Approximation of knapsack problems with conflict and forcing graphs

We study the classical 0–1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is the use of conflict (resp. forcing) graphs. By modifying a recent result of Lokstanov et al. (Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 570–581, 2014) we derive a fairly complicated FPTAS for the knapsack problem on weakly chordal conflict graphs. Next, we show that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs. In particular, we can show that every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph. We point out a number of structural results from the literature which can be used to show the existence of an FPTAS for several graph classes characterized by the exclusion of certain induced subgraphs. Finally, a PTAS for the knapsack problem with H-minor free conflict graph is derived. This includes planar graphs and, more general, graphs of bounded genus. The PTAS is obtained by expanding a general result of Demaine et al. (Proceedings of 46th annual IEEE symposium on foundations of computer science, FOCS 2005, pp 637–646, 2005). The knapsack problem with forcing graphs can be transformed into a minimization knapsack problem with conflict graphs. It follows immediately that all our FPTAS results of the current and a previous paper carry over from conflict graphs to forcing graphs. In contrast, the forcing graph variant is already inapproximable on planar graphs.     

Improving robustness of next-hop routing

A weakness of next-hop routing is that following a link or router failure there may be no routes between some source-destination pairs, or packets may get stuck in a routing loop as the protocol operates to establish new routes. In this article, we address these weaknesses by describing mechanisms to choose alternate next hops. Our first contribution is to model the scenario as the following tree augmentation problem. Consider a mixed graph where some edges are directed and some undirected. The directed edges form a spanning tree pointing towards the common destination node. Each directed edge represents the unique next hop in the routing protocol. Our goal is to direct the undirected edges so that the resulting graph remains acyclic and the number of nodes with outdegree two or more is maximized. These nodes represent those with alternative next hops in their routing paths. We show that tree augmentation is NP-hard in general and present a simple \(\frac{1}{2}\)-approximation algorithm. We also study 3 special cases. We give exact polynomial-time algorithms for when the input spanning tree consists of exactly 2 directed paths or when the input graph has bounded treewidth. For planar graphs, we present a polynomial-time approximation scheme when the input tree is a breadth-first search tree. To the best of our knowledge, tree augmentation has not been previously studied.     

Degree-constrained orientations of embedded graphs

We investigate the problem of orienting the edges of an embedded graph in such a way that the resulting digraph fulfills given in-degree specifications both for the vertices and for the faces of the embedding. This primal-dual orientation problem was first proposed by Frank for the case of planar graphs, in conjunction with the question for a good characterization of the existence of such orientations. We answer this question by showing that a feasible orientation of a planar embedding, if it exists, can be constructed by combining certain parts of a primally feasible orientation and a dually feasible orientation. For the general case of arbitrary embeddings, we show that the number of feasible orientations is bounded by \(2^{2g}\), where \(g\) is the genus of the embedding. Our proof also yields a fixed-parameter algorithm for determining all feasible orientations parameterized by the genus. In contrast to these positive results, however, we also show that the problem becomes \(N\!P\)-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.     

Fast searching games on graphs

Given a graph, suppose that intruders hide on vertices or along edges of the graph. The fast searching problem is to find the minimum number of searchers required to capture all the intruders satisfying the constraint that every edge is traversed exactly once and searchers are not allowed to jump. In this paper, we prove lower bounds on the fast search number. We present a linear time algorithm to compute the fast search number of Halin graphs and their extensions. We present a quadratic time algorithm to compute the fast search number of cubic graphs.     

Partitioning 2-edge-colored complete multipartite graphs into monochromatic cycles, paths and trees

In this paper we consider the problem of partitioning complete multipartite graphs with edges colored by 2 colors into the minimum number of vertex disjoint monochromatic cycles, paths and trees, respectively. For general graphs we simply address the decision version of these three problems the 2-PGMC, 2-PGMP and 2-PGMT problems, respectively. We show that both 2-PGMC and 2-PGMP problems are NP-complete for complete multipartite graphs and the 2-PGMT problem is NP-complete for bipartite graphs. This also implies that all these three problems are NP-complete for general graphs, which solves a question proposed by the authors in a previous paper. Nevertheless, we show that the 2-PGMT problem can be solved in polynomial time for complete multipartite graphs. Research supported by NSFC.     

Critical edges/nodes for the minimum spanning tree problem: complexity and approximation

In this paper, we study the complexity and the approximation of the k most vital edges (nodes) and min edge (node) blocker versions for the minimum spanning tree problem (MST). We show that the k most vital edges MST problem is NP-hard even for complete graphs with weights 0 or 1 and 3-approximable for graphs with weights 0 or 1. We also prove that the k most vital nodes MST problem is not approximable within a factor n ^1?? , for any ?>0, unless NP=ZPP, even for complete graphs of order n with weights 0 or 1. Furthermore, we show that the min edge blocker MST problem is NP-hard even for complete graphs with weights 0 or 1 and that the min node blocker MST problem is NP-hard to approximate within a factor 1.36 even for graphs with weights 0 or 1.     

