A novel approach for detecting multiple rumor sources in networks with partial observations

Locating source of information diffusion in networks has very important applications such as locating the sources of epidemics, news/rumors in social networks or online computer virus. In this paper, we consider detecting multiple rumor sources from a deterministic point of view by modeling it as the set resolving set (SRS) problem. Let G be a network on n nodes. A node subset K is an SRS of G if all detectable node sets are distinguishable by K. The problem of multiple rumor source detection (MRSD) in the network can be modeled as finding an SRS K with the smallest cardinality. In this paper, we propose a polynomial-time greedy algorithm for finding a minimum SRS in a general network, achieving performance ratio \(O(\ln n)\). This is the first work establishing a relation between the MRSD problem and a deterministic concept of SRS, and a first work to give the minimum SRS problem a polynomial-time performance-guaranteed approximation algorithm. Our framework suggests a robust and efficient approach for estimating multiple rumor sources independent of diffusion models in networks.     

Optimum turn-restricted paths,nested compatibility,and optimum convex polygons

We consider two apparently unrelated classes of combinatorial and geometric optimization problems. First, we give compact extended formulations, i.e., polynomial-size linear programming formulations with integer optima, for optimum path problems with turn restrictions satisfying a nested compatibility condition in acyclic digraphs. We then apply these results to optimum convex polygon problems in the plane, by interpreting certain dynamic programming algorithms as sequences of optimum turn-restricted path problems with nested compatibility in acyclic digraphs. As a result, we derive compact extended formulations for these geometric problems as well.     

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.     

FPT-algorithms for some problems related to integer programming

This paper studies a new version of the location problem called the mixed center location problem. Let P be a set of n points in the plane. We first consider the mixed 2-center problem, where one of the centers must be in P, and we solve it in \(O(n^2\log n)\) time. Second, we consider the mixed k-center problem, where m of the centers are in P, and we solve it in \(O(n^{m+O(\sqrt{k-m})})\) time. Motivated by two practical constraints, we propose two variations of the problem. Third, we present a 2-approximation algorithm and three heuristics solving the mixed k-center problem (\(k>2\)).     

On tree representations of relations and graphs: symbolic ultrametrics and cograph edge decompositions

Tree representations of (sets of) symmetric binary relations, or equivalently edge-colored undirected graphs, are of central interest, e.g. in phylogenomics. In this context symbolic ultrametrics play a crucial role. Symbolic ultrametrics define an edge-colored complete graph that allows to represent the topology of this graph as a vertex-colored tree. Here, we are interested in the structure and the complexity of certain combinatorial problems resulting from considerations based on symbolic ultrametrics, and on algorithms to solve them.This includes, the characterization of symbolic ultrametrics that additionally distinguishes between edges and non-edges of arbitrary edge-colored graphs G and thus, yielding a tree representation of G, by means of so-called cographs. Moreover, we address the problem of finding “closest” symbolic ultrametrics and show the NP-completeness of the three problems: symbolic ultrametric editing, completion and deletion. Finally, as not all graphs are cographs, and hence, do not have a tree representation, we ask, furthermore, what is the minimum number of cotrees needed to represent the topology of an arbitrary non-cograph G. This is equivalent to find an optimal cograph edge k-decomposition \(\{E_1,\dots ,E_k\}\) of E so that each subgraph \((V,E_i)\) of G is a cograph. We investigate this problem in full detail, resulting in several new open problems, and NP-hardness results.For all optimization problems proven to be NP-hard we will provide integer linear program formulations to solve them.     

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.     

Modified linear programming and class 0 bounds for graph pebbling

Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move removes two pebbles from some vertex and places one pebble on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert’s paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every n-vertex Class 0 graph has at least \(\frac{5}{3}n - \frac{11}{3}\) edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to \(2n - 5\), which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.     

List-edge-coloring of planar graphs without 6-cycles with three chords

A graph G is edge-k-choosable if, whenever we are given a list L(e) of colors with \(|L(e)|\ge k\) for each \(e\in E(G)\), we can choose a color from L(e) for each edge e such that no two adjacent edges receive the same color. In this paper we prove that if G is a planar graph, and each 6-cycle contains at most two chords, then G is edge-k-choosable, where \(k=\max \{8,\Delta (G)+1\}\), and edge-t-choosable, where \(t=\max \{10,\Delta (G)\}\).     

