arXiv subjects

Fabrizio Grandoni

Publications and source records attributed to Fabrizio Grandoni.

At least 37 records · Page 2Linked to original sources

arXiv2022

Unsplittable Euclidean Capacitated Vehicle Routing: A $(2+ε)$-Approximation Algorithm

In the unsplittable capacitated vehicle routing problem, we are given a metric space with a vertex called depot and a set of vertices called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the depot such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1. Our main result is a polynomial-time $(2+ε)$-approximation algorithm for this problem in the two-dimensional Euclidean plane, i.e., for the special case where the terminals and the depot are associated with points in the Euclidean plane and their distances are defined accordingly. This improves on recent work by Blauth, Traub, and Vygen [IPCO'21] and Friggstad, Mousavi, Rahgoshay, and Salavatipour [IPCO'22].

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arXiv2022

Breaching the 2-Approximation Barrier for the Forest Augmentation Problem

The basic goal of survivable network design is to build cheap networks that guarantee the connectivity of certain pairs of nodes despite the failure of a few edges or nodes. A celebrated result by Jain [Combinatorica'01] provides a 2-approximation for a wide class of these problems. However nothing better is known even for very basic special cases, raising the natural question whether any improved approximation factor is possible at all. In this paper we address one of the most basic problems in this family for which 2 is still the best-known approximation factor, the Forest Augmentation Problem (FAP): given an undirected unweighted graph (that w.l.o.g. is a forest) and a collection of extra edges (links), compute a minimum cardinality subset of links whose addition to the graph makes it 2-edge-connected. Several better-than-2 approximation algorithms are known for the special case where the input graph is a tree, a.k.a. the Tree Augmentation Problem (TAP). Recently this was achieved also for the weighted version of TAP, and for the k-edge-connectivity generalization of TAP. These results heavily exploit the fact that the input graph is connected, a condition that does not hold in FAP. In this paper we breach the 2-approximation barrier for FAP. Our result is based on two main ingredients. First, we describe a reduction to the Path Augmentation Problem (PAP), the special case of FAP where the input graph is a collection of disjoint paths. Our reduction is not approximation preserving, however it is sufficiently accurate to improve on a factor 2 approximation. Second, we present a better-than-2 approximation algorithm for PAP, an open problem on its own. Here we exploit a novel notion of implicit credits which might turn out to be helpful in future related work.

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arXiv2021

A Refined Approximation for Euclidean k-Means

In the Euclidean $k$-Means problem we are given a collection of $n$ points $D$ in an Euclidean space and a positive integer $k$. Our goal is to identify a collection of $k$ points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in $D$ and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a primal-dual $6.357$ approximation based on a standard LP for the problem [Ahmadian et al. FOCS'17, SICOMP'20]. In this note we show how a minor modification of Ahmadian et al.'s analysis leads to a slightly improved $6.12903$ approximation. As a related result, we also show that the mentioned LP has integrality gap at least $\frac{16+\sqrt{5}}{15}>1.2157$.

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arXiv2021

Maintaining an EDCS in General Graphs: Simpler, Density-Sensitive and with Worst-Case Time Bounds

In their breakthrough ICALP'15 paper, Bernstein and Stein presented an algorithm for maintaining a $(3/2+ε)$-approximate maximum matching in fully dynamic {\em bipartite} graphs with a {\em worst-case} update time of $O_ε(m^{1/4})$; we use the $O_ε$ notation to suppress the $ε$-dependence. Their main technical contribution was in presenting a new type of bounded-degree subgraph, which they named an {\em edge degree constrained subgraph (EDCS)}, which contains a large matching -- of size that is smaller than the maximum matching size of the entire graph by at most a factor of $3/2+ε$. They demonstrate that the EDCS can be maintained with a worst-case update time of $O_ε(m^{1/4})$, and their main result follows as a direct corollary. In their followup SODA'16 paper, Bernstein and Stein generalized their result for general graphs, achieving the same update time of $O_ε(m^{1/4})$, albeit with an amortized rather than worst-case bound. To date, the best {\em deterministic} worst-case update time bound for {\em any} better-than-2 approximate matching is $O(\sqrt{m})$ [Neiman and Solomon, STOC'13], [Gupta and Peng, FOCS'13]; allowing randomization (against an oblivious adversary) one can achieve a much better (still polynomial) update time for approximation slightly below 2 [Behnezhad, Lacki and Mirrokni, SODA'20]. In this work we\footnote{\em quasi nanos, gigantium humeris insidentes} simplify the approach of Bernstein and Stein for bipartite graphs, which allows us to generalize it for general graphs while maintaining the same bound of $O_ε(m^{1/4})$ on the {\em worst-case} update time. Moreover, our approach is {\em density-sensitive}: If the {\em arboricity} of the dynamic graph is bounded by $α$ at all times, then the worst-case update time of the algorithm is $O_ε(\sqrtα)$.