The densest <Emphasis Type="Italic">k</Emphasis>-subgraph problem on clique graphs

The Densest k-Subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The problem is strongly NP-hard, as a generalization of the well known Clique problem and we also know that it does not admit a Polynomial Time Approximation Scheme (PTAS). In this paper we focus on special cases of the problem, with respect to the class of the input graph. Especially, towards the elucidation of the open questions concerning the complexity of the problem for interval graphs as well as its approximability for chordal graphs, we consider graphs having special clique graphs. We present a PTAS for stars of cliques and a dynamic programming algorithm for trees of cliques. M.L. is co-financed within Op. Education by the ESF (European Social Fund) and National Resources. V.Z. is partially supported by the Special Research Grants Account of the University of Athens under Grant 70/4/5821.     

On the max-weight edge coloring problem

We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges, so that the total weight of the partition is minimized. We explore the frontier between polynomial and NP-hard variants of the problem, with respect to the class of the underlying graph, as well as the approximability of NP-hard variants. In particular, we present polynomial algorithms for bounded degree trees and star of chains, as well as an approximation algorithm for bipartite graphs of maximum degree at most twelve which beats the best known approximation ratios.     

New analysis and computational study for the planar connected dominating set problem

The connected dominating set (CDS) problem is a well studied NP-hard problem with many important applications. Dorn et al. (Algorithmica 58:790–810 2010) introduce a branch-decomposition based algorithm design technique for NP-hard problems in planar graphs and give an algorithm (DPBF algorithm) which solves the planar CDS problem in \(O(2^{9.822\sqrt{n}}n+n^3)\) time and \(O(2^{8.11\sqrt{n}}n+n^3)\) time, with a conventional method and fast matrix multiplication in the dynamic programming step of the algorithm, respectively. We show that DPBF algorithm solves the planar CDS problem in \(O(2^{9.8\sqrt{n}}n+n^3)\) time with a conventional method and in \(O(2^{8.08\sqrt{n}}n+n^3)\) time with a fast matrix multiplication. For a graph \(G\), let \({\hbox {bw}}(G)\) be the branchwidth of \(G\) and \(\gamma _c(G)\) be the connected dominating number of \(G\). We prove \({\hbox {bw}}(G)\le 2\sqrt{10\gamma _c(G)}+32\). From this result, the planar CDS problem admits an \(O(2^{23.54\sqrt{\gamma _c(G)}}\gamma _c(G)+n^3)\) time fixed-parameter algorithm. We report computational study results on the practical performance of DPBF algorithm, which show that the size of instances can be solved by the algorithm mainly depends on the branchwidth of the instances, coinciding with the theoretical analysis. For graphs with small or moderate branchwidth, the CDS problem instances with size up to a few thousands edges can be solved in a practical time and memory space.     

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .     

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.     

Visual cryptography on graphs

In this paper, we consider a new visual cryptography scheme that allows for sharing of multiple secret images on graphs: we are given an arbitrary graph (V,E) where every node and every edge are assigned an arbitrary image. Images on the vertices are “public” and images on the edges are “secret”. The problem that we are considering is how to make a construction such that when the encoded images of two adjacent vertices are printed on transparencies and overlapped, the secret image corresponding to the edge is revealed. We define the most stringent security guarantees for this problem (perfect secrecy) and show a general construction for all graphs where the cost (in terms of pixel expansion and contrast of the images) is proportional to the chromatic number of the cube of the underlying graph. For the case of bounded degree graphs, this gives us constant-factor pixel expansion and contrast. This compares favorably to previous works, where pixel expansion and contrast are proportional to the number of images.     

A note on the minimum number of choosability of planar graphs

The problem of minimum number of choosability of graphs was first introduced by Vizing. It appears in some practical problems when concerning frequency assignment. In this paper, we study two important list coloring, list edge coloring and list total coloring. We prove that \(\chi '_{l}(G)=\varDelta \) and \(\chi ''_{l}(G)=\varDelta +1\) for planar graphs with \(\varDelta \ge 8\) and without adjacent 4-cycles.     

