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.     

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.     

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 $\frac{3}{2} -\frac{1}{2m}$ .     

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 $\mathcal {B}$ .     

Spanning 3-connected index of graphs

For an integer $s>0$ and for $u,v\in V(G)$ with $u\ne v$ , an $(s;u,v)$ -path-system of G is a subgraph H of G consisting of s internally disjoint (u, v)-paths, and such an H is called a spanning $(s;u,v)$ -path system if $V(H)=V(G)$ . The spanning connectivity $\kappa ^{*}(G)$ of graph G is the largest integer s such that for any integer k with $1\le k \le s$ and for any $u,v\in V(G)$ with $u\ne v$ , G has a spanning ( $k;u,v$ )-path-system. Let G be a simple connected graph that is not a path, a cycle or a $K_{1,3}$ . The spanning k-connected index of G, written $s_{k}(G)$ , is the smallest nonnegative integer m such that $L^m(G)$ is spanning k-connected. Let $l(G)=\max \{m:\,G$ has a divalent path of length m that is not both of length 2 and in a $K_{3}$ }, where a divalent path in G is a path whose interval vertices have degree two in G. In this paper, we prove that $s_{3}(G)\le l(G)+6$ . The key proof to this result is that every connected 3-triangular graph is 2-collapsible.     

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.     

Distance-d independent set problems for bipartite and chordal graphs

The paper studies a generalization of the Independent Set problem (IS for short). A distance- $d$ independent set for an integer $d\ge 2$ in an unweighted graph $G = (V, E)$ is a subset $S\subseteq V$ of vertices such that for any pair of vertices $u, v \in S$ , the distance between $u$ and $v$ is at least $d$ in $G$ . Given an unweighted graph $G$ and a positive integer $k$ , the Distance- $d$ Independent Set problem (D $d$ IS for short) is to decide whether $G$ contains a distance- $d$ independent set $S$ such that $|S| \ge k$ . D2IS is identical to the original IS. Thus D2IS is $\mathcal{NP}$ -complete even for planar graphs, but it is in $\mathcal{P}$ for bipartite graphs and chordal graphs. In this paper we investigate the computational complexity of D $d$ IS, its maximization version MaxD $d$ IS, and its parameterized version ParaD $d$ IS( $k$ ), where the parameter is the size of the distance- $d$ independent set: (1) We first prove that for any $\varepsilon >0$ and any fixed integer $d\ge 3$ , it is $\mathcal{NP}$ -hard to approximate MaxD $d$ IS to within a factor of $n^{1/2-\varepsilon }$ for bipartite graphs of $n$ vertices, and for any fixed integer $d\ge 3$ , ParaD $d$ IS( $k$ ) is $\mathcal{W}[1]$ -hard for bipartite graphs. Then, (2) we prove that for every fixed integer $d\ge 3$ , D $d$ IS remains $\mathcal{NP}$ -complete even for planar bipartite graphs of maximum degree three. Furthermore, (3) we show that if the input graph is restricted to chordal graphs, then D $d$ IS can be solved in polynomial time for any even $d\ge 2$ , whereas D $d$ IS is $\mathcal{NP}$ -complete for any odd $d\ge 3$ . Also, we show the hardness of approximation of MaxD $d$ IS and the $\mathcal{W}[1]$ -hardness of ParaD $d$ IS( $k$ ) on chordal graphs for any odd $d\ge 3$ .     

(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.     

The adjacent vertex distinguishing total coloring of planar graphs

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ≥14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.     

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.     

Online LPT algorithms for parallel machines scheduling with a single server

We consider two parallel machines scheduling problems with a single server. For the general case we present an online LPT algorithm with competitive ratio 2, and give a lower bound $\frac{\sqrt{5} + 1}{2}$ . We also apply the online LPT algorithm to the special case where all the setup times are equal to 1. We show that the competitive ratio is 1.5, and no online algorithm can has a competitive ratio less than  $\sqrt{2}$ .     

