Improved floor-planning of graphs via adjacency-preserving transformations

Let G=(V,E) and G′=(V′,E′) be two graphs, an adjacency-preserving transformation from G to G′ is a one-to-many and onto mapping from V to V′ satisfying the following: (1) Each vertex v∈V in G is mapped to a non-empty subset \(\mathcal{A}(v)\subset V'\) in G′. The subgraph induced by \(\mathcal{A}(v)\) is a connected subgraph of G′; (2) if u≠v∈V, then \(\mathcal{A}(u)\cap\mathcal{A}(v)=\emptyset\); and (3) two vertices u and v are adjacent to each other in G if and only if subgraphs induced by \(\mathcal{A}(u)\) and \(\mathcal{A}(v)\) are connected in G′.In this paper, we study adjacency-preserving transformations from plane triangulations to irreducible triangulations (which are internally triangulated, with four exterior vertices and no separating triangles). As one shall see, our transformations not only preserve adjacency well, but also preserve the endowed realizers of plane triangulations well in the endowed transversal structures of the image irreducible triangulations, which may be desirable in some applications.We then present such an application in floor-planning of plane graphs. The expected grid size of the floor-plan of our linear time algorithm is improved to \((\frac{5n}{8}+O(1))\times (\frac{23n}{24}+O(1))\), though the worst case grid size bound of the algorithm remains \(\lfloor\frac{2n+1}{3}\rfloor\times(n-1)\), which is the same as the algorithm presented in Liao et al. (J. Algorithms 48:441–451, 2003).     

Two-stage proportionate flexible flow shop to minimize the makespan

We consider a two-stage flexible flow shop problem with a single machine at one stage and m identical machines at the other stage, where the processing times of each job at both stages are identical. The objective is to minimize the makespan. We describe some optimality conditions and show that the problem is NP-hard when m is fixed. Finally, we present an approximation algorithm that has a worst-case performance ratio of $\frac{5}{4}$ for m=2 and $\frac{\sqrt{1+m^{2}}+1+m}{2m}$ for m≥3.     

L(p,q)-labeling of sparse graphs

Let p and q be positive integers. An L(p,q)-labeling of a graph G with a span s is a labeling of its vertices by integers between 0 and s such that adjacent vertices of G are labeled using colors at least p apart, and vertices having a common neighbor are labeled using colors at least q apart. We denote by λ _p,q (G) the least integer k such that G has an L(p,q)-labeling with span k. The maximum average degree of a graph G, denoted by $\operatorname {Mad}(G)$ , is the maximum among the average degrees of its subgraphs (i.e. $\operatorname {Mad}(G) = \max\{\frac{2|E(H)|}{|V(H)|} ; H \subseteq G \}$ ). We consider graphs G with $\operatorname {Mad}(G) < \frac{10}{3}$ , 3 and $\frac{14}{5}$ . These sets of graphs contain planar graphs with girth 5, 6 and 7 respectively. We prove in this paper that every graph G with maximum average degree m and maximum degree Δ has:

λ _p,q (G)≤(2q?1)Δ+6p+10q?8 if $m < \frac{10}{3}$ and p≥2q.

λ _p,q (G)≤(2q?1)Δ+4p+14q?9 if $m < \frac{10}{3}$ and 2q>p.

λ _p,q (G)≤(2q?1)Δ+4p+6q?5 if m<3.

λ _p,q (G)≤(2q?1)Δ+4p+4q?4 if $m < \frac{14}{5}$ .

We give also some refined bounds for specific values of p, q, or Δ. By the way we improve results of Lih and Wang (SIAM J. Discrete Math. 17(2):264–275, 2003).     

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., \(n\rightarrow \infty \).     

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 \(F\cong K_{k+1,k+2}\).     

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 \(|E(F)| \ge \frac{4}{7} (|E(G)| + 1)+\frac{1}{7}|V_{2}(G)|\).     

Semi-online scheduling for jobs with release times

In this paper we consider three semi-online scheduling problems for jobs with release times on m identical parallel machines. The worst case performance ratios of the LS algorithm are analyzed. The objective function is to minimize the maximum completion time of all machines, i.e. the makespan. If the job list has a non-decreasing release times, then $2-\frac{1}{m}$ is the tight bound of the worst case performance ratio of the LS algorithm. If the job list has non-increasing processing times, we show that $2-\frac{1}{2m}$ is an upper bound of the worst case performance ratio of the LS algorithm. Furthermore if the job list has non-decreasing release times and the job list has non-increasing processing times we prove that the LS algorithm has worst case performance ratio not greater than $\frac{3}{2} -\frac{1}{2m}$ .     