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arXiv2021

Approximation Algorithms for Demand Strip Packing

In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+eps)-approximation algorithm for any constant eps>0. We also achieve best-possible approximation factors for some relevant special cases.

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arXiv2021

Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More

In the \textsc{2-Dimensional Knapsack} problem (2DK) we are given a square knapsack and a collection of $n$ rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed non-overlappingly into the knapsack. The currently best known polynomial-time approximation factor for 2DK is $17/9+\varepsilon<1.89$ and there is a $(3/2+\varepsilon)$-approximation algorithm if we are allowed to rotate items by 90 degrees~{[}Gálvez et al., FOCS 2017{]}. In this paper, we give $(4/3+\varepsilon)$-approximation algorithms in polynomial time for both cases, assuming that all input data are {integers polynomially bounded in $n$}. Gálvez et al.'s algorithm for 2DK partitions the knapsack into a constant number of rectangular regions plus \emph{one} L-shaped region and packs items into those {in a structured way}. We generalize this approach by allowing up to a \emph{constant} number of {\emph{more general}} regions that can have the shape of an L, a U, a Z, a spiral, and more, and therefore obtain an improved approximation ratio. {In particular, we present an algorithm that computes the essentially optimal structured packing into these regions. }

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arXiv2020

All-Pairs LCA in DAGs: Breaking through the $O(n^{2.5})$ barrier

Let $G=(V,E)$ be an $n$-vertex directed acyclic graph (DAG). A lowest common ancestor (LCA) of two vertices $u$ and $v$ is a common ancestor $w$ of $u$ and $v$ such that no descendant of $w$ has the same property. In this paper, we consider the problem of computing an LCA, if any, for all pairs of vertices in a DAG. The fastest known algorithms for this problem exploit fast matrix multiplication subroutines and have running times ranging from $O(n^{2.687})$ [Bender et al.~SODA'01] down to $O(n^{2.615})$ [Kowaluk and Lingas~ICALP'05] and $O(n^{2.569})$ [Czumaj et al.~TCS'07]. Somewhat surprisingly, all those bounds would still be $Ω(n^{2.5})$ even if matrix multiplication could be solved optimally (i.e., $ω=2$). This appears to be an inherent barrier for all the currently known approaches, which raises the natural question on whether one could break through the $O(n^{2.5})$ barrier for this problem. In this paper, we answer this question affirmatively: in particular, we present an $\tilde O(n^{2.447})$ ($\tilde O(n^{7/3})$ for $ω=2$) algorithm for finding an LCA for all pairs of vertices in a DAG, which represents the first improvement on the running times for this problem in the last 13 years. A key tool in our approach is a fast algorithm to partition the vertex set of the transitive closure of $G$ into a collection of $O(\ell)$ chains and $O(n/\ell)$ antichains, for a given parameter $\ell$. As usual, a chain is a path while an antichain is an independent set. We then find, for all pairs of vertices, a \emph{candidate} LCA among the chain and antichain vertices, separately. The first set is obtained via a reduction to min-max matrix multiplication. The computation of the second set can be reduced to Boolean matrix multiplication similarly to previous results on this problem. We finally combine the two solutions together in a careful (non-obvious) manner.