Online maximum directed cut

We investigate a natural online version of the well-known Maximum Directed Cut problem on DAGs. We propose a deterministic algorithm and show that it achieves a competitive ratio of $\frac{3\sqrt{3}}{2}\approx 2.5981$ . We then give a lower bound argument to show that no deterministic algorithm can achieve a ratio of $\frac{3\sqrt{3}}{2}-\epsilon$ for any ??>0 thus showing that our algorithm is essentially optimal. Then, we extend our technique to improve upon the analysis of an old result: we show that greedily derandomizing the trivial randomized algorithm for MaxDiCut in general graphs improves the competitive ratio from 4 to 3, and also provide a tight example.     

Colorability of mixed hypergraphs and their chromatic inversions

We solve a long-standing open problem concerning a discrete mathematical model, which has various applications in computer science and several other fields, including frequency assignment and many other problems on resource allocation. A mixed hypergraph $\mathcal H $ is a triple $(X,\mathcal C ,\mathcal D )$ , where $X$ is the set of vertices, and $\mathcal C $ and $\mathcal D $ are two set systems over $X$ , the families of so-called C-edges and D-edges, respectively. A vertex coloring of a mixed hypergraph $\mathcal H $ is proper if every C-edge has two vertices with a common color and every D-edge has two vertices with different colors. A mixed hypergraph is colorable if it has at least one proper coloring; otherwise it is uncolorable. The chromatic inversion of a mixed hypergraph $\mathcal H =(X,\mathcal C ,\mathcal D )$ is defined as $\mathcal H ^c=(X,\mathcal D ,\mathcal C )$ . Since 1995, it was an open problem wether there is a correlation between the colorability properties of a hypergraph and its chromatic inversion. In this paper we answer this question in the negative, proving that there exists no polynomial-time algorithm (provided that $P \ne NP$ ) to decide whether both $\mathcal H $ and $\mathcal H ^c$ are colorable, or both are uncolorable. This theorem holds already for the restricted class of 3-uniform mixed hypergraphs (i.e., where every edge has exactly three vertices). The proof is based on a new polynomial-time algorithm for coloring a special subclass of 3-uniform mixed hypergraphs. Implementation in C++ programming language has been tested. Further related decision problems are investigated, too.     

Improved upper bound for acyclic chromatic index of planar graphs without 4-cycles

Let $\chi'_{a}(G)$ and Δ(G) denote the acyclic chromatic index and the maximum degree of a graph G, respectively. Fiam?ík conjectured that $\chi'_{a}(G)\leq \varDelta (G)+2$ . Even for planar graphs, this conjecture remains open with large gap. Let G be a planar graph without 4-cycles. Fiedorowicz et al. showed that $\chi'_{a}(G)\leq \varDelta (G)+15$ . Recently Hou et al. improved the upper bound to Δ(G)+4. In this paper, we further improve the upper bound to Δ(G)+3.     

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).     

On the outer-connected domination in graphs

A set S of vertices of a graph G is an outer-connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by V?S is connected. The outer-connected domination number $\widetilde{\gamma}_{c}(G)$ is the minimum size of such a set. We prove that if δ(G)≥2 and diam?(G)≤2, then $\widetilde{\gamma}_{c}(G)\le (n+1)/2$ , and we study the behavior of $\widetilde{\gamma}_{c}(G)$ under an edge addition.     

Parity and strong parity edge-colorings of graphs

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. A parity edge-coloring (respectively, strong parity edge-coloring) is an edge-coloring in which there is no nontrivial parity path (respectively, open parity walk). The parity edge-chromatic number p(G) (respectively, strong parity edge-chromatic number $\widehat{p}(G)$ ) is the least number of colors in a parity edge-coloring (respectively, strong parity edge-coloring) of G. Notice that $\widehat{p}(G) \ge p(G) \ge \chi'(G) \ge \Delta(G)$ for any graph G. In this paper, we determine $\widehat{p}(G)$ and p(G) for some complete bipartite graphs and some products of graphs. For instance, we determine $\widehat{p}(K_{m,n})$ and p(K _m,n ) for m≤n with n≡0,?1,?2 (mod 2^?lg?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.     

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).