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 $d\ge 3$ .     

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.     

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 $\mathcal {B}$ .     

On the generalized multiway cut in trees problem

Given a tree $T = (V, E)$ with $n$ vertices and a collection of terminal sets $D = \{S_1, S_2, \ldots , S_c\}$ , where each $S_i$ is a subset of $V$ and $c$ is a constant, the generalized multiway cut in trees problem (GMWC(T)) asks to find a minimum size edge subset $E^{\prime } \subseteq E$ such that its removal from the tree separates all terminals in $S_i$ from each other for each terminal set $S_i$ . The GMWC(T) problem is a natural generalization of the classical multiway cut in trees problem, and has an implicit relation to the Densest $k$ -Subgraph problem. In this paper, we show that the GMWC(T) problem is fixed-parameter tractable by giving an $O(n^2 + 2^k)$ time algorithm, where $k$ is the size of an optimal solution, and the GMWC(T) problem is polynomial time solvable when the problem is restricted in paths.We also discuss some heuristics for the GMWC(T) problem     

Adjacent vertex distinguishing edge colorings of planar graphs with girth at least five

An adjacent vertex distinguishing edge coloring of a graph \(G\) is a proper edge coloring of \(G\) such that any pair of adjacent vertices admit different sets of colors. The minimum number of colors required for such a coloring of \(G\) is denoted by \(\chi ^{\prime }_{a}(G)\) . In this paper, we prove that if \(G\) is a planar graph with girth at least 5 and \(G\) is not a 5-cycle, then \(\chi ^{\prime }_{a}(G)\le \Delta +2\) , where \(\Delta \) is the maximum degree of \(G\) . This confirms partially a conjecture in Zhang et al. (Appl Math Lett 15:623–626, 2002).     

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.     

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.     

Realizations of the game domination number

Domination game is a game on a finite graph which includes two players. First player, Dominator, tries to dominate a graph in as few moves as possible; meanwhile the second player, Staller, tries to hold him back and delay the end of the game as long as she can. In each move at least one additional vertex has to be dominated. The number of all moves in the game in which Dominator makes the first move and both players play optimally is called the game domination number and is denoted by \(\gamma _g\) . The total number of moves in a Staller-start game is denoted by \(\gamma _g^{\prime }\) . It is known that \(|\gamma _g(G)-\gamma _g^{\prime }(G)|\le 1\) for any graph \(G\) . Graph \(G\) realizes a pair \((k,l)\) if \(\gamma _g(G)=k\) and \(\gamma _g^{\prime }(G)=l\) . It is shown that pairs \((2k,2k-1)\) for all \(k\ge 2\) can be realized by a family of 2-connected graphs. We also present 2-connected classes which realize pairs \((k,k)\) and \((k,k+1)\) . Exact game domination number for combs and 1-connected realization of the pair \((2k+1,2k)\) are also given.     

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.     

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  $\sqrt{2}$ .     

On metric dimension of permutation graphs

The metric dimension \(\dim (G)\) of a graph \(G\) is the minimum number of vertices such that every vertex of \(G\) is uniquely determined by its vector of distances to the set of chosen vertices. Let \(G_1\) and \(G_2\) be disjoint copies of a graph \(G\) , and let \(\sigma : V(G_1) \rightarrow V(G_2)\) be a permutation. Then, a permutation graph \(G_{\sigma }=(V, E)\) has the vertex set \(V=V(G_1) \cup V(G_2)\) and the edge set \(E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}\) . We show that \(2 \le \dim (G_{\sigma }) \le n-1\) for any connected graph \(G\) of order \(n\) at least \(3\) . We give examples showing that neither is there a function \(f\) such that \(\dim (G) for all pairs \((G,\sigma )\) , nor is there a function \(g\) such that \(g(\dim (G))>\dim (G_{\sigma })\) for all pairs \((G, \sigma )\) . Further, we characterize permutation graphs \(G_{\sigma }\) satisfying \(\dim (G_{\sigma })=n-1\) when \(G\) is a complete \(k\) -partite graph, a cycle, or a path on \(n\) vertices.     

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.     