Cost sharing and strategyproof mechanisms for set cover games

We develop for set cover games several general cost-sharing methods that are approximately budget-balanced, in the core, and/or group-strategyproof. We first study the cost sharing for a single set cover game, which does not have a budget-balanced mechanism in the core. We show that there is no cost allocation method that can always recover more than $\frac{1}{\ln n}$ of the total cost and in the core. Here n is the number of all players to be served. We give a cost allocation method that always recovers $\frac{1}{\ln d_{\mathit{max}}}$ of the total cost, where d _max is the maximum size of all sets. We then study the cost allocation scheme for all induced subgames. It is known that no cost sharing scheme can always recover more than $\frac{1}{n}$ of the total cost for every subset of players. We give an efficient cost sharing scheme that always recovers at least $\frac{1}{2n}$ of the total cost for every subset of players and furthermore, our scheme is cross-monotone. When the elements to be covered are selfish agents with privately known valuations, we present a strategyproof charging mechanism, under the assumption that all sets are simple sets; further, the total cost of the set cover is no more than ln?d _max times that of an optimal solution. When the sets are selfish agents with privately known costs, we present a strategyproof payment mechanism to them. We also show how to fairly share the payments to all sets among the elements.     

An upper bound on the total restrained domination number of a tree

Let G=(V,E) be a graph. A set of vertices S?V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of $V-\nobreak S$ is adjacent to a vertex in V?S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. A support vertex of a graph is a vertex of degree at least two which is adjacent to a leaf. We show that $\gamma_{\mathit{tr}}(T)\leq\lfloor\frac{n+2s+\ell-1}{2}\rfloor$ where T is a tree of order n≥3, and s and ? are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.     

A note on optimal pebbling of hypercubes

A pebbling move consists of removing two pebbles from one vertex and then placing one pebble at an adjacent vertex. If a distribution δ of pebbles lets us move at least one pebble to each vertex by applying pebbling moves repeatedly(if necessary), then δ is called a pebbling of G. The optimal pebbling number f′(G) of G is the minimum number of pebbles used in a pebbling of G. In this paper, we improve the known upper bound for the optimal pebbling number of the hypercubes Q _n . Mainly, we prove for large n, $f'(Q_{n})=O(n^{3/2}(\frac {4}{3})^{n})$ by a probabilistic argument.     

Adjacent vertex-distinguishing edge coloring of graphs with maximum degree Δ

An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let $\operatorname {mad}(G)$ and Δ(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every graph G with Δ(G)≥5 and $\operatorname{mad}(G) < 3-\frac {2}{\Delta}$ can be avd-colored with Δ(G)+1 colors. This completes a result of Wang and Wang (J. Comb. Optim. 19:471–485, 2010).     

(p,q)-total labeling of complete graphs

Given a graph G and positive integers p,q with p≥q, the (p,q)-total number $\lambda_{p,q}^{T}(G)$ of G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that the labels of any two adjacent vertices are at least q apart, the labels of any two adjacent edges are at least q apart, and the difference between the labels of a vertex and its incident edges is at least p. Havet and Yu (Discrete Math 308:496–513, 2008) first introduced this problem and determined the exact value of $\lambda_{p,1}^{T}(K_{n})$ except for even n with p+5≤n≤6p ²?10p+4. Their proof for showing that $\lambda _{p,1}^{T}(K_{n})\leq n+2p-3$ for odd n has some mistakes. In this paper, we prove that if n is odd, then $\lambda_{p}^{T}(K_{n})\leq n+2p-3$ if p=2, p=3, or $4\lfloor\frac{p}{2}\rfloor+3\leq n\leq4p-1$ . And we extend some results that were given in Havet and Yu (Discrete Math 308:496–513, 2008). Beside these, we give a lower bound for $\lambda_{p,q}^{T}(K_{n})$ under the condition that q<p<2q.     

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.     

Scheduling with interjob communication on parallel processors

Consider a scheduling problem in which a set of tasks needs to be scheduled on m parallel processors. Each task \(T_i\) consists of a set of jobs with interjob communication demands, represented by a weighted, undirected graph \(G_i\). The processors are assumed to be interconnected by a shared communication channel, which can be used by jobs to communicate among each other while being processed in parallel. In each time step, the scheduler assigns jobs to the processors and allows any processed job to use a certain capacity of the channel in order to satisfy (parts of) its communication demands to adjacent jobs processed in the same step. The goal is to find a schedule with minimum length in which the communication demands of all jobs are satisfied. We show that this problem is NP-hard in the strong sense even if the number of processors is constant and the underlying graph is a single path or a forest with arbitrary constant maximum degree. Consequently, we design and analyze approximation algorithms with asymptotic approximation ratio \(\min \{1.8, 1.5 \frac{m}{m-1}\}+1\) if the underlying graph G, the union of the \(G_i\), is a forest. For general graphs it is \(\min \left\{ 1.8, \frac{1.5m}{m-1}\right\} \cdot \left( \text {arb}(G) + \frac{5}{3}\right) \), where \(\text {arb}(G)\) denotes the arboricity of G.     

