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.     

Popular matchings: structure and algorithms

An instance of the popular matching problem (POP-M) consists of a set of applicants and a set of posts. Each applicant has a preference list that strictly ranks a subset of the posts. A matching M of applicants to posts is popular if there is no other matching M′ such that more applicants prefer M′ to M than prefer M to M′. Abraham et al. (SIAM J. Comput. 37:1030–1045, 2007) described a linear time algorithm to determine whether a popular matching exists for a given instance of POP-M, and if so to find a largest such matching. A number of variants and extensions of POP-M have recently been studied. This paper provides a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a structure called the switching graph, a directed graph computable in linear time from the preference lists. We show that the switching graph can be exploited to yield efficient algorithms for a range of associated problems, including the counting and enumeration of the set of popular matchings, generation of a popular matching uniformly at random, finding all applicant-post pairs that can occur in a popular matching, and computing popular matchings that satisfy various additional optimality criteria. Our algorithms for computing such optimal popular matchings improve those described in a recent paper by Kavitha and Nasre (Proceedings of MATCH-UP: Matching Under Preferences—Algorithms and Complexity, 2008).     

Optimization problems with color-induced budget constraints

In 1984, Gabow and Tarjan provided a very elegant and fast algorithm for the following problem: given a matroid defined on a red and blue colored ground set, determine a basis of minimum cost among those with k red elements, or decide that no such basis exists. In this paper, we investigate extensions of this problem from ordinary matroids to the more general notion of poset matroids which take precedence constraints on the ground set into account. We show that the problem on general poset matroids becomes -hard, already if the underlying partially ordered set (poset) consists of binary trees of height two. On the positive side, we present two algorithms: a pseudopolynomial one for integer polymatroids, i.e., the case where the poset consists of disjoint chains, and a polynomial algorithm for the problem to determine a minimum cost ideal of size l with k red elements, i.e., the uniform rank-l poset matroid, on series-parallel posets.     

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.     

Matching colored points with rectangles

Let S be a point set in the plane such that each of its elements is colored either red or blue. A matching of S with rectangles is any set of pairwise-disjoint axis-aligned closed rectangles such that each rectangle contains exactly two points of S. Such a matching is monochromatic if every rectangle contains points of the same color, and is bichromatic if every rectangle contains points of different colors. We study the following two problems: (1) Find a maximum monochromatic matching of S with rectangles. (2) Find a maximum bichromatic matching of S with rectangles. For each problem we provide a polynomial-time approximation algorithm that constructs a matching with at least 1 / 4 of the number of rectangles of an optimal matching. We show that the first problem is \(\mathsf {NP}\)-hard even if either the matching rectangles are restricted to axis-aligned segments or S is in general position, that is, no two points of S share the same x or y coordinate. We further show that the second problem is also \(\mathsf {NP}\)-hard, even if S is in general position. These \(\mathsf {NP}\)-hardness results follow by showing that deciding the existence of a matching that covers all points is \(\mathsf {NP}\)-complete in each case. Additionally, we prove that it is \(\mathsf {NP}\)-complete to decide the existence of a matching with rectangles that cover all points in the case where all the points have the same color, solving an open problem of Bereg et al. (Comput Geom 42(2):93–108, 2009).     

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 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.     

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.     

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.     

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.     

Reconstruction of hidden graphs and threshold group testing

Classical group testing is a search paradigm where the goal is the identification of individual positive elements in a large collection of elements by asking queries of the form “Does a set of elements contain a positive one?”. A graph reconstruction problem that generalizes the classical group testing problem is to reconstruct a hidden graph from a given family of graphs by asking queries of the form “Whether a set of vertices induces an edge”. Reconstruction problems on families of Hamiltonian cycles, matchings, stars and cliques on n vertices have been studied where algorithms of using at most 2nlg?n,(1+o(1))(nlg?n),2n and 2n queries were proposed, respectively. In this paper we improve them to \((1+o(1))(n\lg n),(1+o(1))(\frac{n\lg n}{2}),n+2\lg n\) and n+lg?n, respectively. Threshold group testing is another generalization of group testing which is to identify the individual positive elements in a collection of elements under a more general setting, in which there are two fixed thresholds ? and u, with ?<u, and the response to a query is positive if the tested subset of elements contains at least u positive elements, negative if it contains at most ? positive elements, and it is arbitrarily given otherwise. For the threshold group testing problem with ?=u?1, we show that p positive elements among n given elements can be determined by using O(plg?n) queries, with a matching lower bound.     

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.     

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.     

