Asymmetric Traveling Salesman Path and Directed Latency Problems

We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric and two specified nodes s and t, both problems ask to find an s-t path visiting all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total cost of this path. In the directed latency problem, the objective is to minimize the sum of distances on this path from s to each node. Both of these problems are NP-hard. The best known approximation algorithms for ATSPP had ratio O(log n) until the very recent result that improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the integrality gap of its linear programming relaxation has been known. For directed latency, the best previously known approximation algorithm has a guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new algorithm for the ATSPP problem that has an approximation ratio of O(log n), but whose analysis also bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed by Chekuri and Pal [2007]. We then pursue a deeper study of this linear program and its variations, which leads to an algorithm for the k-person ATSPP (where k s-t paths of minimum total length are sought) and an O(log n)-approximation for the directed latency problem

Approximation Algorithms for Generalized Path Scheduling

Scheduling problems where the machines can be represented as the edges of a network and each job needs to be processed by a sequence of machines that form a path in this network have been the subject of many research articles (e.g. flow shop is the special case where the network as well as the sequence of machines for each job is a simple path). In this paper we consider one such problem, called Generalized Path Scheduling (GPS) problem, which can be defined as follows. Given a set of non-preemptive jobs J and identical machines M ( |J| = n and |M| = m ). The machines are ordered on a path. Each job j = {P_j = {l_j, r_j}, p_j} is defined by its processing time p_j and a sub-path P_j from machine with index l_j to r_j (l_j, r_j ? M, and l_j ? r_j) specifying the order of machines it must go through. We assume each machine has a queue of infinite size where jobs can sit in the queue to resolve conflicts. Two objective functions, makespan and total completion time, are considered. Machines can be identical or unrelated. In the latter case, this problem generalizes the classical Flow shop problem (in which all jobs have to go through all machines from 1 to m in that order). Generalized Path Scheduling has been studied (e.g. see [Ronald Koch et al., 2009; Zachary Friggstad et al., 2019]). In this paper, we present several improved approximation algorithms for both objectives. For the case of number of machines being sub-logarithmic in the number of jobs we present a PTAS for both makespan and total completion time. The PTAS holds even on unrelated machines setting and therefore, generalizes the result of Hall [Leslie A. Hall, 1998] for the classic problem of Flow shop. For the case of identical machines, we present an O((log m)/(log log m))-approximation algorithms for both objectives, which improve the previous best result of [Zachary Friggstad et al., 2019]. We also show that the GPS problem is NP-complete for both makespan and total completion time objectives

Approximating Connected Facility Location with Lower and Upper Bounds via LP Rounding

We consider a lower- and upper-bounded generalization of the classical facility location problem, where each facility has a capacity (upper bound) that limits the number of clients it can serve and a lower bound on the number of clients it must serve if it is opened. We develop an LP rounding framework that exploits a Voronoi diagram-based clustering approach to derive the first bicriteria constant approximation algorithm for this problem with non-uniform lower bounds and uniform upper bounds. This naturally leads to the the first LP-based approximation algorithm for the lower bounded facility location problem (with non-uniform lower bounds). We also demonstrate the versatility of our framework by extending this and presenting the first constant approximation algorithm for some connected variant of the problems in which the facilities are required to be connected as well

Approximation Algorithms for Capacitated k-Travelling Repairmen Problems

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem

Approximation Schemes for Min-Sum k-Clustering

We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G = (V, E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized. The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz, Friggstad, Salavatipour and Sivakumar [Algorithmica 2019] achieve an approximation ratio of O(log |V|) in polynomial time for general metrics and an approximation ratio 2+? in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC. In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics

Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means

We investigate the complexity of solving stable or perturbation-resilient instances of k-Means and k-Median clustering in fixed dimension Euclidean metrics (or more generally doubling metrics). The notion of stable or perturbation resilient instances was introduced by Bilu and Linial [2010] and Awasthi et al. [2012]. In our context we say a k-Means instance is \alpha-stable if there is a unique OPT solution which remains unchanged if distances are (non-uniformly) stretched by a factor of at most \alpha. Stable clustering instances have been studied to explain why heuristics such as Lloyd's algorithm perform well in practice. In this work we show that for any fixed \epsilon>0, (1+\epsilon)-stable instances of k-Means in doubling metrics can be solved in polynomial time. More precisely we show a natural multiswap local search algorithm in fact finds the OPT solution for (1+\epsilon)-stable instances of k-Means and k-Median in a polynomial number of iterations. We complement this result by showing that under a plausible PCP hypothesis this is essentially tight: that when the dimension d is part of the input, there is a fixed \epsilon_0>0 s.t. there is not even a PTAS for (1+\epsilon_0)-stable k-Means in R^d unless NP=RP. To do this, we consider a robust property of CSPs; call an instance stable if there is a unique optimum solution x^* and for any other solution x', the number of unsatisfied clauses is proportional to the Hamming distance between x^* and x'. Dinur et al. have already shown stable QSAT is hard to approximate for some constant Q, our hypothesis is simply that stable QSAT with bounded variable occurrence is also hard. Given this hypothesis, we consider "stability-preserving" reductions to prove our hardness for stable k-Means. Such reductions seem to be more fragile than standard L-reductions and may be of further use to demonstrate other stable optimization problems are hard.Comment: 29 page

Approximations for Throughput Maximization

In this paper we study the classical problem of throughput maximization. In this problem we have a collection $J$ of $n$ jobs, each having a release time $r_j$, deadline $d_j$, and processing time $p_j$. They have to be scheduled non-preemptively on $m$ identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their $[r_j,d_j]$ window. This problem has been studied extensively (even for the case of $m=1$). Several special cases of the problem remain open. Bar-Noy et al. [STOC1999] presented an algorithm with ratio $1-1/(1+1/m)^m$ for $m$ machines, which approaches $1-1/e$ as $m$ increases. For $m=1$, Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation with ratio $1-\frac{1}{e}-\varepsilon$ (for any $\varepsilon>0$). Recently Im-Li-Moseley [IPCO2017] presented an algorithm with ratio $1-1/e-\varepsilon_0$ for some absolute constant $\varepsilon_0>0$ for any fixed $m$. They also presented an algorithm with ratio $1-O(\sqrt{\log m/m})-\varepsilon$ for general $m$ which approaches 1 as $m$ grows. The approximability of the problem for $m=O(1)$ remains a major open question. Even for the case of $m=1$ and $c=O(1)$ distinct processing times the problem is open (Sgall [ESA2012]). In this paper we study the case of $m=O(1)$ and show that if there are $c$ distinct processing times, i.e. $p_j$'s come from a set of size $c$, then there is a $(1-\varepsilon)$-approximation that runs in time $O(n^{mc^7\varepsilon^{-6}}\log T)$, where $T$ is the largest deadline. Therefore, for constant $m$ and constant $c$ this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning

Approximation Algorithms for Minimum-Load k-Facility Location

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F\u27 of k facilities, and assign each client j to a facility f(j) in F\u27 so as to minimize maximum, over all facilities in F\u27, of the sum of distances of clients j assigned to F\u27 to F\u27. We call this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming