Approximation Algorithms for Directed Weighted Spanners

Group parking permit problems

Route planning based on user’s preferences and Points of Interests (POIs) is one of the most popular applications of Location-Based Services (LBS). Variants of route planning consider distance constraints (e.g., the maximum length of the route), origin constraints (e.g., a set of possible starting locations of the route), and category constraints (e.g., a multiset of POI categories that the route must visit). However, the problem of deciding whether a route exists that visits all required POI categories under the distance constraint is known to be NP-hard. Assuming P ≠ NP, this means that there is no efficient (polynomial time) solution to find such paths. Recently, approximate algorithms have been proposed for searching for such a path. This demonstration leverages several of these algorithms to provide a web-based system with a graphical user interface (UI) which allows the users to find a path that: (a) satisfies a distance limit; (b) generates a route to visit a list of POIs, based on the user’s preferred categories; (c) provides a set of hotels (as possible starting locations of the path). If the approximate search algorithms are able to find such a path, it will be displayed on a Mapbox-based map interface that shows: (1) all POIs on a path and (2) alternative paths if any were found. The system then allows a user to explore the returned paths, select a path, or refine their constraints. Moreover, the system allows the users to select which approximate algorithm they would prefer to execute.

We study a generalized busy-time scheduling model on heterogeneous machines. The input to the model includes a set of jobs and a set of machine types. Each job has a size and a time interval during which it should be processed. Each job is to be placed on a machine for execution. Different types of machines have distinct capacities and cost rates. The total size of the jobs running on a machine must always be kept within the machine's capacity, giving rise to placement restrictions for jobs of various sizes among the machine types. Each machine used is charged according to the time duration in which it is busy, i.e., it is processing jobs. The objective is to schedule the jobs into machines to minimize the total cost of all the machines used. We develop an <inline-formula><tex-math notation="LaTeX">$O(1)$</tex-math></inline-formula> -approximation algorithm in the offline setting and an <inline-formula><tex-math notation="LaTeX">$O(\mu)$</tex-math></inline-formula> -competitive algorithm in the online setting (where <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> is the max/min job length ratio), both of which are asymptotically optimal. This article significantly improves the analysis of the algorithms over our preliminary work.

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 work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings and results in randomized $O(\log n)$-competitive algorithms, where $n$ is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get $O(\log k)$ competitive ratios, where $k$ is the number of terminals. In this paper, we apply our approach to obtain deterministic $O(\log k)$-competitive online algorithms for the following problems. - Steiner network with edge duplication. Previously, only a randomized $O(\log n)$-competitive algorithm was known. - Rent-or-buy. Previously, only deterministic $O(\log^2 k)$-competitive and randomized $O(\log k)$-competitive algorithms by Awerbuch, Azar and Bartal (2004) were known. - Connected facility location. Previously, only a randomized $O(\log^2 k)$-competitive algorithm by San Felice, Williamson and Lee (2014) was known. - Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (2011) and give a simpler analysis.

We introduce and study a general scheduling problem that we term the Packing Scheduling problem (PSP). In this problem, jobs can have different arrival times and sizes; a scheduler can process job j at rate xj, subject to arbitrary packing constraints over the set of rates (x) of the outstanding jobs. The PSP framework captures a variety of scheduling problems, including the classical problems of unrelated machines scheduling, broadcast scheduling, and scheduling jobs of different parallelizability. It also captures scheduling constraints arising in diverse modern environments ranging from individual computer architectures to data centers. More concretely, PSP models multidimensional resource requirements and parallelizability, as well as network bandwidth requirements found in data center scheduling. In this paper, we design non-clairvoyant online algorithms for PSP and its special cases -- in this setting, the scheduler is unaware of the sizes of jobs. Our results are summarized as follows. • For minimizing total weighted completion time, we show a O(1)-competitive algorithm. Surprisingly, we achieve this result by applying the well-known Proportional Fairness algorithm (PF) to perform allocations each time instant. Though PF has been extensively studied in the context of maximizing fairness in resource allocation, we present the first analysis in adversarial and general settings for optimizing job latency. Our result is also the first O(1)-competitive algorithm for weighted completion time for several classical non-clairvoyant scheduling problems. •For minimizing total weighted flow time, for any constant e > 0, any O(n1---e)-competitive algorithm requires extra speed (resource augmentation) compared to the offline optimum. We show that PF is a O(log n)-speed O(log n)-competitive non-clairvoyant algorithm, where n is the total number of jobs. We further show that there is an instance of PSP for which no non-clairvoyant algorithm can be O(n1---e)-competitive with o(√log n) speed. •For the classical problem of minimizing total flow time for unrelated machines in the non-clairvoyant setting, we present the first online algorithm which is scalable ((1 + e)-speed O(1)-competitive for any constant e > 0). No non-trivial results were known for this setting, and the previous scalable algorithm could handle only related machines. We develop new algorithmic techniques to handle the unrelated machines setting that build on a new single machine scheduling policy. Since unrelated machine scheduling is a special case of PSP, when contrasted with the lower bound for PSP, our result also shows that PSP is significantly harder than perhaps the most general classical scheduling settings. Our results for PSP show that instantaneous fair scheduling algorithms can also be effective tools for minimizing the overall job latency, even when the scheduling decisions are non-clairvoyant and constrained by general packing constraints.

On Generalizations of the Parking Permit Problem and Network Leasing Problems

Consider the edge-connectivity survivable network design problem (SNDP): given a graph $G = (V,E)$ with edge-costs, and edge-connectivity requirements $r_{ij} \in \mathbb{Z}_{\geq 0}$ for every pair of vertices $i, j \in V$, find an (approximately) minimum-cost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized $\tilde{O}(r_{\max} \log^3 n)$-competitive online algorithm for this edge-connectivity network design problem that runs in time $O(m^{r_{\max}})$, where $r_{\max} = \max_{ij} r_{ij}$. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. As a consequence of using these random embeddings, our algorithms are competitive only against oblivious adversaries. Our results for the online problem give us approximation algorithms that admit strict cost-shares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rent-or-buy version and (b) the (two-stage) stochastic version of the edge-connected network design problem with independent arrivals. If we are in the case when the underlying graph is complete and the edge-costs are metric (i.e., the triangle inequality is satisfied), we improve on our results to give an $O(\log n)$-competitive deterministic online algorithm for the rooted version of the problem, and constant-factor approximation algorithms for the rent-or-buy and stochastic variants of SNDP.

The problem of item selling with the objective of maximizing the revenue is studied. Given a seller with k types of items and n single-minded buyers, i.e., each buyer is only interested in a particular bundle of items, to maximize the revenue, the seller must carefully assign some amount of bundles to each buyer with respect to the buyer’s accepted price. Each buyer $$b_i$$ is associated with a value function $$v_i(\cdot )$$ such that $$v_i(x)$$ is the accepted unit bundle price $$b_i$$ is willing to pay for x bundles. We show that the single-minded item selling problem is NP-hard. Moreover, we give an $$O(\sqrt{k})$$ -approximation algorithm. For the online version, i.e., the buyers come one by one and the decision on each buyer must be made before the arrival of the next buyer, an $$O(\sqrt{k}\cdot (\log h + \log k))$$ -competitive algorithm is achieved, where h is the highest unit item price among all buyers.

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 work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings --- embed the entire graph into a tree at the beginning and then solve the problem on the tree --- and results in randomized O(log n)-competitive algorithms, where n is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get O(log k) competitive ratios, where k is the number of terminals.In this paper, we apply our approach to obtain deterministic O(log k)-competitive online algorithms for the following problems.1. Steiner network with edge duplication. Previously, only a randomized O(log n)-competitive algorithm was known.2. Rent-or-buy. Previously, only deterministic O(log2k)-competitive and randomized O(log k)-competitive algorithms by Awerbuch, Azar and Bartal (Theoretical Computer Science 2004) were known.3. Connected facility location. Previously, only a randomized O(log2k)-competitive algorithm of San Felice, Williamson and Lee (LATIN 2014) was known.4. Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (ICALP 2011) and give a simpler analysis.Our competitive ratios are optimal up to constant factors as these problems capture the online Steiner tree problem which has a lower bound of Ω(log k).