Graphs with small balanced decomposition numbers

A balanced coloring of a graph \(G\) is an ordered pair \((R,B)\) of disjoint subsets \(R,B \subseteq V(G)\) with \(|R|=|B|\) . The balanced decomposition number  \(f(G)\) of a connected graph \(G\) is the minimum integer \(f\) such that for any balanced coloring \((R,B)\) of \(G\) there is a partition \(\mathcal{P}\) of \(V(G)\) such that \(S\) induces a connected subgraph with \(|S| \le f\) and \(|S \cap R| = |S \cap B|\) for \(S \in \mathcal{P}\) . This paper gives a short proof for the result by Fujita and Liu (2010) that a graph \(G\) of \(n\) vertices has \(f(G)=3\) if and only if \(G\) is \(\lfloor \frac{n}{2} \rfloor \) -connected but is not a complete graph.     

Minimum number of disjoint linear forests covering a planar graph

Graph coloring has interesting real-life applications in optimization, computer science and network design, such as file transferring in a computer network, computation of Hessians matrix and so on. In this paper, we consider one important coloring, linear arboricity, which is an improper edge coloring. Moreover, we study linear arboricity on planar graphs with maximum degree \(\varDelta \ge 7\) . We have proved that the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \) , if for each vertex \(v\in V(G)\) , there are two integers \(i_v,j_v\in \{3,4,5,6,7,8\}\) such that any two cycles of length \(i_v\) and \(j_v\) , which contain \(v\) , are not adjacent. Clearly, if \(i_v=i, j_v=j\) for each vertex \(v\in V(G)\) , then we can easily get one corollary: for two fixed integers \(i,j\in \{3,4,5,6,7,8\}\) , if there is no adjacent cycles with length \(i\) and \(j\) in \(G\) , then the linear arboricity of \(G\) is \(\lceil \frac{\varDelta }{2}\rceil \) .     

On certain geometric properties of the Yao–Yao graphs

We show that, for any constant $\rho > 1$ , there exists an integer constant $k$ such that the Yao–Yao graph with parameter $k$ defined on a civilized unit disk graph is a geometric spanner of stretch factor $\rho $ . This improves the results of Wang and Li in several aspects, as described in the paper. This partially answers an open problem posed by Demaine, Mitchell and O’Rourke about the spanner properties of Yao–Yao graphs. We also show that the Yao–Yao graph with parameter $k=4$ defined on the complete Euclidean graph is not plane.     

Efficient polynomial-time algorithms for the constrained LCS problem with strings exclusion

In this paper, we revisit a recent variant of the longest common subsequence (LCS) problem, the string-excluding constrained LCS (STR-EC-LCS) problem, which was first addressed by Chen and Chao (J Comb Optim 21(3):383–392, 2011). Given two sequences \(X\) and \(Y\) of lengths \(m\) and \(n,\) respectively, and a constraint string \(P\) of length \(r,\) we are to find a common subsequence \(Z\) of \(X\) and \(Y\) which excludes \(P\) as a substring and the length of \(Z\) is maximized. In fact, this problem cannot be correctly solved by the previously proposed algorithm. Thus, we give a correct algorithm with \(O(mnr)\) time to solve it. Then, we revisit the STR-EC-LCS problem with multiple constraints \(\{ P_1, P_2, \ldots , P_k \}.\) We propose a polynomial-time algorithm which runs in \(O(mnR)\) time, where \(R = \sum _{i=1}^{k} |P_i|,\) and thus it overthrows the previous claim of NP-hardness.     

Efficient algorithms for the max k -vertex cover problem

Given a graph  \(G(V,E)\) of order  \(n\) and a constant \(k \leqslant n\) , the max  \(k\) -vertex cover problem consists of determining  \(k\) vertices that cover the maximum number of edges in  \(G\) . In its (standard) parameterized version, max  \(k\) -vertex cover can be stated as follows: “given  \(G,\) \(k\) and parameter  \(\ell ,\) does  \(G\) contain  \(k\) vertices that cover at least  \(\ell \) edges?”. We first devise moderately exponential exact algorithms for max  \(k\) -vertex cover, with time-complexity exponential in  \(n\) but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for max  \(k\) -vertex cover with complexity bounded above by the maximum among  \(c^k\) and  \(\gamma ^{\tau },\) for some \(\gamma < 2,\) where  \(\tau \) is the cardinality of a minimum vertex cover of  \(G\) (note that \({\textsc {max}}\,\) k \({\textsc {\!-vertex cover}}{} \notin \mathbf{FPT}\) with respect to parameter  \(k\) unless \(\mathbf{FPT} = \mathbf{W[1]}\) ), using polynomial space. We finally study approximation of max  \(k\) -vertex cover by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catch-up on polynomial inapproximability, by providing algorithms achieving, with worst-case running times importantly smaller than those needed for exact computation, approximation ratios unachievable in polynomial time.     

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.