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.     

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.     

Distance domination in graphs with given minimum and maximum degree

For an integer \(k \ge 1\), a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V(G) is at distance at most k from some vertex of S. The distance k-domination number \(\gamma _k(G)\) of G is the minimum cardinality of a distance k-dominating set of G. In this paper, we establish an upper bound on the distance k-domination number of a graph in terms of its order, minimum degree and maximum degree. We prove that for \(k \ge 2\), if G is a connected graph with minimum degree \(\delta \ge 2\) and maximum degree \(\Delta \) and of order \(n \ge \Delta + k - 1\), then \(\gamma _k(G) \le \frac{n + \delta - \Delta }{\delta + k - 1}\). This result improves existing known results.     

On nonlinear multi-covering problems

In this paper we define the exact k-coverage problem, and study it for the special cases of intervals and circular-arcs. Given a set system consisting of a ground set of n points with integer demands \(\{d_0,\dots ,d_{n-1}\}\) and integer rewards, subsets of points, and an integer k, select up to k subsets such that the sum of rewards of the covered points is maximized, where point i is covered if exactly \(d_i\) subsets containing it are selected. Here we study this problem and some related optimization problems. We prove that the exact k-coverage problem with unbounded demands is NP-hard even for intervals on the real line and unit rewards. Our NP-hardness proof uses instances where some of the natural parameters of the problem are unbounded (each of these parameters is linear in the number of points). We show that this property is essential, as if we restrict (at least) one of these parameters to be a constant, then the problem is polynomial time solvable. Our polynomial time algorithms are given for various generalizations of the problem (in the setting where one of the parameters is a constant).     

Equitable colorings of Cartesian products of square of cycles and paths with complete bipartite graphs

A graph G is said to be equitably k-colorable if the vertex set of G can be divided into k independent sets for which any two sets differ in size at most one. The equitable chromatic number of G, \(\chi _{=}(G)\), is the minimum k for which G is equitably k-colorable. The equitable chromatic threshold of G, \(\chi _{=}^{*}(G)\), is the minimum k for which G is equitably \(k'\)-colorable for all \(k'\ge k\). In this paper, the exact values of \(\chi _{=}^{*}(G\Box H)\) and \(\chi _{=}(G\Box H)\) are obtained when G is the square of a cycle or a path and H is a complete bipartite graph.     

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.     

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.     

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.     

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 (\(k>2\)).     

A practical greedy approximation for the directed Steiner tree problem

The directed Steiner tree (DST) NP-hard problem asks, considering a directed weighted graph with n nodes and m arcs, a node r called root and a set of k nodes X called terminals, for a minimum cost directed tree rooted at r spanning X. The best known polynomial approximation ratio for DST is a \(O(k^\varepsilon )\)-approximation greedy algorithm. However, a much faster k-approximation, returning the shortest paths from r to X, is generally used in practice. We give two new algorithms : a fast k-approximation called Greedy\(_\text {FLAC}\) running in \(O(m \log (n)k + \min (m, nk)nk^2)\) and a \(O(\sqrt{k})\)-approximation called Greedy\(_\text {FLAC}^\triangleright \) running in \(O(nm + n^2 \log (n)k +n^2 k^3)\). We provide computational results to show that, Greedy\(_\text {FLAC}\) rivals in practice with the running time of the fast k-approximation and returns solution with smaller cost in practice.     

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 \(\mu +1\).     

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.     

Approximating <Emphasis Type="Italic">k</Emphasis>-generalized connectivity via collapsing HSTs

An instance of the k -generalized connectivity problem consists of an undirected graph G=(V,E), whose edges are associated with non-negative costs, and a collection \({\mathcal{D}}=\{(S_{1},T_{1}),\ldots,(S_{d},T_{d})\}\) of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph ??G connects a demand (S _i ,T _i ) when it contains a path with one endpoint in S _i and the other in T _i . Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in \({\mathcal{D}}\).Alon, Awerbuch, Azar, Buchbinder and Naor (SODA ’04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiple-choice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log?² nlog?² d) approximation due to Chekuri, Even, Gupta and Segev (SODA ’08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA ’06) related its inapproximability to that of dense k -subgraph.In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n ^2/3?polylog(n,k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.     

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.     

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.     

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.     

Socioemotional wealth’s implications in the calculus of the minimum rate of return required by family businesses’ owners

This paper demonstrates that the minimum rate of return (k _e ) required by family business shareholders is inversely related to the emotional endowment presented in these firms. After reviewing the socioemotional wealth (SEW) literature, we find empirical support to justify that different SEW dimensions influence k _e . Findings from a population of 207 family firms show that the identification of family members with the firm and the renewal of family bonds with the firm through dynastic succession have consistently negative impacts on k _e , while family control and influence have significantly positive impacts on k _e .     

On the vertex cover $$P_3$$ problem parameterized by treewidth

Consider a graph G. A subset of vertices, F, is called a vertex cover \(P_t\) (\(VCP_t\)) set if every path of order t contains at least one vertex in F. Finding a minimum \(VCP_t\) set in a graph is is NP-hard for any integer \(t\ge 2\) and is called the \(MVCP_3\) problem. In this paper, we study the parameterized algorithms for the \(MVCP_3\) problem when the underlying graph G is parameterized by the treewidth. Given an n-vertex graph together with its tree decomposition of width at most p, we present an algorithm running in time \(4^{p}\cdot n^{O(1)}\) for the \(MVCP_3\) problem. Moreover, we show that for the \(MVCP_3\) problem on planar graphs, there is a subexponential parameterized algorithm running in time \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) where k is the size of the optimal solution.     

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.     

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.