Quasi-polynomial Time Approximation Scheme Research Articles

Approximation schemes for Min-Sum [formula omitted]-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 et al. (2019) achieve an approximation ratio of O(log|V|) in polynomial time for general metrics and an approximation ratio 2+ϵ 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.

Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás’ Path Argument

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S t,t,t as an induced subgraph and provide a subexponential-time algorithm with improved running time \(2^{\mathcal {O}(\sqrt {nt}\log n)}\) and a quasipolynomial-time approximation scheme with improved running time \(2^{\mathcal {O}(\varepsilon ^{-1}t \log ^{5} n)}\) . The Gyárfás’ path argument, a powerful tool that is the main building block for many algorithms in P t -free graphs, ensures that given an n -vertex P t -free graph, in polynomial time we can find a set P of at most t -1 vertices such that every connected component of G-N[P] has at most n /2 vertices. Our main technical contribution is an analog of this result for S t,t,t -free graphs: given an n -vertex S t,t,t -free graph, in polynomial time we can find a set P of \(\mathcal {O}(t \log n)\) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n /2 vertices.

Quasi-polynomial time approximation schemes for assortment optimization under Mallows-based rankings

Quasi-polynomial time approximation schemes for assortment optimization under Mallows-based rankings

Approximation Schemes for Capacitated Vehicle Routing on Graphs of Bounded Treewidth, Bounded Doubling, or Highway Dimension

In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [ 14 ], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V , and a vehicle of bounded capacity Q . The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand d v and the total demand of each tour must be no more than Q . Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed \(ε \gt \frac{1}{3000}\) , where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [ 5 ] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [ 15 ] presented an approximation scheme with time n log O(1/ε) n for Euclidean plane ℝ 2 . No other approximation scheme is known for any other class of metrics (without further restrictions on Q ). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time n O(log 6 n/ε 5 ) .

Min-max coverage problems on tree-like metrics

We consider a number of min-max coverage problems. In each problem, the input is an unweighted graph G and an integer k, and possibly some additional information, such as a root vertex r. In the Min-Max Path Cover problem, the task is to cover all vertices of the graph by k walks, minimizing the length of the longest walk. The variant of Min-Max Path Cover in which all walks start and end at the same prescribed root vertex r is called the k-Traveling Salesmen Problem. In the Min-Max Tree Cover problem, the task is to cover all vertices of the graph by k trees, minimizing the size (number of edges) of the largest tree. In the rooted version, Min-Max k-Rooted Tree Cover, the input also contains k roots r1, ., rk, and the ith tree must contain the root ri. These four problems are all known to be APX-hard and to admit a constant-factor approximation. In this paper, we initiate the systematic study of these problems on trees and, more generally, on graphs of constant treewidth. As opposed to most graph problems, all four of the above coverage problems remain NP-hard even when G is a tree. We obtain an nO(k)-time exact algorithm for all four problems on graphs of bounded treewidth. Our main contribution is a quasi-polynomial-time approximation scheme (QPTAS) for the k-Traveling Salesmen Problem, Min-Max Path Cover, and Min-Max Tree Cover on graphs of bounded treewidth.

Approximations for generalized unsplittable flow on paths with application to power systems optimization

The Unsplittable Flow on a Path (UFP) problem has garnered considerable attention as a challenging combinatorial optimization problem with notable practical implications. Steered by its pivotal applications in power engineering, the present work formulates a novel generalization of UFP, wherein demands and capacities in the input instance are monotone step functions over the set of edges. As an initial step towards tackling this generalization, we draw on and extend ideas from prior research to devise a quasi-polynomial time approximation scheme (QPTAS) under the premise that the demands and capacities lie in a quasi-polynomial range. Second, retaining the same assumption, an efficient logarithmic approximation is introduced for the single-source variant of the problem. Finally, we round up the contributions by designing a (kind of) black-box reduction that, under some mild conditions, allows to translate LP-based approximation algorithms for the studied problem into their counterparts for the Alternating Current Optimal Power Flow (AC OPF) problem -- a fundamental workflow in operation and control of power systems.

Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment

Resource Minimization Fire Containment (RMFC) is a natural model for optimal inhibition of harmful spreading phenomena on a graph. In the RMFC problem on trees, we are given an undirected tree G, and a vertex r where the fire starts at, called root. At each time step, the firefighters can protect up to B vertices of the graph while the fire spreads from burning vertices to all their neighbors that have not been protected so far. The task is to find the smallest B that allows for saving all the leaves of the tree. The problem is hard to approximate up to any factor better than 2 even on trees unless P = NP (King and MacGillivray in Discret Math 310(3):614–621, 2010). Chalermsook and Chuzhoy (In: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms, SODA 2010, Austin, Texas, USA, 17–19 Jan 2010, SIAM, pp 1334–1349, 2010) presented a Linear Programming (LP) based \(O(\log ^* n)\) approximation for RMFC on trees that matches the integrality gap of the natural Linear Programming relaxation. This was recently improved by Adjiashvili et al. (ACM Trans Algorithms 15(2):20:1–20:33, 2019) to a 12-approximation through a combination of LP rounding along with several new techniques. In this paper we present an asymptotic QPTAS for RMFC on trees. More specifically, let \(\epsilon >0\), and \(\mathcal {I}\) be an instance of RMFC where the optimum number of firefighters to save all the leaves is \(OPT(\mathcal {I})\). We present an algorithm which uses at most \(\lceil (1+\epsilon )OPT(\mathcal {I})\rceil \) many firefighters at each time step and runs in time \(n^{O(\log \log n/\epsilon )}\). This suggests that the existence of an asymptotic PTAS is plausible especially since the exponent is \(O(\log \log n)\), not \(O(\log n)\). Our result combines a more refined height reduction lemma than the one in Adjiashvili et al. (2019) with LP rounding and dynamic programming to find the solution. We also apply our height reduction lemma to the algorithm provided in Adjiashvili et al. (2019) plus a more careful analysis to improve their 12-approximation and provide a polynomial time (\(5+\epsilon \))-approximation.

Approximation algorithms for the generalized incremental knapsack problem

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of n items, each associated with a non-negative weight, and T time periods with non-decreasing capacities \(W_1 \le \dots \le W_T\). When item i is inserted at time t, we gain a profit of \(p_{it}\); however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time \((\frac{1}{2}-\epsilon )\)-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys–Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and new structural properties of nearly-optimal solutions, we turn our algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.

Quasi-Polynomial Time Approximation Schemes for Assortment Optimization under Mallows-Based Rankings

Quasi-Polynomial Time Approximation Schemes for Assortment Optimization under Mallows-Based Rankings

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter roughly measures how many central nodes are visited by all shortest paths of a certain length. It has been shown that transportation networks, on which TSP and STP naturally occur for various applications in logistics, typically have a small highway dimension. While it was previously shown that these problems admit a quasi-polynomial time approximation scheme on graphs of constant highway dimension, we demonstrate that a significant improvement is possible in the special case when the highway dimension is 1. Specifically, we present a fully-polynomial time approximation scheme (FPTAS). We also prove that both TSP and STP are weakly \({\mathsf {NP}}\)-hard for these restricted graphs.

Efficient Approximation of the Capacitated Vehicle Routing Problem in a Metric Space of an Arbitrary Fixed Doubling Dimension

In this paper, for the first time, we provide a quasi-polynomial time approximation scheme for the well-known capacitated vehicle routing problem formulated in metric spaces of an arbitrary fixed doubling dimension.

Lipschitz Continuity and Approximate Equilibria

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in L_1 and L_2^2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs

We consider two optimization problems in planar graphs. In Maximum Weight Independent Set of Objects we are given a graph G and a family $${\mathcal {D}}$$ of objects, each being a connected subgraph of G with a prescribed weight, and the task is to find a maximum-weight subfamily of $${\mathcal {D}}$$ consisting of pairwise disjoint objects. In Minimum Weight Distance Set Cover we are given a graph G in which the edges might have different lengths, two sets $${\mathcal {D}},{\mathcal {C}}$$ of vertices of G, where vertices of $${\mathcal {D}}$$ have prescribed weights, and a nonnegative radius r. The task is to find a minimum-weight subset of $${\mathcal {D}}$$ such that every vertex of $${\mathcal {C}}$$ is at distance at most r from some selected vertex. Via simple reductions, these two problems generalize a number of geometric optimization tasks, notably Maximum Weight Independent Set for polygons in the plane and Weighted Geometric Set Cover for unit disks and unit squares. We present quasi-polynomial time approximation schemes (QPTASs) for both of the above problems in planar graphs: given an accuracy parameter $$\epsilon >0$$ we can compute a solution whose weight is within multiplicative factor of $$(1+\epsilon )$$ from the optimum in time $$2^{{\mathrm {poly}}(1/\epsilon ,\log |{\mathcal {D}}|)}\cdot n^{{\mathcal {O}}(1)}$$, where n is the number of vertices of the input graph. We note that a QPTAS for Maximum Weight Independent Set of Objects would follow from existing work. However, our main contribution is to provide a unified framework that works for both problems in both a planar and geometric setting and to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek and Wiese (in Proceedings of the FOCS 2013, IEEE, 2013; in Proceedings of the SODA 2014, SIAM, 2014) and Har-Peled and Sariel (in Proceedings of the SOCG 2014, SIAM, 2014) to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods as a phenomenon in planar graphs.

