An approximation algorithm for maximum internal spanning tree

Given a graph G, the maximum internal spanning tree problem (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of cost-efficient communication networks and water supply networks and hence has been extensively studied in the literature. MIST is NP-hard and hence a number of polynomial-time approximation algorithms have been designed for MIST in the literature. The previously best polynomial-time approximation algorithm for MIST achieves a ratio of \(\frac{3}{4}\). In this paper, we first design a simpler algorithm that achieves the same ratio and the same time complexity as the previous best. We then refine the algorithm into a new approximation algorithm that achieves a better ratio (namely, \(\frac{13}{17}\)) with the same time complexity. Our new algorithm explores much deeper structure of the problem than the previous best. The discovered structure may be used to design even better approximation or parameterized algorithms for the problem in the future.     

A linear-time algorithm for clique-coloring problem in circular-arc graphs

A maximal clique of G is a clique not properly contained in any other clique. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no maximal clique with at least two vertices is monochromatic. The smallest integer k admitting a k-clique-coloring of G is called clique-coloring number of G. Cerioli and Korenchendler (Electron Notes Discret Math 35:287–292, 2009) showed that there is a polynomial-time algorithm to solve the clique-coloring problem in circular-arc graphs and asked whether there exists a linear-time algorithm to find an optimal clique-coloring in circular-arc graphs or not. In this paper we present a linear-time algorithm of the optimal clique-coloring in circular-arc graphs.     

Directed Steiner trees with diffusion costs

Given a directed arc-weighted graph G with n nodes, a root r and k terminals, the directed steiner tree problem (DST) consists in finding a minimum-weight tree rooted at r and spanning all the terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are unable to duplicate packets. We define a more general problem, namely the directed steiner tree with a limited number of diffusing nodes (DSTLD), that enables us to model multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a \(\left\lceil \frac{k-1}{d} \right\rceil \)-approximation algorithm for DST that is XP in d. We deduce from that result a strong inapproximability property. In particular, we prove that, under the assumption that NP \(\not \subseteq \) ZTIME \([n^{\log ^{O(1)}n}]\), there is no polynomial-time approximation algorithm for DSTLD with ratio \(\varOmega \left( \frac{k}{d}\right) \). We finally give an evaluation of performances of an exact algorithm dedicated to the case \(d \le 3\).     

Safe sets in graphs: Graph classes and structural parameters

A safe set of a graph \(G=(V,E)\) is a non-empty subset S of V such that for every component A of G[S] and every component B of \(G[V {\setminus } S]\), we have \(|A| \ge |B|\) whenever there exists an edge of G between A and B. In this paper, we show that a minimum safe set can be found in polynomial time for trees. We then further extend the result and present polynomial-time algorithms for graphs of bounded treewidth, and also for interval graphs. We also study the parameterized complexity. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between the tree-depth and the vertex cover number. We then conclude the paper by showing hardness for split graphs and planar graphs.     

On the minimum routing cost clustered tree problem

For an edge-weighted graph \(G=(V,E,w)\), in which the vertices are partitioned into k clusters \(\mathcal {R}=\{R_1,R_2,\ldots ,R_k\}\), a spanning tree T of G is a clustered spanning tree if T can be cut into k subtrees by removing \(k-1\) edges such that each subtree is a spanning tree for one cluster. In this paper, we show the inapproximability of finding a clustered spanning tree with minimum routing cost, where the routing cost is the total distance summed over all pairs of vertices. We present a 2-approximation for the case that the input is a complete weighted graph whose edge weights obey the triangle inequality. We also study a variant in which the objective function is the total distance summed over all pairs of vertices of different clusters. We show that the problem is polynomial-time solvable when the number of clusters k is 2 and NP-hard for \(k=3\). Finally, we propose a polynomial-time 2-approximation algorithm for the case of three clusters.     

Edge-disjoint spanning trees and the number of maximum state circles of a graph

Motivated by the connection with the genus of unoriented alternating links, Jin et al. (Acta Math Appl Sin Engl Ser, 2015) introduced the number of maximum state circles of a plane graph G, denoted by \(s_{\max }(G)\), and proved that \(s_{\max }(G)=\max \{e(H)+2c(H)-v(H)|\) H is a spanning subgraph of \(G\}\), where e(H), c(H) and v(H) denote the size, the number of connected components and the order of H, respectively. In this paper, we show that for any (not necessarily planar) graph G, \(s_{\max }(G)\) can be achieved by the spanning subgraph H of G whose each connected component is a maximal subgraph of G with two edge-disjoint spanning trees. Such a spanning subgraph is proved to be unique and we present a polynomial-time algorithm to find such a spanning subgraph for any graph G.     

