A Polynomial Time Approximation Scheme for Optimal Product-Requirement Communication Spanning Trees

Approximation algorithms for some optimum communication spanning tree problems

For an undirected graph G¿=¿(V,E), we say that for l,u,v¿¿¿V, l separates u from v if the distance between u and l differs from the distance from v to l. A set of vertices L¿¿¿V is a feasible solution if for every pair of vertices u,v¿¿¿V there is l¿¿¿L that separates u from v. The metric dimension of a graph is the minimum cardinality of such a feasible solution. Here, we extend this well-studied problem to the case where each vertex v has a non-negative cost, and the goal is to find a feasible solution with a minimum total cost. This weighted version is NP-hard since the unweighted variant is known to be NP-hard. We show polynomial time algorithms for the cases where G is a path, a tree, a cycle, a cograph, a k-edge-augmented tree (that is, a tree with additional k edges) for a constant value of k, and a (not necessarily complete) wheel. The results for paths, trees, cycles, and complete wheels extend known polynomial time algorithms for the unweighted version, whereas the other results are the first known polynomial time algorithms for these classes of graphs even for the unweighted version. Next, we extend the set of graph classes for which computing the unweighted metric dimension of a graph is known to be NPC by showing that for split graphs, bipartite graphs, co-bipartite graphs, and line graphs of bipartite graphs, the problem of computing the unweighted metric dimension of the graph is NPC.

For an undirected graph $$G=(V,E)$$G=(V,E), a vertex $$\tau \in V$$źźV separates vertices u and v (where $$u,v\in V$$u,vźV, $$u\ne v$$uźv) if their distances to $$\tau $$ź are not equal. Given an integer parameter $$k \ge 1$$kź1, a set of vertices $$L\subseteq V$$L⊆V is a feasible solution, if for every pair of distinct vertices, u, v, there are at least k distinct vertices $$\tau _{1},\tau _{2},\ldots ,\tau _{k}\in L$$ź1,ź2,ź,źkźL, each separating u and v. Such a feasible solution is called a landmark set, and the k-metric dimension of a graph is the minimal cardinality of a landmark set for the parameter k. The case $$k=1$$k=1 is a classic problem, where in its weighted version, each vertex v has a non-negative cost, and the goal is to find a landmark set with minimal total cost. We generalize the problem for $$k \ge 2$$kź2, introducing two models, and we seek for solutions to both the weighted version and the unweighted version of this more general problem. In the model of all-pairs (AP), k separations are needed for every pair of distinct vertices of V, while in the non-landmarks model (NL), such separations are required only for pairs of distinct vertices in $$V \setminus L$$V\L. We study the weighted and unweighted versions for both models (AP and NL), for path graphs, complete graphs, complete bipartite graphs, and complete wheel graphs, for all values of $$k \ge 2$$kź2. We present algorithms for these cases, thus demonstrating the difference between the two new models, and the differences between the cases $$k=1$$k=1 and $$k \ge 2$$kź2.

Combined column-and-row-generation for the optimal communication spanning tree problem

A central question in computational biology is the design of genetic markers to distinguish between two given sets of (DNA) sequences. This question is formalized as the NP-complete Distinguishing Substring Selection problem (DSSS for short) which asks, given a set of "good" strings and a set of "bad" strings, for a solution string which is, with respect to the Hamming metric, "away" from the good strings and "close" to the bad strings. More precisely, given integers dg, db, and L, we ask for a length-L string s such that s has Hamming distance at least dg to every length-L substring of the good strings and such that every bad string has a length-L substring with Hamming distance at most db to s. Studying the parameterized complexity of DSSS, we show that, already for binary alphabet, DSSS is W[1]-hard with respect to its natural parameters. This, in particular, implies that a recently given polynomial-time approximation scheme (PTAS) by Deng et al. cannot be replaced by a so-called efficient polynomial-time approximation scheme (EPTAS) unless an unlikely collapse in parameterized complexity theory occurs. This is seemingly the first computational biology problem for which such a border between PTAS (which exists) and EPTAS (which is unlikely to exist) could be established. By way of contrast, for a special case of DSSS, we present an exact fixed-parameter algorithm solving the problem efficiently. In this way we also exhibit a sharp border between fixed-parameter tractability and intractability results.

