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.     

Optimal strategies for the one-round discrete Voronoi game on a line

The one-round discrete Voronoi game, with respect to a n-point user set  $\mathcal {U}$ , consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set $\mathcal{F}_{1}$ of m facilities following which P2 chooses another set $\mathcal{F}_{2}$ of m facilities, disjoint from  $\mathcal{F}_{1}$ , where m(=O(1)) is a positive constant. The payoff of P2 is defined as the cardinality of the set of points in $\mathcal{U}$ which are closer to a facility in $\mathcal{F}_{2}$ than to every facility in $\mathcal{F}_{1}$ , and the payoff of P1 is the difference between the number of users in $\mathcal{U}$ and the payoff of P2. The objective of both the players in the game is to maximize their respective payoffs. In this paper, we address the case where the points in $\mathcal{U}$ are located along a line. We show that if the sorted order of the points in $\mathcal{U}$ along the line is known, then the optimal strategy of P2, given any placement of facilities by P1, can be computed in O(n) time. We then prove that for m≥2 the optimal strategy of P1 in the one-round discrete Voronoi game, with the users on a line, can be computed in $O(n^{m-\lambda_{m}})$ time, where 0<λ _m <1, is a constant depending only on m.     

Popularity at minimum cost

We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph $G = (\mathcal{A}\cup\mathcal{B},E)$ , where $\mathcal{A}$ is a set of people, $\mathcal{B}$ is a set of items, and each person $a \in\mathcal{A}$ ranks a subset of items in order of preference, with ties allowed. The popular matching problem seeks to compute a matching M ^? between people and items such that there is no matching M where more people are happier with M than with M ^?. Such a matching M ^? is called a popular matching. However, there are simple instances where no popular matching exists. Here we consider the following natural extension to the above problem: associated with each item $b \in\mathcal{B}$ is a non-negative price cost(b), that is, for any item b, new copies of b can be added to the input graph by paying an amount of cost(b) per copy. When G does not admit a popular matching, the problem is to “augment” G at minimum cost such that the new graph admits a popular matching. We show that this problem is NP-hard; in fact, it is NP-hard to approximate it within a factor of $\sqrt{n_{1}}/2$ , where n ₁ is the number of people. This problem has a simple polynomial time algorithm when each person has a preference list of length at most 2. However, if we consider the problem of constructing a graph at minimum cost that admits a popular matching that matches all people, then even with preference lists of length 2, the problem becomes NP-hard. On the other hand, when the number of copies of each item is fixed, we show that the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn ₁) time, where m is the number of edges.     

Separating NE from some nonuniform nondeterministic complexity classes

We investigate the question whether NE can be separated from the reduction closures of tally sets, sparse sets and NP. We show that (1) \(\mathrm{NE}\not\subseteq R^{\mathrm{NP}}_{n^{o(1)}-T}(\mathrm{TALLY})\); (2) \(\mathrm{NE}\not\subseteq R^{SN}_{m}(\mathrm{SPARSE})\); (3) \(\mathrm{NEXP}\not\subseteq \mathrm{P}^{\mathrm{NP}}_{n^{k}-T}/n^{k}\) for all k≥1; and (4) \(\mathrm{NE}\not\subseteq \mathrm{P}_{btt}(\mathrm{NP}\oplus\mathrm{SPARSE})\). Result (3) extends a previous result by Mocas to nonuniform reductions. We also investigate how different an NE-hard set is from an NP-set. We show that for any NP subset A of a many-one-hard set H for NE, there exists another NP subset A′ of H such that A′? A and A′?A is not of sub-exponential density.     

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

Minimum diameter cost-constrained Steiner trees

Given an edge-weighted undirected graph $G=(V,E,c,w)$ where each edge $e\in E$ has a cost $c(e)\ge 0$ and another weight $w(e)\ge 0$ , a set $S\subseteq V$ of terminals and a given constant $\mathrm{C}_0\ge 0$ , the aim is to find a minimum diameter Steiner tree whose all terminals appear as leaves and the cost of tree is bounded by $\mathrm{C}_0$ . The diameter of a tree refers to the maximum weight of the path connecting two different leaves in the tree. This problem is called the minimum diameter cost-constrained Steiner tree problem, which is NP-hard even when the topology of the Steiner tree is fixed. In this paper, we deal with the fixed-topology restricted version. We prove the restricted version to be polynomially solvable when the topology is not part of the input and propose a weakly fully polynomial time approximation scheme (weakly FPTAS) when the topology is part of the input, which can find a $(1+\epsilon )$ –approximation of the restricted version problem for any $\epsilon >0$ with a specific characteristic.     