Approximation schemes for r-weighted Minimization Knapsack problems

Stimulated by salient applications arising from power systems, this paper studies a class of non-linear Knapsack problems with non-separable quadratic constrains, formulated in either binary or integer form. These problems resemble the duals of the corresponding variants of 2-weighted Knapsack problem (a.k.a., complex-demand Knapsack problem) which has been studied in the extant literature under the paradigm of smart grids. Nevertheless, the employed techniques resulting in a polynomial-time approximation scheme (PTAS) for the 2-weighted Knapsack problem are not amenable to its minimization version. We instead propose a greedy geometry-based approach that arrives at a quasi PTAS (QPTAS) for the minimization variant with boolean variables. As for the integer formulation, a linear programming-based method is developed that obtains a PTAS. In view of the curse of dimensionality, fast greedy heuristic algorithms are presented, additionally to QPTAS. Their performance is corroborated extensively by empirical simulations under diverse settings and scenarios.

Approximation Schemes for Capacitated Geometric Network Design

We study a capacitated network design problem in a geometric setting. The input consists of an integral edge capacity k and two sets of points on the Euclidean plane, sources, and sinks, with an integral demand for each point. The demand of each source specifies the amount of flow that has to be shipped from the source, and the demand of each sink specifies the amount of flow that has to be shipped to the sink. The goal is to construct a minimum-length network that allows one to route the requested flow from the sources to the sinks and where each edge in the network has capacity k. The vertices of the network are not constrained to the sets of sinks and sources-any point on the Euclidean plane can be used as a vertex. The flow is splittable and parallel edges are allowed. The capacitated geometric network design problem generalizes, among others, the geometric Steiner tree problem, and as such it is NP-hard. We show that if the demands are polynomially bounded and the edge capacity k is not too large, the single-sink capacitated geometric network design problem admits a polynomial time approximation scheme. If the capacity is arbitrarily large, then we design a quasi-polynomial time approximation scheme for the capacitated geometric network design problem allowing for an arbitrary number of sinks. Our results rely on a derivation of an upper bound on the number of vertices different from sources and sinks (the so-called Steiner vertices) in an optimal network. The bound is polynomial in the total demand of the sources. (Less)

A $(1+\varepsilon)$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E') that distorts shortest path distances of G by at most a 1 + ${\varepsilon}$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyse the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.

Hardness of Approximation for Strip Packing

Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, for example, in scheduling and stock-cutting, and has been studied extensively. When the dimensions of the objects are allowed to be exponential in the total input size, it is known that the problem cannot be approximated within a factor better than 3/2, unless P= NP. However, there was no corresponding lower bound for polynomially bounded input data. In fact, Nadiradze and Wiese [SODA 2016] have recently proposed a (1.4 + ϵ)-approximation algorithm for this variant, thus showing that strip packing with polynomially bounded data can be approximated better than when exponentially large values are allowed in the input. Their result has subsequently been improved to a (4/3 + ϵ)-approximation by two independent research groups [FSTTCS 2016, WALCOM 2017]. This raises a question whether strip packing with polynomially bounded input data admits a quasi-polynomial time approximation scheme, as is the case for related two-dimensional packing problems like maximum independent set of rectangles or two-dimensional knapsack. In this article, we answer this question in negative by proving that it is NP-hard to approximate strip packing within a factor better than 12/11, even when restricted to polynomially bounded input data. In particular, this shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless NP} ⊑ DTIME(2 polylog (n) ).

The triangle scheduling problem

This paper introduces a novel scheduling problem, where jobs occupy a triangular shape on the time line. This problem is motivated by scheduling jobs with different criticality levels. A measure is introduced, namely the binary tree ratio. It is shown that the Greedy algorithm solves the problem to optimality when the binary tree ratio of the input instance is at most 2. We also show that the problem is unary NP-hard for instances with binary tree ratio strictly larger than 2 and provide a quasi-polynomial time approximation scheme. The approximation ratio of Greedy on general instances is shown to be between 1.5 and 1.05.

Zero-Free Regions of Partition Functions with Applications to Algorithms and Graph Limits

Based on a technique of Barvinok [4,5,6] and Barvinok and Soberon [8,9] we identify a class of edge-coloring models whose partition functions do not evaluate to zero on bounded degree graphs. Subsequently we give a quasi-polynomial time approximation scheme for computing these partition functions. As another application we show that the normalised partition functions of these models are continuous with respect to the Benjamini-Schramm topology on bounded degree graphs. We moreover give quasi-polynomial time approximation schemes for evaluating a large class of graph polynomials, including the Tutte polynomial, on bounded degree graphs.

Tractable Algorithms for Approximate Nash Equilibria in Generalized Graphical Games with Tree Structure

We provide the first fully polynomial time approximation scheme (FPTAS) for computing an approximate mixed-strategy Nash equilibrium in graphical multi-hypermatrix games (GMhGs), which are generalizations of normal-form games, graphical games, graphical polymatrix games, and hypergraphical games. Computing an exact mixed-strategy Nash equilibria in graphical polymatrix games is PPAD complete and thus generally believed to be intractable. In contrast, to the best of our knowledge, we are the first to establish an FPTAS for tree polymatrix games as well as tree graphical games when the number of actions is bounded by a constant. As a corollary, we give a quasi-polynomial time approximation scheme (quasi-PTAS) when the number of actions is bounded by a logarithm of the number of players.