Online Constrained Forest and Prize-Collecting Network Design
Jiawei Qian, Seeun William Umboh, David P. Williamson
备注：
A preliminary version [20] appeared in ICALP 2011 but was later discovered to be flawed. This paper presents a corrected version of the algorithm and proofs, and a generalisation of the prize-collecting problem considered in [20]
In this paper, we study a very general type of online network design problem, and generalize two different previous algorithms, one for an online network design problem due to Berman and Coulston [4] and one for (offline) general network design problems due to Goemans and Williamson [9]; we give an O(log k)-competitive algorithm, where k is the number of nodes that must be connected. We also consider a further generalization of the problem that allows us to pay penalties in exchange for violating connectivity constraints; we give an online O(log k)-competitive algorithm for this case as well.
We present a 1.91457-approximation algorithm for the prize-collecting travelling salesman problem. This is obtained by combining a randomized variant of a rounding algorithm of Bienstock et al. and a primal-dual algorithm of Goemans and Williamson.
We obtain polynomial-time approximation-preserving reductions (up to a factor of 1 + ε) from the prize-collecting Steiner tree and prize-collecting Steiner forest problems in planar graphs to the corresponding problems in graphs of bounded treewidth. We also give an exact algorithm for the prize-collecting Steiner tree problem that runs in polynomial time for graphs of bounded treewidth. This, combined with our reductions, yields a PTAS for the prize-collecting Steiner tree problem in planar graphs and generalizes the PTAS of Borradaile, Klein and Mathieu for the Steiner tree problem in planar…
We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a ($2.88 + ε$)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest…
In this paper, we consider Steiner forest and its generalizations, prize-collecting Steiner forest and k-Steiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomial-time approximation scheme (PTAS) of Borradaile et al. [12] for the Euclidean Steiner forest problem. This is done by proving a new structural property and modifying the dynamic programming by adding a new piece of information to each dynamic programming state. Next we develop a PTAS for a well-motivated case, i…
In the network activation problem, each edge in a graph is associated with an activation function, that decides whether the edge is activated from node-weights assigned to its end-nodes. The feasible solutions of the problem are the node-weights such that the activated edges form graphs of required connectivity, and the objective is to find a feasible solution minimizing its total weight. In this paper, we consider a prize-collecting version of the network activation problem, and present first non- trivial approximation algorithms. Our algorithms are based on a new LP relaxation of the problem…
In this paper, we study locating charging station problem as facility location problem and its variants ($k$-Median, $k$-Facility location and $k$-center). We study the connectivity and the capacity constraints in these problem. Capacity and connectivity constraints have been studied in the literature separately for all these problems. We give first constant factor approximations when both the constraints are present. Extending/modifying the techniques used for connected variants of the problem to include capacities or for capacitated variants of problem to include connectivity is a tedious an…
In the node-weighted prize-collecting Steiner tree problem (NW-PCST) we are given an undirected graph $G=(V,E)$, non-negative costs $c(v)$ and penalties $π(v)$ for each $v \in V$. The goal is to find a tree $T$ that minimizes the total cost of the vertices spanned by $T$ plus the total penalty of vertices not in $T$. This problem is well-known to be set-cover hard to approximate. Moss and Rabani (STOC'01) presented a primal-dual Lagrangean-multiplier-preserving $O(\ln |V|)$-approximation algorithm for this problem. We show a serious problem with the algorithm, and present a new, fundamentally …
In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF) and more generally Submodular Prize-Collecting Steiner Forest (SPCSF) on planar graphs (and more generally bounded-genus graphs) to the same problems on graphs of bounded treewidth. More precisely, we show any $α$-approximation algorithm for these problems on graphs of bounded treewidth gives an $(α+ ε)$-approximation algorithm for these problems on planar graphs (and more generally bounded-genus graphs), for any constant $ε>…
We present a unified polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space, whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be connected for PCSTP. Before our work, even a QPTAS fo…
We consider the rooted prize-collecting walks (PCW) problem, wherein we seek a collection $C$ of rooted walks having minimum prize-collecting cost, which is the (total cost of walks in $C$) + (total node-reward of nodes not visited by any walk in $C$). This problem arises naturally as the Lagrangian relaxation of both orienteering, where we seek a length-bounded walk of maximum reward, and the $\ell$-stroll problem, where we seek a minimum-length walk covering at least $\ell$ nodes. Our main contribution is to devise a simple, combinatorial algorithm for the PCW problem in directed graphs that…
The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2-1/(n-1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n^3 log n) time. it applies the primal-dual scheme once for each of the n vertices of the graph. Johnson, Minkoff and Phillips proposed a faster implementation of Goemans and Williamson's algorithm. We give a proof that the approximation ratio of this implementation is exactly 2.
We consider the $k$-prize-collecting Steiner tree problem. An instance is composed of an integer $k$ and a graph $G$ with costs on edges and penalties on vertices. The objective is to find a tree spanning at least $k$ vertices which minimizes the cost of the edges in the tree plus the penalties of vertices not in the tree. This is one of the most fundamental network design problems and is a common generalization of the prize-collecting Steiner tree and the $k$-minimum spanning tree problems. Our main result is a 2-approximation algorithm, which improves on the currently best known approximatio…
We study the behavior of an algorithm derived from the cavity method for the Prize-Collecting Steiner Tree (PCST) problem on graphs. The algorithm is based on the zero temperature limit of the cavity equations and as such is formally simple (a fixed point equation resolved by iteration) and distributed (parallelizable). We provide a detailed comparison with state-of-the-art algorithms on a wide range of existing benchmarks networks and random graphs. Specifically, we consider an enhanced derivative of the Goemans-Williamson heuristics and the DHEA solver, a Branch and Cut Linear/Integer Progra…
In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph $G=(V,E)$, edge costs $\{c_e\geq 0\}_{e\in E}$, terminal pairs $\{(s_i,t_i)\}_{i=1}^k$, and penalties $\{π_i\}_{i=1}^k$ for each terminal pair; the goal is to find a forest $F$ to minimize $c(F)+\sum_{i: (s_i,t_i)\text{ not connected in }F}π_i$. The Steiner forest problem can be viewed as the special case where $π_i=\infty$ for all $i$. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF is at most 2. We dispel this belief by showing …
We consider a multistage framework introduced recently where, given a time horizon t=1,2,...,T, the input is a sequence of instances of a (static) combinatorial optimization problem I_1,I_2,...,I_T, (one for each time step), and the goal is to find a sequence of solutions S_1,S_2,...,S_T (one for each time step) reaching a tradeoff between the quality of the solutions in each time step and the stability/similarity of the solutions in consecutive time steps. For several polynomial-time solvable problems, such as Minimum Cost Perfect Matching, the multistage variant becomes hard to approximate (…
We consider a variant of the prize collecting Steiner tree problem in which we are given a \emph{directed graph} $D=(V,A)$, a monotone submodular prize function $p:2^V \rightarrow \mathbb{R}^+ \cup \{0\}$, a cost function $c:V \rightarrow \mathbb{Z}^{+}$, a root vertex $r \in V$, and a budget $B$. The aim is to find an out-subtree $T$ of $D$ rooted at $r$ that costs at most $B$ and maximizes the prize function. We call this problem \emph{Directed Rooted Submodular Tree} (\textbf{DRSO}). Very recently, Ghuge and Nagarajan [SODA\ 2020] gave an optimal quasi-polynomial-time $O\left(\frac{\log n'}…
Moss and Rabani[12] study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(log n)-approximation algorithm for the prize-collecting node-weighted Steiner tree problem (PCST). They use the algorithm for PCST to obtain a bicriteria (2, O(log n))-approximation algorithm for the Budgeted node-weighted Steiner tree problem. Their solution may cost up to twice the budget, but collects a factor Omega(1/log n) of the optimal prize. We improve these results from at least two aspects. Our firs…
We study the problem of finding a minimum weight connected subgraph spanning at least $k$ vertices on planar, node-weighted graphs. We give a $(4+\eps)$-approximation algorithm for this problem. We achieve this by utilizing the recent LMP primal-dual $3$-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an approach by Chudak et al. (Math.\ Prog.\ '04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013)…
We design and analyse approximation algorithms for the minimum-cost connected T-join problem: given an undirected graph G = (V;E) with nonnegative costs on the edges, and a subset of nodes T, find (if it exists) a spanning connected subgraph H of minimum cost such that every node in T has odd degree and every node not in T has even degree; H may have multiple copies of any edge of G. Two well-known special cases are the TSP (|T| = 0) and the s-t path TSP (|T| = 2). Recently, An, Kleinberg, and Shmoys [STOC 2012] improved on the long-standing 5/3-approximation guarantee for the latter problem a…
This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal-dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set …
We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held…
We present a unified framework for minimizing average completion time for many seemingly disparate online scheduling problems, such as the traveling repairperson problems (TRP), dial-a-ride problems (DARP), and scheduling on unrelated machines. We construct a simple algorithm that handles all these scheduling problems, by computing and later executing auxiliary schedules, each optimizing a certain function on already seen prefix of the input. The optimized function resembles a prize-collecting variant of the original scheduling problem. By a careful analysis of the interplay between these auxi…
We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our wo…
The Team Orienteering Problem (TOP) is an NP-hard routing problem in which a fleet of identical vehicles aims at collecting rewards (prizes) available at given locations, while satisfying restrictions on the travel times. In TOP, each location can be visited by at most one vehicle, and the goal is to maximize the total sum of rewards collected by the vehicles within a given time limit. In this paper, we propose a generalization of TOP, namely the Steiner Team Orienteering Problem (STOP). In STOP, we provide, additionally, a subset of mandatory locations. In this sense, STOP also aims at maximi…
We consider various {\em multi-vehicle versions of the minimum latency problem}. There is a fleet of $k$ vehicles located at one or more depot nodes, and we seek a collection of routes for these vehicles that visit all nodes so as to minimize the total latency incurred, which is the sum of the client waiting times. We obtain an $8.497$-approximation for the version where vehicles may be located at multiple depots and a $7.183$-approximation for the version where all vehicles are located at the same depot, both of which are the first improvements on this problem in a decade. Perhaps more signif…
The Belief Propagation approximation, or cavity method, has been recently applied to several combinatorial optimization problems in its zero-temperature implementation, the max-sum algorithm. In particular, recent developments to solve the edge-disjoint paths problem and the prize-collecting Steiner tree problem on graphs have shown remarkable results for several classes of graphs and for benchmark instances. Here we propose a generalization of these techniques for two variants of the Steiner trees packing problem where multiple "interacting" trees have to be sought within a given graph. Depen…
