Maximum coverage problem with group budget constraints

We study the maximum coverage problem with group budget constraints (MCG). The input consists of a ground set X, a collection \(\psi \) of subsets of X each of which is associated with a combinatorial structure such that for every set \(S_j\in \psi \), a cost \(c(S_j)\) can be calculated based on the combinatorial structure associated with \(S_j\), a partition \(G_1,G_2,\ldots ,G_l\) of \(\psi \), and budgets \(B_1,B_2,\ldots ,B_l\), and B. A solution to the problem consists of a subset H of \(\psi \) such that \(\sum _{S_j\in H} c(S_j) \le B\) and for each \(i \in {1,2,\ldots ,l}\), \(\sum _{S_j \in H\cap G_i}c(S_j)\le B_i\). The objective is to maximize \(|\bigcup _{S_j\in H}S_j|\). In our work we use a new and improved analysis of the greedy algorithm to prove that it is a \((\frac{\alpha }{3+2\alpha })\)-approximation algorithm, where \(\alpha \) is the approximation ratio of a given oracle which takes as an input a subset \(X^{new}\subseteq X\) and a group \(G_i\) and returns a set \(S_j\in G_i\) which approximates the optimal solution for \(\max _{D\in G_i}\frac{|D\cap X^{new}|}{c(D)}\). This analysis that is shown here to be tight for the greedy algorithm, improves by a factor larger than 2 the analysis of the best known approximation algorithm for MCG.     

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 \(\max _{1\le i\le k}\{e(V_{i})\}\le \frac{m}{k^{2}}+\frac{k-1}{2k^{2}}h(m)\).     

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 \(0 \le \alpha \le 1\).     

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.     

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 \(\rho _2(G)>\frac{1}{363}\).     

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.     

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 \(F\cong K_{k+1,k+2}\).     

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.     

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.     

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.     

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.

     

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 \( 1 \le \rho \le \displaystyle \frac{\sqrt{5} + 1}{2}\).     

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., \(n\rightarrow \infty \).     

On the performance of mildly greedy players in cut games

We continue the study of the performance of mildly greedy players in cut games initiated by Christodoulou et al. (Theoret Comput Sci 438:13–27, 2012), where a mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves by a factor of more than \(1+\epsilon \), for some given \(\epsilon \ge 0\). Hence, in presence of mildly greedy players, the classical concepts of pure Nash equilibria and best-responses generalize to those of \((1+\epsilon )\)-approximate pure Nash equilibria and \((1+\epsilon )\)-approximate best-responses, respectively. We first show that the \(\epsilon \)-approximate price of anarchy, that is the price of anarchy of \((1+\epsilon )\)-approximate pure Nash equilibria, is at least \(\frac{1}{2+\epsilon }\) and that this bound is tight for any \(\epsilon \ge 0\). Then, we evaluate the approximation ratio of the solutions achieved after a \((1+\epsilon )\)-approximate one-round walk starting from any initial strategy profile, where a \((1+\epsilon )\)-approximate one-round walk is a sequence of \((1+\epsilon )\)-approximate best-responses, one for each player. We improve the currently known lower bound on this ratio from \(\min \left\{ \frac{1}{4+2\epsilon },\frac{\epsilon }{4+2\epsilon }\right\} \) up to \(\min \left\{ \frac{1}{2+\epsilon },\frac{2\epsilon }{(1+\epsilon )(2+\epsilon )}\right\} \) and show that this is again tight for any \(\epsilon \ge 0\). An interesting and quite surprising consequence of our results is that the worst-case performance guarantee of the very simple solutions generated after a \((1+\epsilon )\)-approximate one-round walk is the same as that of \((1+\epsilon )\)-approximate pure Nash equilibria when \(\epsilon \ge 1\) and of that of subgame perfect equilibria (i.e., Nash equilibria for greedy players with farsighted, rather than myopic, rationality) when \(\epsilon =1\).     

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 \((1+\alpha )x^{2}+\alpha x=1+\alpha \).     