A PTAS for minimum weighted connected vertex cover $$P_3$$ problem in 3-dimensional wireless sensor networks

Given a connected and weighted graph \(G=(V, E)\) with each vertex v having a nonnegative weight w(v), the minimum weighted connected vertex cover \(P_{3}\) problem \((MWCVCP_{3})\) is required to find a subset C of vertices of the graph with minimum total weight, such that each path with length 2 has at least one vertex in C, and moreover, the induced subgraph G[C] is connected. This kind of problem has many applications concerning wireless sensor networks and ad hoc networks. When homogeneous sensors are deployed into a three-dimensional space instead of a plane, the mathematical model for the sensor network is a unit ball graph instead of a unit disk graph. In this paper, we propose a new concept called weak c-local and give the first polynomial time approximation scheme (PTAS) for \(MWCVCP_{3}\) in unit ball graphs when the weight is smooth and weak c-local.     

A tighter insertion-based approximation of the crossing number

Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of \(G+F\) with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F. We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee—depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph \(G+F\), while computing the crossing number of \(G+F\) exactly is NP-hard already when \(|F|=1\). Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.     

A simpler PTAS for connected <Emphasis Type="Italic">k</Emphasis>-path vertex cover in homogeneous wireless sensor network

Because of its application in the field of security in wireless sensor networks, k-path vertex cover (\(\hbox {VCP}_k\)) has received a lot of attention in recent years. Given a graph \(G=(V,E)\), a vertex set \(C\subseteq V\) is a k-path vertex cover (\(\hbox {VCP}_k\)) of G if every path on k vertices has at least one vertex in C, and C is a connected k-path vertex cover of G (\(\hbox {CVCP}_k\)) if furthermore the subgraph of G induced by C is connected. A homogeneous wireless sensor network can be modeled as a unit disk graph. This paper presents a new PTAS for \(\hbox {MinCVCP}_k\) on unit disk graphs. Compared with previous PTAS given by Liu et al., our method not only simplifies the algorithm and reduces the time-complexity, but also simplifies the analysis by a large amount.     

Algorithms for connected <Emphasis Type="Italic">p</Emphasis>-centdian problem on block graphs

We consider the facility location problem of locating a set \(X_p\) of p facilities (resources) on a network (or a graph) such that the subnetwork (or subgraph) induced by the selected set \(X_p\) is connected. Two problems on a block graph G are proposed: one problem is to minimizes the sum of its weighted distances from all vertices of G to \(X_p\), another problem is to minimize the maximum distance from each vertex that is not in \(X_p\) to \(X_p\) and, at the same time, to minimize the sum of its distances from all vertices of G to \(X_p\). We prove that the first problem is linearly solvable on block graphs with unit edge length. For the second problem, it is shown that the set of Pareto-optimal solutions of the two criteria has cardinality not greater than n, and can be obtained in \(O(n^2)\) time, where n is the number of vertices of the block graph G.     

Is there any polynomial upper bound for the universal labeling of graphs?

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation \(\ell \) of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from \(\{1,2,\ldots , k\}\) denoted it by \(\overrightarrow{\chi _{u}}(G) \). We have \(2\Delta (G)-2 \le \overrightarrow{\chi _{u}} (G)\le 2^{\Delta (G)}\), where \(\Delta (G)\) denotes the maximum degree of G. In this work, we offer a provocative question that is: “Is there any polynomial function f such that for every graph G, \(\overrightarrow{\chi _{u}} (G)\le f(\Delta (G))\)?”. Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree T, \(\overrightarrow{\chi _{u}}(T)={\mathcal {O}}(\Delta ^3) \). Next, we show that for a given 3-regular graph G, the universal labeling number of G is 4 if and only if G belongs to Class 1. Therefore, for a given 3-regular graph G, it is an \( {{\mathbf {N}}}{{\mathbf {P}}} \)-complete to determine whether the universal labeling number of G is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem.     

Maximum flows in generalized processing networks

Processing networks (cf. Koene in Minimal cost flow in processing networks: a primal approach, 1982) and manufacturing networks (cf. Fang and Qi in Optim Methods Softw 18:143–165, 2003) are well-studied extensions of traditional network flow problems that allow to model the decomposition or distillation of products in a manufacturing process. In these models, so called flow ratios \(\alpha _e \in [0,1]\) are assigned to all outgoing edges of special processing nodes. For each such special node, these flow ratios, which are required to sum up to one, determine the fraction of the total outgoing flow that flows through the respective edges. In this paper, we generalize processing networks to the case that these flow ratios only impose an upper bound on the respective fractions and, in particular, may sum up to more than one at each node. We show that a flow decomposition similar to the one for traditional network flows is possible and can be computed in strongly polynomial time. Moreover, we show that there exists a fully polynomial-time approximation scheme (FPTAS) for the maximum flow problem in these generalized processing networks if the underlying graph is acyclic and we provide two exact algorithms with strongly polynomial running-time for the problem on series–parallel graphs. Finally, we study the case of integral flows and show that the problem becomes \({\mathcal {NP}}\)-hard to solve and approximate in this case.     