The closest string and substring problems find applications in PCR primer design, genetic probe design, motif finding, and antisense drug design. For their importance, the two problems have been extensively studied recently in computational biology. Unfortunately both problems are NP-complete. Researchers have developed both fixed-parameter algorithms and approximation algorithms for the two problems. In terms of fixed-parameter, when the radius d is the parameter, the best-known fixed-parameter algorithm for closest string has time complexity O(ndd+1), which is still superpolynomial even if d = O(log n). In this paper we provide an O(n|Σ|O(d)) algorithm where Σ is the alphabet. This gives a polynomial time algorithm when d = O(log n) and Σ has constant size. Using the same technique, we additionally provide a more efficient subexponential time algorithm for the closest substring problem. In terms of approximation, both closest string and closest substring problems admit polynomial time approximation schemes (PTAS). The best known time complexity of the PTAS is O(nO(Ɛ-2 log 1/Ɛ)). In this paper we present a PTAS with time complexity O(nO(Ɛ-2)). At last, we prove that a restricted version of the closest substring has the same parameterized complexity as closest substring, answering an open question in the literature.

Given an undirected graph G and two vertex subsets H 1 and H 2, the smallest bi-level augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertex-disjoint paths between every pair of vertices in H 1 and two edgedisjoint paths between every pair of vertices in H 2. We solve the bi-level augmentation problem in O(n + m) time, where n and m are the numbers of vertices and edges in G. Our algorithm can be parallelized to run in O(log2 n) time using n + m processors on an EREW PRAM.KeywordsCritical NodeGraph ProblemVertex SubsetConnectivity RequirementAugmentation ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A tournament T = (V , A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed ϵ > 0, the kernelized instance has at most (2 + ϵ)k vertices. Our result improves the previous known bound of O(k2 ) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k- FAST.

The expansion of the online retailing platform leads to the growth of logistic business. The integration between the main two activities in logistic decision which are inventory control and transportation management. The idea here is to improve overall efficiency in term of total cost of the logistic system. This research studies a multi-retailer multi-depot distribution system with multiple items. Decisions in this system include replenishment quantities of items at each retailer and its corresponding delivery route. The main objective is to provide the minimum total inventory and routing cost during considered time horizon. Three heuristic algorithms, SIOH AIRRH and AIRVH, are proposed to solve this problem. The performances of the heuristics are tested by comparing to solution obtained by IBM ILOG CPLEX 12.6 for small-sized problem and Lot-for-Lot policy for medium and large-sized problem. The computational results show that the proposed heuristics can achieve the optimal solution in small-sized problem. Moreover the heuristic AIRVH show improvement from Lot-for-Lot policy between 15-25% in average for the medium and large-sized problem.

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, which is to determine the probability that, in a undirected graph, assuming each edge fails independently, the remaining graph is still connected. Our main contribution is to confirm a conjecture by Gorodezky and Pak (Random Struct. Algorithms, 2014), that the expected running time of the "cluster-popping" algorithm in bi-directed graphs is bounded by a polynomial in the size of the input.

For a simple graph [Formula: see text] and for a pair of vertices [Formula: see text], we say that a vertex [Formula: see text] resolves [Formula: see text] and [Formula: see text] if the shortest path from [Formula: see text] to [Formula: see text] is of a different length than the shortest path from [Formula: see text] to [Formula: see text]. A set of vertices [Formula: see text] is a resolving set if for every pair of vertices [Formula: see text] and [Formula: see text] in [Formula: see text], there exists a vertex [Formula: see text] that resolves [Formula: see text] and [Formula: see text]. The minimum weight resolving set problem is to find a resolving set [Formula: see text] for a weighted graph [Formula: see text] such that [Formula: see text] is minimum, where [Formula: see text] is the weight of vertex [Formula: see text]. In this paper, we explore the possible solutions of this problem for grid graphs [Formula: see text] where [Formula: see text]. We give a complete characterization of solutions whose cardinalities are 2 or 3, and show that the maximum cardinality of a solution is [Formula: see text]. We show that the grid has the property that given a landmark set, we only need to investigate whether or not all pairs of vertices that share common neighbors are resolved to determine if the whole graph is resolved. We use this result to provide a characterization of a class of minimals whose cardinalities range from [Formula: see text] to [Formula: see text] and show that the number of such minimals is [Formula: see text].

The complexity of minimum convex coloring

A support or realization of a hypergraph $\mathcal{H}$ is a graph \(G\) on the same vertex set as \(\mathcal{H}\) such that for each hyperedge of $\mathcal{H}$ it holds that its vertices induce a connected subgraph of $G$. The NP-hard problem of finding a planar support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins---pairs of vertices that are in precisely the same hyperedges---can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with \(m\) hyperedges to have an \(r\)-outerplanar support, which depends only on \(r\) and \(m\). Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing \(r\)-outerplanar supports for hypergraphs with \(m\) hyperedges if $m$ and $r$ are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters $m$ and $r$.

Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width

We consider an augmentation problem on undirected and directed graphs, where given a directed (an undirected) graph G and p pairs of vertices $$P=\left\{ {\left( {s_1 ,t_1 } \right) ,\ldots ,\left( {s_p ,t_p } \right) } \right\} $$P=s1,t1,ź,sp,tp, 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.