We present online algorithms for directed spanners and directed Steiner forests. These are well-studied network connectivity problems that fall under the unifying framework of online covering and packing linear programming formulations. This framework was developed in the seminal work of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and is based on primal-dual techniques. Specifically, our results include the following: - For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{4/5}) for graphs with general edge lengths, where n is the number of vertices of the given graph. For graphs with uniform edge lengths, we give an efficient randomized algorithm with competitive ratio Õ(n^{2/3+ε}), and an efficient deterministic algorithm with competitive ratio Õ(k^{1/2+ε}), where k is the number of terminal pairs. To the best of our knowledge, these are the first online algorithms for directed spanners. In the offline version, the current best approximation ratio for uniform edge lengths is Õ(n^{3/5 + ε}), due to Chlamt{á}č, Dinitz, Kortsarz, and Laekhanukit (SODA 2017, TALG 2020). - For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{2/3 + ε}). The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is Õ(k^{1/2 + ε})-competitive. In the offline version, the current best approximation ratio with uniform costs is Õ(n^{26/45 + ε}), due to Abboud and Bodwin (SODA 2018). To obtain efficient and competitive online algorithms, we observe that a small modification of the online covering and packing framework by Buchbinder and Naor implies a polynomial-time implementation of the primal-dual approach with separation oracles, which a priori might perform exponentially many calls to the oracle. We convert the online spanner problem into an online covering problem and complete the rounding-step analysis in a problem-specific fashion.

This paper studies a generalized busy-time scheduling model on heterogeneous machines. The input to the model includes a set of jobs and a set of machine types. Each job has a size and a time interval during which it should be processed. Each job is to be placed on a machine for execution. Different types of machines have distinct capacities and cost rates. The total size of the jobs running on a machine must always be kept within the machine's capacity, giving rise to placement restrictions for jobs of various sizes among the machine types. Each machine used is charged according to the time duration in which it is busy, i.e., it is processing jobs. The objective is to schedule the jobs onto machines to minimize the total cost of all the machines used. We develop an $O(1)$-approximation algorithm in the offline setting and an $O(\mu)$-competitive algorithm in the online setting (where $\mu$ is the max/min job length ratio), both of which are asymptotically optimal.

We study a general network design problem with additional degree constraints. Given connectivity requirements ruv for all pairs of vertices, a Steiner network is a graph in which there are at least ruv edge-disjoint paths between u and v for all pairs of vertices u,v. In the MINIMUM BOUNDED-DEGREE STEINER NETWORK problem, we are given an undirected graph G with an edge cost for each edge, a connectivity requirement ruv for each pair of vertices u and v, and a degree upper bound for each vertex v. The task is to find a minimum cost Steiner network which satisfies all the degree upper bounds. The aim of this paper is to design approximation algorithms that minimize the total cost and the degree violation simultaneously. Our main results are the following: There is a polynomial time algorithm which returns a Steiner forest of cost at most 2 OPT and the degree violation at each vertex is at most 3 ,where OPT is the cost of an optimal solution which satisfies all the degree bounds. There is a polynomial time algorithm which returns a Steiner network of cost at most 2 OPT and the degree violation at each vertex is at most 6r max +3 ,where OPT is the cost of an optimal solution which satisfies all the degree bounds, and r max := max u,v {r uv}. These results achieve the best known guarantees for both the total cost and the degree violation simultaneously. As corollaries, these results provide the first additive approximation algorithms for finding low degree subgraphs including Steiner forests, k -edge-connected subgraphs, and Steiner networks. The algorithms develop on the iterative relaxation method applied to a natural linear programming relaxation as in [10, 16, 22]. The new algorithms avoid paying a multiplicative factor of two on the degree bounds even though the algorithm can only pick edges with fractional value 1/2 . This is based on a stronger characterization of the basic so-algorithm is nearly tight.