Possible winner problems on partial tournaments: a parameterized study

We study possible winner problems related to the uncovered set and the Banks set on partial tournaments from the viewpoint of parameterized complexity. We first study a problem where given a partial tournament D and a subset X of vertices, we are asked to add some arcs to D such that all vertices in X are included in the uncovered set. We focus on two parameterizations: parameterized by |X| and parameterized by the number of arcs to be added. In addition, we study a parameterized variant of the problem which is to determine whether all vertices of X can be included in the uncovered set by reversing at most k arcs. Finally, we study some parameterizations of a possible winner problem on partial tournaments, where we are given a partial tournament D and a distinguished vertex p, and asked whether D has a maximal transitive subtournament with p being the 0-indegree vertex. These parameterized problems are related to the Banks set. We achieve \(\mathcal {XP}\) results, \(\mathcal {W}\)-hardness results as well as \(\mathcal {FPT}\) results along with a kernelization lower bound for the problems studied in this paper.     

Corporate social responsibility: the organizational view

Designing strategies for corporate social responsibility (CSR)-practice nowadays has become essential for organizations. Notwithstanding, how organizations appear internally in a socially responsible context toward their employees has been insufficiently investigated. This study aims at ascertaining how Internal CSR can be conceptualized as well as how it affects employees’ commitment. To do so, the manifestations of Internal CSR are discussed based on social identity theory and further literature, while the developed research model is checked for suitability through a survey generating 2081 employee responses from an international pharmaceutical company. As one result, it can be stated that the factors proposed to constitute Internal CSR are confirmed. Further, the findings entail the following conclusions: Internal CSR has a notable influence on employees’ Affective Organizational Commitment whilst relatively moderate impacting Normative Organizational Commitment. Additionally, Affective Organizational Commitment adopts a mediating function regarding Normative Organizational Commitment.     

Two extremal problems related to orders

We consider two extremal problems related to total orders on all subsets of \({\mathbb N}\). The first one is to maximize the Lagrangian of hypergraphs among all hypergraphs with m edges for a given positive integer m. In 1980’s, Frankl and Füredi conjectured that for a given positive integer m, the r-uniform hypergraph with m edges formed by taking the first m r-subsets of \({\mathbb N}\) in the colex order has the largest Lagrangian among all r-uniform hypergraphs with m edges. We provide some partial results for 4-uniform hypergraphs to this conjecture. The second one is for a given positive integer m, how to minimize the cardinality of the union closure families generated by edge sets of the r-uniform hypergraphs with m edges. Leck, Roberts and Simpson conjectured that the union closure family generated by the first m r-subsets of \({\mathbb N}\) in order U has the minimum cardinality among all the union closure families generated by edge sets of the r-uniform hypergraphs with m edges. They showed that the conjecture is true for graphs. We show that a similar result holds for non-uniform hypergraphs whose edges contain 1 or 2 vertices.     

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.     

A compact representation for minimizers of <Emphasis Type="Italic">k</Emphasis>-submodular functions

A k-submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a k-submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando–Fujishige’s signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from k-submodular functions. We give algorithms to construct the elementary PIP of minimizers of a k-submodular function f for three cases: (i) a minimizing oracle of f is available, (ii) f is network-representable, and (iii) f arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts k-submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.     

On (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>)-relaxed strong edge-coloring of graphs

In this paper, we introduce a new relaxation of strong edge-coloring. Let G be a graph. For two nonnegative integers s and t, an (s, t)-relaxed strong k-edge-coloring is an assignment of k colors to the edges of G, such that for any edge e, there are at most s edges adjacent to e and t edges which are distance two apart from e assigned the same color as e. The (s, t)-relaxed strong chromatic index, denoted by \({\chi '}_{(s,t)}(G)\), is the minimum number k of an (s, t)-relaxed strong k-edge-coloring admitted by G. This paper studies the (s, t)-relaxed strong edge-coloring of graphs, especially trees. For a tree T, the tight upper bounds for \({\chi '}_{(s,0)}(T)\) and \({\chi '}_{(0,t)}(T)\) are given. And the (1, 1)-relaxed strong chromatic index of an infinite regular tree is determined. Further results on \({\chi '}_{(1,0)}(T)\) are also presented.     