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.     

Minimum covering with travel cost

Given a polygon and a visibility range, the Myopic Watchman Problem with Discrete Vision (MWPDV) asks for a closed path P and a set of scan points $\mathcal{S}$ , such that (i) every point of the polygon is within visibility range of a scan point; and (ii) path length plus weighted sum of scan number along the tour is minimized. Alternatively, the bicriteria problem (ii??) aims at minimizing both scan number and tour length. We consider both lawn mowing (in which tour and scan points may leave?P) and milling (in which tour, scan points and visibility must stay within P) variants for the MWPDV; even for simple special cases, these problems are NP-hard. We show that this problem is NP-hard, even for the special cases of rectilinear polygons and L _?? scan range 1, and negligible small travel cost or negligible travel cost. For rectilinear MWPDV milling in grid polygons we present a 2.5-approximation with unit scan range; this holds for the bicriteria version, thus for any linear combination of travel cost and scan cost. For grid polygons and circular unit scan range, we describe a bicriteria 4-approximation. These results serve as stepping stones for the general case of circular scans with scan radius r and arbitrary polygons of feature size a, for which we extend the underlying ideas to a $\pi(\frac{r}{a}+\frac{r+1}{2})$ bicriteria approximation algorithm. Finally, we describe approximation schemes for MWPDV lawn mowing and milling of grid polygons, for fixed ratio between scan cost and travel cost.     

Minimum degree, edge-connectivity and radius

Let G be a connected graph on n≥4 vertices with minimum degree δ and radius r. Then $\delta r\leq4\lfloor\frac{n}{2}\rfloor-4$ , with equality if and only if one of the following holds:

1. G is K ₅,
2. G?K _n ?M, where M is a perfect matching, if n is even,
3. δ=n?3 and Δ≤n?2, if n is odd.

This solves a conjecture on the product of the edge-connectivity and radius of a graph, which was posed by Sedlar, Vuki?evi?, Aouchice, and Hansen.     

On the total outer-connected domination in graphs

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph induced by V?S is connected. The total outer-connected domination number γ _toc (G) is the minimum size of such a set. We give some properties and bounds for γ _toc in general graphs and in trees. For graphs of order n, diameter 2 and minimum degree at least 3, we show that $\gamma_{toc}(G)\le \frac{2n-2}{3}$ and we determine the extremal graphs.     

Objective functions with redundant domains

Let $(E,{ \mathcal{A}})$ be a set system consisting of a finite collection ${ \mathcal{A}}$ of subsets of a ground set E, and suppose that we have a function ? which maps ${ \mathcal{A}}$ into some set S. Now removing a subset K from E gives a restriction ${ \mathcal{A}}(\bar{K})$ to those sets of ${ \mathcal{A}}$ disjoint from K, and we have a corresponding restriction $\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{K})}$ of our function ?. If the removal of K does not affect the image set of ?, that is $\mbox {Im}(\phi|_{\hspace {.02in}{ \mathcal{A}}(\bar{X})})=\mbox {Im}(\phi)$ , then we will say that K is a kernel set of ${ \mathcal{A}}$ with respect to ?. Such sets are potentially useful in optimisation problems defined in terms of ?. We will call the set of all subsets of E that are kernel sets with respect to ? a kernel system and denote it by $\mathrm {Ker}_{\phi}({ \mathcal{A}})$ . Motivated by the optimisation theme, we ask which kernel systems are matroids. For instance, if ${ \mathcal{A}}$ is the collection of forests in a graph G with coloured edges and ? counts how many edges of each colour occurs in a forest then $\mathrm {Ker}_{\phi}({ \mathcal{A}})$ is isomorphic to the disjoint sum of the cocycle matroids of the differently coloured subgraphs; on the other hand, if ${ \mathcal{A}}$ is the power set of a set of positive integers, and ? is the function which takes the values 1 and 0 on subsets according to whether they are sum-free or not, then we show that $\mathrm {Ker}_{\phi}({ \mathcal{A}})$ is essentially never a matroid.     

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.