using the bi-directional links is connected, that is, there is a path between every pair of sensors, and (ii) the sum of the transmit powers of all the sensors is minimum. This problem is known to be NP-hard. In this paper, we study a special case of the SMET problem, namely , the \(k\)-strong minimum energy hierarchical topology (\(k\)-SMEHT) problem. Given a set of \(n\) sensors and an integer \(k\), the \(k\)-SMEHT problem is to assign transmission powers to all sensors such that (i) the graph induced using only bi-directional links is connected, (ii) at most \(k\) nodes of the graph induced using only bi-directional links have two or more neighbors, that is they are non-pendant nodes, and (iii) the sum of the transmit powers of all the sensors in \(G\) is minimum. We show that \(k\)-SMEHT problem is NP-hard for arbitrary \(k\). However, we propose a \(\frac{k+1}{2}\)-approximation algorithm for \(k\)-SMEHT problem, when \(k\) is a fixed constant. Finally, we propose a polynomial time algorithm for the \(k\)-SMEHT problem for \(k=2\).     

Even factors of graphs

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Favaron and Kouider (J Gr Theory 77:58–67, 2014) showed that if a simple graph G has an even factor, then it has an even factor F with \(|E(F)| \ge \frac{7}{16} (|E(G)| + 1)\). This ratio was improved to \(\frac{4}{7}\) recently by  Chen and Fan (J Comb Theory Ser B 119:237–244, 2016), which is the best possible. In this paper, we take the set of vertices of degree 2 (say \(V_{2}(G)\)) into consideration and further strengthen this lower bound. Our main result is to show that for any simple graph G having an even factor, G has an even factor F with \(|E(F)| \ge \frac{4}{7} (|E(G)| + 1)+\frac{1}{7}|V_{2}(G)|\).     

Online covering salesman problem

Given a graph \(G=(V,E,D,W)\), the generalized covering salesman problem (CSP) is to find a shortest tour in G such that each vertex \(i\in D\) is either on the tour or within a predetermined distance L to an arbitrary vertex \(j\in W\) on the tour, where \(D\subset V\),\(W\subset V\). In this paper, we propose the online CSP, where the salesman will encounter at most k blocked edges during the traversal. The edge blockages are real-time, meaning that the salesman knows about a blocked edge when it occurs. We present a lower bound \(\frac{1}{1 + (k + 2)L}k+1\) and a CoverTreeTraversal algorithm for online CSP which is proved to be \(k+\alpha \)-competitive, where \(\alpha =0.5+\frac{(4k+2)L}{OPT}+2\gamma \rho \), \(\gamma \) is the approximation ratio for Steiner tree problem and \(\rho \) is the maximal number of locations that a customer can be served. When \(\frac{L}{\texttt {OPT}}\rightarrow 0\), our algorithm is near optimal. The problem is also extended to the version with service cost, and similar results are derived.     

On Barnette’s conjecture and the $$H^{+-}$$ property

Let \(\mathcal{B}\) denote the class of all 3-connected cubic bipartite plane graphs. A conjecture of Barnette states that every graph in \(\mathcal{B}\) has a Hamilton cycle. A cyclic sequence of big faces is a cyclic sequence of different faces, each bounded by at least six edges, such that two faces from the sequence are adjacent if and only if they are consecutive in the sequence. Suppose that \({F_1, F_2, F_3}\) is a proper 3-coloring of the faces of \(G^*\in \mathcal{B}\). We prove that if every cyclic sequence of big faces of \(G^*\) has a face belonging to \(F_1\) and a face belonging to \(F_2\), then \(G^*\) has the following properties: \(H^{+-}\): If any two edges are chosen on the same face, then there is a Hamilton cycle through one and avoiding the other, \(H^{--}\): If any two edges are chosen which are an even distance apart on the same face, then there is a Hamilton cycle that avoids both. Moreover, let \(X\) and \(Y\) partition the set of big faces of \(G^*\) such that all such faces of \(F_1\) are in \(X\) and all such faces of \(F_2\) are in \(Y\). We prove that if every cyclic sequence of big faces has a face belonging to \(X\) and a face belonging to \(Y\), then \(G^*\) has a Hamilton cycle.