Anti-forcing spectra of perfect matchings of graphs

Let M be a perfect matching of a graph G. The smallest number of edges whose removal to make M as the unique perfect matching in the resulting subgraph is called the anti-forcing number of M. The anti-forcing spectrum of G is the set of anti-forcing numbers of all perfect matchings in G, denoted by \(\hbox {Spec}_{af}(G)\). In this paper, we show that any finite set of positive integers can be the anti-forcing spectrum of a graph. We present two classes of hexagonal systems whose anti-forcing spectra are integer intervals. Finally, we show that determining the anti-forcing number of a perfect matching of a bipartite graph with maximum degree four is a NP-complete problem.     

A new graph parameter and a construction of larger graph without increasing radio <Emphasis Type="Italic">k</Emphasis>-chromatic number

For a positive integer \(k\ge 2\), the radio k-coloring problem is an assignment L of non-negative integers (colors) to the vertices of a finite simple graph G satisfying the condition \(|L(u)-L(v)| \ge k+1-d(u,v)\), for any two distinct vertices u, v of G and d(u, v) being distance between u, v. The span of L is the largest integer assigned by L, while 0 is taken as the smallest color. An \(rc_k\)-coloring on G is a radio k-coloring on G of minimum span which is referred as the radio k-chromatic number of G and denoted by \(rc_k(G)\). An integer h, \(0<h<rc_k(G)\), is a hole in a \(rc_k\)-coloring on G if h is not assigned by it. In this paper, we construct a larger graph from a graph of a certain class by using a combinatorial property associated with \((k-1)\) consecutive holes in any \(rc_k\)-coloring of a graph. Exploiting the same property, we introduce a new graph parameter, referred as \((k-1)\)-hole index of G and denoted by \(\rho _k(G)\). We also explore several properties of \(\rho _k(G)\) including its upper bound and relation with the path covering number of the complement \(G^c\).     

Two-agent scheduling of time-dependent jobs

We consider the problem of scheduling deteriorating jobs or shortening jobs with two agents A and B. We are interested in generating all Pareto-optimal schedules for the two criteria: (1) the total completion time of the jobs in A and the maximum cost of the jobs in B, and (2) the maximum cost of the jobs in A and the maximum cost of the jobs in B. We show that all Pareto-optimal schedules for both problems can be generated in polynomial time, whether the jobs are deteriorating or shortening.     

The <Emphasis Type="Italic">k</Emphasis>-hop connected dominating set problem: approximation and hardness

Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G there is a vertex u in D such that the distance between v and u in G is at most k. We study the problem of finding a minimum k-hop connected dominating set of a graph (\({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\)). We prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {NP}\)-hard on planar bipartite graphs of maximum degree 4. We also prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {APX}\)-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) on bipartite and on (1, 2)-split graphs. Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices. We also discuss the complexity of computing this graph parameter. On the positive side, we show an approximation algorithm for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\). Finally, when \(k=1\), we present two new approximation algorithms for the weighted version of the problem restricted to graphs with a polynomially bounded number of minimal separators.     

Neighbor sum distinguishing index of 2-degenerate graphs

We consider proper edge colorings of a graph G using colors in \(\{1,\ldots ,k\}\). Such a coloring is called neighbor sum distinguishing if for each pair of adjacent vertices u and v, the sum of the colors of the edges incident with u is different from the sum of the colors of the edges incident with v. The smallest value of k in such a coloring of G is denoted by \({\mathrm ndi}_{\Sigma }(G)\). In this paper we show that if G is a 2-degenerate graph without isolated edges, then \({\mathrm ndi}_{\Sigma }(G)\le \max \{\Delta (G)+2,7\}\).     

Erdős–Gallai-type results for colorful monochromatic connectivity of a graph

A path in an edge-colored graph is called a monochromatic path if all the edges on the path are colored with one same color. An edge-coloring of G is a monochromatic connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in G. For a connected graph G, the monochromatic connection number of G, denoted by mc(G), is defined to be the maximum number of colors used in an MC-coloring of G. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we study two kinds of Erd?s–Gallai-type problems for mc(G), and completely solve them.