We consider the over-time version of the Max-Min Fair Allocation problem. Given a time horizon $t=1,2,ldots, T$, with at each time t a set of demands and a set of available resources that may change over the time defining instance $I_t$, we seek a sequence of solutions $S_1, S_2, ldots , S_T$ that (1) are near-optimal at each time t , and (2) as stable as possible (inducing small modification costs). We focus on the impact of the knowledge of the future on the quality and the stability of the returned solutions by distinguishing three settings: the off-line setting where the whole set of instances through the time horizon is known in advance, the on-line setting where no future instance is known, and the k -lookahead setting where at time t , the instances at times $t+1, ldots, t+k$ are known. We first consider the case without restrictions where the set of resources and the set of agents are the same for all instances and where every resource can be allocated to any agent. For the off-line setting, we show that the over-time version of the problem is much harder than the static one, since it becomes $\mathcalNP $-hard even for families of instances for which the static problem is trivial. Then, we provide a $\fracρ ρ+1 $-approximation algorithm for the off-line setting using as subroutine a ρ-approximation algorithm for the static version. We also give a $\fracρ ρ+1 $-competitive algorithm for the online setting using also as subroutine a ρ-approximation algorithm for the static version. Furthermore, for the case with restrictions, we show that in the off-line setting it is possible to get a polynomial-time algorithm with the same approximation ratio as in the case without restrictions. For the online setting, we prove that it is not possible to find an online algorithm with bounded competitive ratio. For the 1-lookahead setting however, we give a $\fracρ 2 (2 ρ + 1) $-approximation algorithm using as subroutine a ρ-approximation algorithm for the static version.

Let F be a function on pairs of vertices. An F-labeling scheme is composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute F(u, v) for any two vertices u and v directly from their labels. As applications for labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. This paper investigates labeling schemes for dynamic trees. We consider two dynamic tree models, namely, the leaf-dynamic tree model in which at each step a leaf can be added to or removed from the tree and the leaf-increasing tree model in which the only topological event that may occur is that a leaf joins the tree. A general method for constructing labeling schemes for dynamic trees (under the above mentioned dynamic tree models) was previously developed in Korman et al. (Theory Comput Syst 37:49–75, 2004). This method is based on extending an existing static tree labeling scheme to the dynamic setting. This approach fits many natural functions on trees, such as distance, separation level, ancestry relation, routing (in both the adversary and the designer port models), nearest common ancestor etc.. Their resulting dynamic schemes incur overheads (over the static scheme) on the label size and on the communication complexity. In particular, all their schemes yield a multiplicative overhead factor of Ω(log n) on the label sizes of the static schemes. Following (Korman et al., Theory Comput Syst 37:49–75, 2004), we develop a different general method for extending static labeling schemes to the dynamic tree settings. Our method fits the same class of tree functions. In contrast to the above paper, our trade-off is designed to minimize the label size, sometimes at the expense of communication. Informally, for any function k(n) and any static F-labeling scheme on trees, we present an F-labeling scheme on dynamic trees incurring multiplicative overhead factors (over the static scheme) of $$O(\log_{k(n)} n)$$ on the label size and $$O(k(n)\log_{k(n)} n)$$ on the amortized message complexity. In particular, by setting $$k(n) = n^{\epsilon}$$ for any $$0 < \epsilon < 1$$ , we obtain dynamic labeling schemes with asymptotically optimal label sizes and sublinear amortized message complexity for the ancestry relation, the id-based and label-based nearest common ancestor relation and the routing function.

We consider the k-Directed Steiner Forest (k-DSF) problem: given a directed graph G = (V, E) with edge costs, a collection D ∈ V x V of ordered node pairs, and an integer k ≤ |D|, find a min-cost subgraph H of G that contains an st-path for (at least) k pairs (s, t) ∈ D. When k = |D|, we get the Directed Steiner Forest (DSF) problem. The best known approximation ratios for these problems are: O(k2/3) for k-DSF by Charikar et al. [2], and O(k1/2+e) for DSF by Chekuri et al. [3].For DSF we give an O(ne·min {n4/5,m2/3})-approximation scheme using a novel LP-relaxation seeking to connect pairs via cheap paths. This is the first sublinear (in terms of n = |V|) approximation ratio for the problem. For k-DSF we give a simple greedy O(k1/2+e)-approximation scheme, improving the best known ratio O(k2/3) by Charikar et al. [2], and (almost) matching, in terms of k, the best ratio known for the undirected variant [11]. Even when used for the particular case DSF, our algorithm favorably compares to the one of [3], which repeatedly solves linear programs, and uses complex time and space consuming transformations.