The matching extension problem in general graphs is co-NP-complete

A simple connected graph G with 2n vertices is said to be k-extendable for an integer k with \(0<k<n\) if G contains a perfect matching and every matching of cardinality k in G is a subset of some perfect matching. Lakhal and Litzler (Inf Process Lett 65(1):11–16, 1998) discovered a polynomial algorithm that decides whether a bipartite graph is k-extendable. For general graphs, however, it has been an open problem whether there exists a polynomial algorithm. The new result presented in this paper is that the extendability problem is co-NP-complete.     

A solution to a conjecture on the generalized connectivity of graphs

The generalized k-connectivity \(\kappa _k(G)\) of a graph G was introduced by Chartrand et al. in (Bull Bombay Math Colloq 2:1–6, 1984), which is a nice generalization of the classical connectivity. Recently, as a natural counterpart, Li et al. proposed the concept of generalized edge-connectivity for a graph. In this paper, we consider the computational complexity of the generalized connectivity and generalized edge-connectivity of a graph. We give a confirmative solution to a conjecture raised by S. Li in Ph.D. Thesis (2012). We also give a polynomial-time algorithm to a problem of generalized k-edge-connectivity.     

The connected disk covering problem

Let P be a convex polygon with n vertices. We consider a variation of the K-center problem called the connected disk covering problem (CDCP), i.e., finding K congruent disks centered in P whose union covers P with the smallest possible radius, while a connected graph is generated by the centers of the K disks whose edge length can not exceed the radius. We give a 2.81-approximation algorithm in O(Kn) time.     

On the algorithmic aspects of strong subcoloring

A partition of the vertex set V(G) of a graph G into \(V(G)=V_1\cup V_2\cup \cdots \cup V_k\) is called a k-strong subcoloring if \(d(x,y)\ne 2\) in G for every \(x,y\in V_i\), \(1\le i \le k\) where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as \(\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k\)-\(\text {strong subcoloring}\}\). In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of \(K_{1,r}\)-free split graphs, StrongSubcoloring is in P when \(r\le 3\) and NP-complete when \(r>3\) and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of \(P_4\); otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every \(x,y\in S\), \(d(x,y)= 2\) in G and the cardinality of a maximum strong set in G is denoted by \(\alpha _{s}(G)\). Clearly, \(\alpha _{s}(G)\le \chi _{sc}(G)\). We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) \(K_{1,4}\)-free split graphs, and it is polynomial time solvable for (i) trees and (ii) \(P_4\)-free graphs.     

The popular matching and condensation problems under matroid constraints

The popular matching problem introduced by Abraham, Irving, Kavitha, and Mehlhorn is a matching problem in which there exist applicants and posts, and applicants have preference lists over posts. A matching M is said to be popular, if there exists no other matching N such that the number of applicants that prefer N to M is larger than the number of applicants that prefer M to N. The goal of this problem is to decide whether there exists a popular matching, and find a popular matching if one exists. In this paper, we first consider a matroid generalization of the popular matching problem with strict preference lists, and give a polynomial-time algorithm for this problem. In the second half of this paper, we consider the problem of transforming a given instance of a matroid generalization of the popular matching problem with strict preference lists by deleting a minimum number of applicants so that it has a popular matching. This problem is a matroid generalization of the popular condensation problem with strict preference lists introduced by Wu, Lin, Wang, and Chao. By using the results in the first half, we give a polynomial-time algorithm for this problem.     

Minimum choosability of planar graphs

The problem of list coloring of graphs appears in practical problems concerning channel or frequency assignment. In this paper, we study the minimum number of choosability of planar graphs. A graph G is edge-k-choosable if whenever every edge x is assigned with a list of at least k colors, L(x)), there exists an edge coloring \(\phi \) such that \(\phi (x) \in L(x)\). Similarly, A graph G is toal-k-choosable if when every element (edge or vertex) x is assigned with a list of at least k colors, L(x), there exists an total coloring \(\phi \) such that \(\phi (x) \in L(x)\). We proved \(\chi '_{l}(G)=\Delta \) and \(\chi ''_{l}(G)=\Delta +1\) for a planar graph G with maximum degree \(\Delta \ge 8\) and without chordal 6-cycles, where the list edge chromatic number \(\chi '_{l}(G)\) of G is the smallest integer k such that G is edge-k-choosable and the list total chromatic number \(\chi ''_{l}(G)\) of G is the smallest integer k such that G is total-k-choosable.     

On computing a minimum secure dominating set in block graphs

In a graph \(G=(V,E)\), a set \(D \subseteq V\) is said to be a dominating set of G if for every vertex \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\). A secure dominating set of the graph G is a dominating set D of G such that for every \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\) and \((D{\setminus }\{v\})\cup \{u\}\) is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite 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 \(\mathrm{sd}_{\gamma_{t}}(G)=\lfloor\frac{2n}{3}\rfloor\).