Capacitated Vertex Cover Research Articles

XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure

AbstractIn this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna (ACM Trans Comput Theory 9:1–36, 2018), that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, Capacitated Vertex Cover and Bipartite Bandwidth.

Essentially Tight Kernels for (Weakly) Closed Graphs

We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number c and weak closure number gamma (Fox et al. SIAM J Comput 49(2):448–464, 2020) in addition to the standard parameter solution size k. The weak closure number gamma of a graph is upper-bounded by the minimum of its closure number c and its degeneracy d. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size k^{mathcal {O}(gamma )}, k^{mathcal {O}(gamma )}, and (gamma k)^{mathcal {O}(gamma )}, respectively. This extends previous results on the kernelization of these problems on degenerate graphs. These kernels are essentially tight as these problems are unlikely to admit kernels of size k^{o(gamma )} by previous results on their kernelization complexity on degenerate graphs (Cygan et al. ACM Trans Algorithms 13(3):43:1–43:22, 2017). For Capacitated Vertex Cover, we show that even a kernel of size k^{o(c)} is unlikely. In contrast, for Connected Vertex Cover, we obtain a kernel with mathcal {O}(ck^2) vertices. Moreover, we prove that searching for an induced subgraph of order at least k belonging to a hereditary graph class mathcal {G} admits a kernel of size k^{mathcal {O}(gamma )} when mathcal {G} contains all complete and all edgeless graphs. Finally, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number c and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.

Algorithmic Applications of Tree-Cut Width

The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper, we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems that are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions and provide FPT algorithms for the showcase problems Capacitated Vertex Cover, Capacitated Dominating Set, and Imbalance parameterized by the tree-cut width of an input graph. On the other hand, we show that List Coloring, Precoloring Extension, and Boolean CSP (the last parameterized by the tree-cut width of the incidence graph) are W[1]-hard and hence unlikely to be FPT when parameterized by tree-cut width.

Self-Stabilizing Capacitated Vertex Cover Algorithms for Internet-of-Things-Enabled Wireless Sensor Networks.

Wireless sensor networks (WSNs) achieving environmental sensing are fundamental communication layer technologies in the Internet of Things. Battery-powered sensor nodes may face many problems, such as battery drain and software problems. Therefore, the utilization of self-stabilization, which is one of the fault-tolerance techniques, brings the network back to its legitimate state when the topology is changed due to node leaves. In this technique, a scheduler decides on which nodes could execute their rules regarding spatial and temporal properties. A useful graph theoretical structure is the vertex cover that can be utilized in various WSN applications such as routing, clustering, replica placement and link monitoring. A capacitated vertex cover is the generalized version of the problem which restricts the number of edges covered by a vertex by applying a capacity constraint to limit the covered edge count. In this paper, we propose two self-stabilizing capacitated vertex cover algorithms for WSNs. To the best of our knowledge, these algorithms are the first attempts in this manner. The first algorithm is stabilized under an unfair distributed scheduler (that is, the scheduler which does not grant all enabled nodes to make their moves but guarantees the global progress of the system) at most step, where n is the count of nodes. The second algorithm assumes 2-hop (degree 2) knowledge about the network and runs under the unfair scheduler, which subsumes the synchronous and distributed fair scheduler and stabilizes itself after moves in step, which is acceptable for most WSN setups. We theoretically analyze the algorithms to provide proof of correctness and their step complexities. Moreover, we provide simulation setups by applying IRIS sensor node parameters and compare our algorithms with their counterparts. The gathered measurements from the simulations revealed that the proposed algorithms are faster than their competitors, use less energy and offer better vertex cover solutions.

O(f) Bi-criteria Approximation for Capacitated Covering with Hard Capacities

We consider capacitated vertex cover with hard capacity constraints (VC-HC) on hypergraphs. In this problem we are given a hypergraph $$G=(V,E)$$ with a maximum edge size f. Each edge is associated with a demand and each vertex is associated with a weight (cost), a capacity, and an available multiplicity. The objective is to find a minimum-weight vertex multiset such that the demands of the edges can be covered by the capacities of the vertices and the multiplicity of each vertex does not exceed its available multiplicity. In this paper we present an O(f) bi-criteria approximation for VC-HC that gives a trade-off on the number of augmented multiplicity and the cost of the resulting cover. In particular, we show that, by augmenting the available multiplicity by a factor of $$k \ge 2$$ , a cover with a cost ratio of $$\left( 1+\frac{1}{k-1}\right) (f-1)$$ to the optimal cover for the original instance can be obtained. This improves over the previously best known guarantee, which has a cost ratio of $$f^2$$ via augmenting the available multiplicity by a factor of f.

