No Arabic abstract
This study considers the (soft) capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph $G=(V,E)$, which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex $v$ in the cover, the number of $v$s incident edges covered by the copy is up to a given capacity of $v$. We extend Bhattacharya et al.s work [SODA15 and ICALP15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with $O(log n / epsilon)$ amortized update time, where $n$ is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a nonuniform and unsplittable demand, and (2) the more general capacitated set cover problem.
We study the Capacitated k-Median problem, for which all the known constant factor approximation algorithms violate either the number of facilities or the capacities. While the standard LP-relaxation can only be used for algorithms violating one of the two by a factor of at least two, Shi Li [SODA15, SODA16] gave algorithms violating the number of facilities by a factor of 1+{epsilon} exploring properties of extended relaxations. In this paper we develop a constant factor approximation algorithm for Uniform Capacitated k-Median violating only the capacities by a factor of 1+{epsilon}. The algorithm is based on a configuration LP. Unlike in the algorithms violating the number of facilities, we cannot simply open extra few facilities at selected locations. Instead, our algorithm decides about the facility openings in a carefully designed dependent rounding process.
In the Survivable Network Design problem (SNDP), we are given an undirected graph $G(V,E)$ with costs on edges, along with a connectivity requirement $r(u,v)$ for each pair $u,v$ of vertices. The goal is to find a minimum-cost subset $E^*$ of edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version we need to ensure that there are $r(u,v)$ edge-disjoint paths for every pair $u, v$ of vertices, while in the vertex-connectivity version the paths are required to be vertex-disjoint. The edge-connectivity version of SNDP is known to have a 2-approximation. However, no non-trivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an $O(k^3 log n)$-approximation for this problem, where $k$ denotes the maximum connectivity requirement, and $n$ denotes the number of vertices. We also give a simple proof of the recently discovered $O(k^2 log n)$-approximation result for the single-source version of vertex-connectivity SNDP. We note that in both cases, our analysis in fact yields slightly better guarantees in that the $log n$ term in the approximation guarantee can be replaced with a $log tau$ term where $tau$ denotes the number of distinct vertices that participate in one or more pairs with a positive connectivity requirement.
We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless $NP subseteq coNP/poly$, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an $O(2^{O(pk)}n^{O(1)})$-time algorithm for the $p$-Edge-Connected VC and an $O(2^{O(k^2)}n^{O(1)})$-time algorithm for the $p$-Vertex-Connected VC. Finally, we describe a $2(p+1)$-approximation algorithm for the $p$-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning $p$-vertex/edge-connected subgraph of a $p$-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).
We present a massively parallel algorithm, with near-linear memory per machine, that computes a $(2+varepsilon)$-approximation of minimum-weight vertex cover in $O(loglog d)$ rounds, where $d$ is the average degree of the input graph. Our result fills the key remaining gap in the state-of-the-art MPC algorithms for vertex cover and matching problems; two classic optimization problems, which are duals of each other. Concretely, a recent line of work---by Czumaj et al. [STOC18], Ghaffari et al. [PODC18], Assadi et al. [SODA19], and Gamlath et al. [PODC19]---provides $O(loglog n)$ time algorithms for $(1+varepsilon)$-approximate maximum weight matching as well as for $(2+varepsilon)$-approximate minimum cardinality vertex cover. However, the latter algorithm does not work for the general weighted case of vertex cover, for which the best known algorithm remained at $O(log n)$ time complexity.
We present a local algorithm (constant-time distributed algorithm) for finding a 3-approximate vertex cover in bounded-degree graphs. The algorithm is deterministic, and no auxiliary information besides port numbering is required.