On the approximability of positive influence dominating set in social networks

In social networks, there is a tendency for connected users to match each other’s behaviors. Moreover, a user likely adopts a behavior, if a certain fraction of his family and friends follows that behavior. Identifying people who have the most influential effect to the others is of great advantages, especially in politics, marketing, behavior correction, and so on. Under a graph-theoretical framework, we study the positive influence dominating set (PIDS) problem that seeks for a minimal set of nodes $\mathcal{P}$ such that all other nodes in the network have at least a fraction ρ>0 of their neighbors in $\mathcal{P}$ . We also study a different formulation, called total positive influence dominating set (TPIDS), in which even nodes in $\mathcal{P}$ are required to have a fraction ρ of neighbors inside $\mathcal{P}$ . We show that neither of these problems can be approximated within a factor of (1??)lnmax{Δ,|V|^1/2}, where Δ is the maximum degree. Moreover, we provide a simple proof that both problems can be approximated within a factor lnΔ+O(1). In power-law networks, where the degree sequence follows a power-law distribution, both problems admit constant factor approximation algorithms. Finally, we present a linear-time exact algorithms for trees.     

The partition method for poset-free families

Given a finite poset P, let ${\rm La}(n,P)$ denote the largest size of a family of subsets of an n-set that does not contain P as a (weak) subposet. We employ a combinatorial method, using partitions of the collection of all full chains of subsets of the n-set, to give simpler new proofs of the known asymptotic behavior of ${\rm La}(n,P)$ , as n→∞, when P is the r-fork $\mathcal {V}_{r}$ , the four-element N poset $\mathcal {N}$ , and the four-element butterfly-poset $\mathcal {B}$ .     

Radiation hybrid map construction problem parameterized

In this paper, we study the Radiation hybrid map construction ( $\mathsf{{RHMC} }$ ) problem which is about reconstructing a genome from a set of gene clusters. The problem is known to be $\mathsf{{NP} }$ -complete even when all gene clusters are of size two and the corresponding problem ( $\mathsf{{RHMC}_2 }$ ) admits efficient constant-factor approximation algorithms. In this paper, for the first time, we consider the more general case when the gene clusters can have size either two or three ( $\mathsf{{RHMC}_3 }$ ). Let ${p\text{- }\mathsf {RHMC} }$ be a parameterized version of $\mathsf{{RHMC} }$ where the parameter is the size of solution. We present a linear kernel for ${p\text{- }\mathsf {RHMC}_3 }$ of size $22k$ that when combined with a bounded search-tree algorithm, gives an FPT algorithm running in $O(6^kk+n)$ time. For ${p\text{- }\mathsf {RHMC}_3 }$ we present a bounded search tree algorithm which runs in $O^*(2.45^k)$ time, greatly improving the previous bound using weak kernels.     

On the odd girth and the circular chromatic number of generalized Petersen graphs

A class \(\mathcal{G}\) of simple graphs is said to be girth-closed (odd-girth-closed) if for any positive integer g there exists a graph \(\mathrm {G} \in \mathcal{G}\) such that the girth (odd-girth) of \(\mathrm {G}\) is \(\ge g\). A girth-closed (odd-girth-closed) class \(\mathcal{G}\) of graphs is said to be pentagonal (odd-pentagonal) if there exists a positive integer \(g^*\) depending on \(\mathcal{G}\) such that any graph \(\mathrm {G} \in \mathcal{G}\) whose girth (odd-girth) is greater than \(g^*\) admits a homomorphism to the five cycle (i.e. is \(\mathrm {C}_{_{5}}\)-colourable). Although, the question “Is the class of simple 3-regular graphs pentagonal?” proposed by Ne?et?il (Taiwan J Math 3:381–423, 1999) is still a central open problem, Gebleh (Theorems and computations in circular colourings of graphs, 2007) has shown that there exists an odd-girth-closed subclass of simple 3-regular graphs which is not odd-pentagonal. In this article, motivated by the conjecture that the class of generalized Petersen graphs is odd-pentagonal, we show that finding the odd girth of generalized Petersen graphs can be transformed to an integer programming problem, and using the combinatorial and number theoretic properties of this problem, we explicitly compute the odd girth of such graphs, showing that the class is odd-girth-closed. Also, we obtain upper and lower bounds for the circular chromatic number of these graphs, and as a consequence, we show that the subclass containing generalized Petersen graphs \(\mathrm {Pet}(n,k)\) for which either k is even, n is odd and \(n\mathop {\equiv }\limits ^{k-1}\pm 2\) or both n and k are odd and \(n\ge 5k\) is odd-pentagonal. This in particular shows the existence of nontrivial odd-pentagonal subclasses of 3-regular simple graphs.     