Approximation of Partial Capacitated Vertex Cover

We study the partial capacitated vertex cover problem (PCVC) in which the input consists of a graph $G$ and a covering requirement $L$. Each edge $e$ in $G$ is associated with a demand (or load) $\ell(e)$, and each vertex $v$ is associated with a (soft) capacity $c(v)$ and a weight $w(v)$. A feasible solution is an assignment of edges to vertices such that the total demand of assigned edges is at least $L$. The weight of a solution is $\sum_{v}\alpha(v)w(v)$, where $\alpha(v)$ is the number of copies of $v$ required to cover the demand of the edges that are assigned to $v$. The goal is to find a solution of minimum weight. We consider three variants of PCVC. In PCVC with separable demands the only requirement is that the total demand of edges assigned to $v$ is at most $\alpha(v)c(v)$. In PCVC with inseparable demands there is an additional requirement that if an edge is assigned to $v$, then it must be assigned to one of its copies. The third variant is the unit demands version. We present 3-approximation algorithms for both PCVC with separable demands and PCVC with inseparable demands. We also present a 2-approximation algorithm for PCVC with unit demands. We show that similar results can be obtained for PCVC in hypergraphs and for the prize collecting version of capacitated vertex cover. Our algorithms are based on a unified approach for designing and analyzing approximation algorithms for capacitated covering problems. This approach yields simple algorithms whose analyses rely on the local ratio technique and sophisticated charging schemes.

A Primal-Dual Bicriteria Distributed Algorithm for Capacitated Vertex Cover

In this paper we consider the capacitated vertex cover problem, which is the variant of vertex cover where each node is allowed to cover a limited number of edges. We present an efficient, deterministic, distributed approximation algorithm for the problem. Our algorithm computes a $(2+\epsilon)$-approximate solution which violates the capacity constraints by a factor of $(4+\epsilon)$ in a polylogarithmic number of communication rounds. On the other hand, we also show that every efficient distributed approximation algorithm for this problem must violate the capacity constraints. Our result is achieved in two steps. We first develop a 2-approximate, sequential primal-dual algorithm that violates the capacity constraints by a factor of 2. Subsequently, we present a distributed version of this algorithm. We demonstrate that the sequential algorithm has an inherent need for synchronization which forces any naive distributed implementation to use a linear number of communication rounds. The challenge in this step is therefore to achieve a reduction of the communication complexity to a polylogarithmic number of rounds without worsening the approximation guarantee.

A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k u . A solution consists of a function x:V→ℕ0 and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x u k u . Our objective is to find a cover that minimizes ∑v∈V x v w v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time.

Parameterized Complexity of Vertex Cover Variants

Important variants of theVERTEX COVER problem (among others, CONNECTED VERTEX COVER, CAPACITATED VERTEX COVER, and MAXIMUM PARTIAL VERTEX COVER) have been intensively studied in terms of polynomial-time approximability. By way of contrast, their parameterized complexity has so far been completely open. We close this gap here by showing that, with the size of the desired vertex cover as the parameter, CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER are both fixed-parameter tractable while MAXIMUM PARTIAL VERTEX COVER is W[1]-complete. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered variants of VERTEX COVER behave very similar in terms of constant factor approximability, they display a wide range of different characteristics when investigating their parameterized complexities.

Dependent rounding and its applications to approximation algorithms

We develop a new randomized rounding approach for fractional vectors defined on the edge-sets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include:---low congestion multi-path routing;---richer random-graph models for graphs with a given degree-sequence;---improved approximation algorithms for: (i) throughput-maximization in broadcast scheduling, (ii) delay-minimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and---fair scheduling of jobs on unrelated parallel machines.

An improved approximation algorithm for vertex cover with hard capacities

We study the capacitated vertex cover problem, a generalization of the well-known vertex-cover problem. Given a graph G = ( V , E ) , the goal is to cover all the edges by picking a minimum cover using the vertices. When we pick a vertex, we can cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is clearly NP-hard as it generalizes the well-known vertex-cover problem. Previously, approximation algorithms with an approximation factor of 2 were developed with the assumption that an arbitrary number of copies of a vertex may be chosen in the cover. If we are allowed to pick at most a fixed number of copies of each vertex, the approximation algorithm becomes much more complex. Chuzhoy and Naor (FOCS, 2002) have shown that the weighted version of this problem is at least as hard as set cover; in addition, they developed a 3-approximation algorithm for the unweighted version. We give a 2-approximation algorithm for the unweighted version, improving the Chuzhoy–Naor bound of three and matching (up to lower-order terms) the best approximation ratio known for the vertex-cover problem.

Capacitated vertex covering

In this paper we study the capacitated vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G=( V, E) with weights on the vertices, the goal is to cover all the edges by picking a cover of minimum weight from the vertices. When we pick a copy of a vertex, we pay the weight of the vertex and cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is NP-hard. We give a primal–dual based approximation algorithm with an approximation guarantee of 2, and study several generalizations, as well as the problem restricted to trees.