An $$O(|E(G)|^2)$$ algorithm for recognizing Pfaffian graphs of a type of bipartite graphs

A graph \(G=(V,E)\) with even number vertices is called Pfaffian if it has a Pfaffian orientation, namely it admits an orientation such that the number of edges of any M-alternating cycle which have the same direction as the traversal direction is odd for some perfect matching M of the graph G. In this paper, we obtain a necessary and sufficient condition of Pfaffian graphs in a type of bipartite graphs. Then, we design an \(O(|E(G)|^2)\) algorithm for recognizing Pfaffian graphs in this class and constructs a Pfaffian orientation if the graph is Pfaffian. The results improve and generalize some known results.     

Approximating capacitated tree-routings in networks

Let G=(V,E) be a connected graph such that each edge e∈E is weighted by a nonnegative real w(e). Let s be a vertex designated as a sink, M?V be a set of terminals with a demand function q:M→R ⁺, κ>0 be a routing capacity, and λ≥1 be an integer edge capacity. The capacitated tree-routing problem (CTR) asks to find a partition ?={Z ₁,Z ₂,…,Z _? } of M and a set \({\mathcal{T}}=\{T_{1},T_{2},\ldots,T_{\ell}\}\) of trees of G such that each T _i contains Z _i ∪{s} and satisfies \(\sum_{v\in Z_{i}}q(v)\leq \kappa\). A single copy of an edge e∈E can be shared by at most λ trees in \({\mathcal{T}}\); any integer number of copies of e are allowed to be installed, where the cost of installing a copy of e is w(e). The objective is to find a solution \(({\mathcal{M}},{\mathcal{T}})\) that minimizes the total installing cost. In this paper, we propose a (2+ρ _ST )-approximation algorithm to CTR, where ρ _ST is any approximation ratio achievable for the Steiner tree problem.     

Covering directed graphs by in-trees

Given a directed graph D=(V,A) with a set of d specified vertices S={s ₁,…,s _d }?V and a function f : S→? where ? denotes the set of positive integers, we consider the problem which asks whether there exist ∑ _i=1 ^d f(s _i ) in-trees denoted by \(T_{i,1},T_{i,2},\ldots,T_{i,f(s_{i})}\) for every i=1,…,d such that \(T_{i,1},\ldots,T_{i,f(s_{i})}\) are rooted at s _i , each T _i,j spans vertices from which s _i is reachable and the union of all arc sets of T _i,j for i=1,…,d and j=1,…,f(s _i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑ _i=1 ^d f(s _i ) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite 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 \(d \le 3\).     

Reconfiguration of dominating sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of k, we consider properties of \(D_k(G)\), the graph consisting of a node for each dominating set of size at most k and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that \(D_{\varGamma (G)+1}(G)\) is not necessarily connected, for \(\varGamma (G)\) the maximum cardinality of a minimal dominating set in G. The result holds even when graphs are constrained to be planar, of bounded tree-width, or b-partite for \(b \ge 3\). Moreover, we construct an infinite family of graphs such that \(D_{\gamma (G)+1}(G)\) has exponential diameter, for \(\gamma (G)\) the minimum size of a dominating set. On the positive side, we show that \(D_{n-\mu }(G)\) is connected and of linear diameter for any graph G on n vertices with a matching of size at least \(\mu +1\).     

Incremental Facility Location Problem and Its Competitive Algorithms

We consider the incremental version of the k-Facility Location Problem, which is a common generalization of the facility location and the k-median problems. The objective is to produce an incremental sequence of facility sets F ₁?F ₂?????F _n , where each F _k contains at most k facilities. An incremental facility sequence or an algorithm producing such a sequence is called c -competitive if the cost of each F _k is at most c times the optimum cost of corresponding k-facility location problem, where c is called competitive ratio. In this paper we present two competitive algorithms for this problem. The first algorithm produces competitive ratio 8α, where α is the approximation ratio of k-facility location problem. By recently result (Zhang, Theor. Comput. Sci. 384:126–135, 2007), we obtain the competitive ratio \(16+8\sqrt{3}+\epsilon\). The second algorithm has the competitive ratio Δ+1, where Δ is the ratio between the maximum and minimum nonzero interpoint distances. The latter result has its self interest, specially for the small metric space with Δ≤8α?1.     

Integer programming approach to static monopolies in graphs

A subset M of vertices of a graph is called a static monopoly, if any vertex v outside M has at least \(\lceil \tfrac{1 }{2}\deg (v)\rceil \) neighbors in M. The minimum static monopoly problem has been extensively studied in graph theoretical context. We study this problem from an integer programming point of view for the first time and give a linear formulation for it. We study the facial structure of the corresponding polytope, classify facet defining inequalities of the integer programming formulation and introduce some families of valid inequalities. We show that in the presence of a vertex cut or an edge cut in the graph, the problem can be solved more efficiently by adding some strong valid inequalities. An algorithm is given that solves the minimum monopoly problem in trees and cactus graphs in linear time. We test our methods by performing several experiments on randomly generated graphs. A software package is introduced that solves the minimum monopoly problem using open source integer linear programming solvers.