Signed Roman domination in graphs

In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G=(V,E) is a function f:V→{?1,1,2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f(u)=?1 is adjacent to at least one vertex v for which f(v)=2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that $\gamma _{\mathrm{sR}}(G) \ge\frac{3}{\sqrt{2}} \sqrt{n} - n$ and that γ _sR(G)≥(3n?4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that $\gamma_{\mathrm{sR}}(G) \ge3\sqrt{n+1} - n - 3$ , and we characterize the extremal graphs.     

Bounding the total domination subdivision number of a graph in terms of its order

The total domination subdivision number \(\mathrm{sd}_{\gamma _{t}}(G)\) of a graph G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that \(\mathrm{sd}_{\gamma_{t}}(G)\leq \lfloor\frac{2n}{3}\rfloor\) for any simple connected graph G of order n≥3 other than K ₄. We also determine all simple connected graphs G with \(\mathrm{sd}_{\gamma_{t}}(G)=\lfloor\frac{2n}{3}\rfloor\).     

The total domination subdivision number in graphs with no induced 3-cycle and 5-cycle

A set S of vertices of a graph G=(V,E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $\mathrm{sd}_{\gamma_{t}}(G)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Favaron, Karami, Khoeilar and Sheikholeslami (J. Comb. Optim. 20:76–84, 2010a) conjectured that: For any connected graph G of order n≥3, $\mathrm{sd}_{\gamma_{t}}(G)\le \gamma_{t}(G)+1$ . In this paper we use matching to prove this conjecture for graphs with no 3-cycle and 5-cycle. In particular this proves the conjecture for bipartite graphs.     

The hamiltonian numbers in digraphs

In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digraph D is a closed spanning directed walk with minimum length in D. The length of a hamiltonian walk of a digraph D is called the hamiltonian number of D, denoted h(D). We prove that if a digraph D of order n is strongly connected, then $n\leq h(D)\leq\lfloor\frac{(n+1)^{2}}{4} \rfloor$ , and hence characterize the strongly connected digraphs of order n with hamiltonian number $\lfloor\frac{(n+1)^{2}}{4} \rfloor$ . In addition, we show that for each k with $4\leq n\leq k\leq\lfloor \frac{(n+1)^{2}}{4} \rfloor$ , there exists a digraph with order n and hamiltonian number k. Furthermore, we also study the hamiltonian spectra of graphs.     

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.     

The minimum value of geometric-arithmetic index of graphs with minimum degree 2

The geometric-arithmetic index was introduced in the chemical graph theory and it has shown to be applicable. The aim of this paper is to obtain the extremal graphs with respect to the geometric-arithmetic index among all graphs with minimum degree 2. Let G(2, n) be the set of connected simple graphs on n vertices with minimum degree 2. We use linear programming formulation and prove that the minimum value of the first geometric-arithmetic \((GA_{1})\) index of G(2, n) is obtained by the following formula:

$$\begin{aligned} GA_1^* = \left\{ \begin{array}{ll} n&{}\quad n \le 24, \\ \mathrm{{24}}\mathrm{{.79}}&{}\quad n = 25, \\ \frac{{4\left( {n - 2} \right) \sqrt{2\left( {n - 2} \right) } }}{n}&{}\quad n \ge 26. \\ \end{array} \right. \end{aligned}$$

     

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 $\frac{3}{2} -\frac{1}{2m}$ .     

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).     

A continuous characterization of the maximum vertex-weighted clique in hypergraphs

For a simple graph G on n vertices with adjacency matrix A, Motzkin and Strauss established a remarkable connection between the clique number and the global maximum value of the quadratic programm: \(\textit{max}\{ \mathbf {x}^T A \mathbf {x}\}\) on the standard simplex: \(\{\sum _{i=1}^{n} x_i =1, x_i \ge 0 \}\). In Gibbons et al. (Math Oper Res 122:754–768, 1997), an extension of the Motzkin–Straus formulation was provided for the vertex-weighted clique number of a graph. In this paper, we provide a continuous characterization of the maximum vertex-weighted clique problem for vertex-weighted uniform hypergraphs.