Improved approximation for spanning star forest in dense graphs

A spanning subgraph of a graph G is called a spanning star forest of G if it is a collection of node-disjoint trees of depth at most 1. The size of a spanning star forest is the number of leaves in all its components. The goal of the spanning star forest problem is to find the maximum-size spanning star forest of a given graph. In this paper, we study the spanning star forest problem on c-dense graphs, where for any fixed c∈(0,1), a graph of n vertices is called c-dense if it contains at least cn ²/2 edges. We design a $(\alpha+(1-\alpha)\sqrt{c}-\epsilon)$ -approximation algorithm for spanning star forest in c-dense graphs for any ?>0, where $\alpha=\frac{193}{240}$ is the best known approximation ratio of the spanning star forest problem in general graphs. Thus, our approximation ratio outperforms the best known bound for this problem when dealing with c-dense graphs. We also prove that, for any constant c∈(0,1), approximating spanning star forest in c-dense graphs is APX-hard. We then demonstrate that for weighted versions (both node- and edge-weighted) of this problem, we cannot get any approximation algorithm with strictly better performance guarantee on c-dense graphs than on general graphs. Finally, we give strong inapproximability results for a closely related problem, namely the minimum dominating set problem, restricted on c-dense graphs.     

The Travelling Salesman Problem on Permuted Monge Matrices

We consider traveling salesman problems (TSPs) with a permuted Monge matrix as cost matrix where the associated patching graph has a specially simple structure: a multistar, a multitree or a planar graph. In the case of multistars, we give a complete, concise and simplified presentation of Gaikov's theory. These results are then used for designing an O(m3 + mn) algorithm in the case of multitrees, where n is the number of cities and m is the number of subtours in an optimal assignment. Moreover we show that for planar patching graphs, the problem of finding an optimal subtour patching remains NP-complete.     

Augmenting weighted graphs to establish directed point-to-point connectivity

We consider an augmentation problem on undirected and directed graphs, where given a directed (an undirected) graph G and p pairs of vertices \(P=\left\{ {\left( {s_1 ,t_1 } \right) ,\ldots ,\left( {s_p ,t_p } \right) } \right\} \), one has to find the minimum weight set of arcs (edges) to be added to the graph so that the resulting graph has (can be oriented to have) directed paths between the specified pairs of vertices. In the undirected case, we present an FPT-algorithm with respect to the number of new edges. Also, we have implemented and evaluated the algorithm on some real-world networks to show its efficiency in decreasing the size of input graphs and converting them to much smaller kernels. In the directed case, we consider the complexity of the problem with respect to the various parameters and present some parameterized algorithms and parameterized complexity results for it.     

Balancing Inventory and Stockout Risk in Retail Supply Chains Using Fast‐Ship

In this study, we consider a two‐retailer, one‐supplier supply chain in which retailers satisfy excess demand by offering to directly ship out‐of‐stock items on an expedited basis at no extra cost to customers. This practice is referred to as the fast‐ship option. We consider two scenarios along with the fast‐ship option. In the first scenario, retailers transship when out of stock, whereas in the second scenario, they do not. If they do not transship, some customers may perform the search on their own. In each scenario, the wholesale prices are either exogenous, or chosen endogenously by the supplier. For both cases, we determine the supplier's and the retailers’ optimal decisions. The key research question we ask and answer is the following: which of the two scenarios is preferred by either player when all decisions are made optimally? We show that when fewer customers are willing to search on their own and wholesale prices are exogenous, both the supplier and the retailers prefer to transship. In addition, the decision maker in a centralized chain will have the exact same preference as that of players in a decentralized setting when the retailers’ and the supplier's preferences coincide and wholesale prices are exogenous. This preference concordance does not hold if wholesale prices are endogenous.     

Coloring vertices of claw-free graphs in three colors

We study the computational complexity of the vertex 3-colorability problem in the class of claw-free graphs. Both the problem and the class received much attention in the literature, separately of each other. However, very little is known about the 3-colorability problem restricted to the class of claw-free graphs beyond the fact the problem is NP-complete under this restriction. In this paper we first strengthen this negative fact by revealing various further restrictions under which the problem remains NP-complete. Then we derive a number of positive results that deal with polynomially solvable cases of the problem in the class of claw-free graphs.     

The minimum vulnerability problem on specific graph classes

Suppose that each edge e of an undirected graph G is associated with three nonnegative integers \(\mathsf{cost}(e)\), \(\mathsf{vul}(e)\) and \(\mathsf{cap}(e)\), called the cost, vulnerability and capacity of e, respectively. Then, we consider the problem of finding \(k\) paths in G between two prescribed vertices with the minimum total cost; each edge e can be shared without any cost by at most \(\mathsf{vul}(e)\) paths, and can be shared by more than \(\mathsf{vul}(e)\) paths if we pay \(\mathsf{cost}(e)\), but cannot be shared by more than \(\mathsf{cap}(e)\) paths even if we pay the cost for e. This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number \(k\) of required paths.     

An optimal square coloring of planar graphs

The square coloring of a graph is to color the vertices of a graph at distance at most 2 with different colors. In 1977, Wegner posed a conjecture on square coloring of planar graphs. The conjecture is still open. In this paper, we prove that Wegner’s conjecture is true for planar graphs with girth at least?6.