A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

Let \(G=(V,E)\) be a graph and \(\phi \) be a total \(k\)-coloring of \(G\) using the color set \(\{1,\ldots , k\}\). Let \(\sum _\phi (u)\) denote the sum of the color of the vertex \(u\) and the colors of all incident edges of \(u\). A \(k\)-neighbor sum distinguishing total coloring of \(G\) is a total \(k\)-coloring of \(G\) such that for each edge \(uv\in E(G)\), \(\sum _\phi (u)\ne \sum _\phi (v)\). By \(\chi ^{''}_{nsd}(G)\), we denote the smallest value \(k\) in such a coloring of \(G\). Pil?niak and Wo?niak first introduced this coloring and conjectured that \(\chi _{nsd}^{''}(G)\le \Delta (G)+3\) for any simple graph \(G\). In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with \(\Delta (G)\ge 7\). Moreover, we also show that \(\chi _{nsd}^{''}(G)\le \Delta (G)+2\) for planar graphs without intersecting triangles with \(\Delta (G) \ge 9\). Our approach is based on the Combinatorial Nullstellensatz and the discharging method.     

List 2-distance $$\varDelta +3$$-coloring of planar graphs without 4,5-cycles

Let \(\chi _2(G)\) and \(\chi _2^l(G)\) be the 2-distance chromatic number and list 2-distance chromatic number of a graph G, respectively. Wegner conjectured that for each planar graph G with maximum degree \(\varDelta \) at least 4, \(\chi _2(G)\le \varDelta +5\) if \(4\le \varDelta \le 7\), and \(\chi _2(G)\le \lfloor \frac{3\varDelta }{2}\rfloor +1\) if \(\varDelta \ge 8\). Let G be a planar graph without 4,5-cycles. We show that if \(\varDelta \ge 26\), then \(\chi _2^l(G)\le \varDelta +3\). There exist planar graphs G with girth \(g(G)=6\) such that \(\chi _2^l(G)=\varDelta +2\) for arbitrarily large \(\varDelta \). In addition, we also discuss the list L(2, 1)-labeling number of G, and prove that \(\lambda _l(G)\le \varDelta +8\) for \(\varDelta \ge 27\).     

On the difference of two generalized connectivities of a graph

The concept of k-connectivity \(\kappa '_{k}(G)\) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity \(\kappa _{k}(G)\), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if \(\kappa '_k(G)\ne n-k+1\) where \(k\ge 3\), then \(0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1\); otherwise, \(-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k\). Moreover, all of these bounds are sharp. Some specific study is focused for the case \(k=3\). As results, we characterize the graphs with \(\kappa '_3(G)=\kappa _3(G)=t\) for \(t\in \{1, n-3, n-2\}\), and give a necessary condition for \(\kappa '_3(G)=\kappa _3(G)\) by showing that for a connected graph G of order n and size m, if \(\kappa '_3(G)=\kappa _3(G)=t\) where \(1\le t\le n-3\), then \(m\le {n-2\atopwithdelims ()2}+2t\). Moreover, the unique extremal graph is given for the equality to hold.     

A primal–dual online algorithm for the <Emphasis Type="Italic">k</Emphasis>-server problem on weighted HSTs

In this paper, we show that there is a \(\frac{5}{2}\ell \cdot \ln (1+k)\)-competitive randomized algorithm for the k-sever problem on weighted Hierarchically Separated Trees (HSTs) with depth \(\ell \) when \(n=k+1\) where n is the number of points in the metric space, which improved previous best competitive ratio \(12 \ell \ln (1+4\ell (1+k))\) by Bansal et al. (FOCS, pp 267–276, 2011).     

On minimally 2-connected graphs with generalized connectivity $$\kappa _{3}=2$$

For \(S\subseteq G\), let \(\kappa (S)\) denote the maximum number r of edge-disjoint trees \(T_1, T_2, \ldots , T_r\) in G such that \(V(T_i)\cap V(T_j)=S\) for any \(i,j\in \{1,2,\ldots ,r\}\) and \(i\ne j\). For every \(2\le k\le n\), the k-connectivity of G, denoted by \(\kappa _k(G)\), is defined as \(\kappa _k(G)=\hbox {min}\{\kappa (S)| S\subseteq V(G)\ and\ |S|=k\}\). Clearly, \(\kappa _2(G)\) corresponds to the traditional connectivity of G. In this paper, we focus on the structure of minimally 2-connected graphs with \(\kappa _{3}=2\). Denote by \(\mathcal {H}\) the set of minimally 2-connected graphs with \(\kappa _{3}=2\). Let \(\mathcal {B}\subseteq \mathcal {H}\) and every graph in \(\mathcal {B}\) is either \(K_{2,3}\) or the graph obtained by subdividing each edge of a triangle-free 3-connected graph. We obtain that \(H\in \mathcal {H}\) if and only if \(H\in \mathcal {B}\) or H can be constructed from one or some graphs \(H_{1},\ldots ,H_{k}\) in \(\mathcal {B}\) (\(k\ge 1\)) by applying some operations recursively.