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arXiv2020

Online Edge Coloring Algorithms via the Nibble Method

Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured that an online $(1+o(1))Δ$-edge-coloring algorithm exists for $n$-node graphs of maximum degree $Δ=ω(\log n)$. This conjecture remains open in general, though it was recently proven for bipartite graphs under \emph{one-sided vertex arrivals} by Cohen et al.~[FOCS'19]. In a similar vein, we study edge coloring under widely-studied relaxations of the online model. Our main result is in the \emph{random-order} online model. For this model, known results fall short of the Bar-Noy et al.~conjecture, either in the degree bound [Aggarwal et al.~FOCS'03], or number of colors used [Bahmani et al.~SODA'10]. We achieve the best of both worlds, thus resolving the Bar-Noy et al.~conjecture in the affirmative for this model. Our second result is in the adversarial online (and dynamic) model with \emph{recourse}. A recent algorithm of Duan et al.~[SODA'19] yields a $(1+ε)Δ$-edge-coloring with poly$(\log n/ε)$ recourse. We achieve the same with poly$(1/ε)$ recourse, thus removing all dependence on $n$. Underlying our results is one common offline algorithm, which we show how to implement in these two online models. Our algorithm, based on the Rödl Nibble Method, is an adaptation of the distributed algorithm of Dubhashi et al.~[TCS'98]. The Nibble Method has proven successful for distributed edge coloring. We display its usefulness in the context of online algorithms.

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arXiv2020

Improved Approximation Algorithms for Stochastic-Matching Problems

We consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. In this problem, we are given an undirected graph. Each edge is assigned a known, independent probability of existence and a positive weight (or profit). We must probe an edge to discover whether or not it exists. Each node is assigned a positive integer called a timeout (or a patience). On this random graph we are executing a process, which probes the edges one-by-one and gradually constructs a matching. The process is constrained in two ways. First, if a probed edge exists, it must be added irrevocably to the matching (the query-commit model). Second, the timeout of a node $v$ upper-bounds the number of edges incident to $v$ that can be probed. The goal is to maximize the expected weight of the constructed matching. For this problem, Bansal et al. (Algorithmica 2012) provided a $0.33$-approximation algorithm for bipartite graphs and a $0.25$-approximation for general graphs. We improve the approximation factors to $0.39$ and $0.269$, respectively. The main technical ingredient in our result is a novel way of probing edges according to a not-uniformly-random permutation. Patching this method with an algorithm that works best for large-probability edges (plus additional ideas) leads to our improved approximation factors.

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arXiv2019

Breaching the 2-Approximation Barrier for Connectivity Augmentation: a Reduction to Steiner Tree

The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of the most basic problems in this area: given a $k$(-edge)-connected graph $G$ and a set of extra edges (links), select a minimum cardinality subset $A$ of links such that adding $A$ to $G$ increases its edge connectivity to $k+1$. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is $2$, and this can be achieved with multiple approaches (the first such result is in [Frederickson and Jájá'81]). It is known [Dinitz et al.'76] that CAP can be reduced to the case $k=1$, a.k.a. the Tree Augmentation Problem (TAP), for odd $k$, and to the case $k=2$, a.k.a. the Cactus Augmentation Problem (CacAP), for even $k$. Several better than $2$ approximation algorithms are known for TAP, culminating with a recent $1.458$ approximation [Grandoni et al.'18]. However, for CacAP the best known approximation is $2$. In this paper we breach the $2$ approximation barrier for CacAP, hence for CAP, by presenting a polynomial-time $2\ln(4)-\frac{967}{1120}+ε<1.91$ approximation. Previous approaches exploit properties of TAP that do not seem to generalize to CacAP. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al.'14]. This reduction is not approximation preserving, and using the current best approximation factor for Steiner tree [Byrka et al.'13] as a black-box would not be good enough to improve on $2$. To achieve the latter goal, we ``open the box'' and exploit the specific properties of the instances of Steiner tree arising from CacAP.

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arXiv2019

Fully Dynamic $(Δ+1)$-Coloring in Constant Update Time

The problem of (vertex) $(Δ+1)$-coloring a graph of maximum degree $Δ$ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA'18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized data structure for maintaining a $(Δ+1)$-coloring with $O(\log Δ)$ expected amortized update time. In this paper, we present a $(Δ+1)$-coloring data structure that achieves a constant amortized update time and show that this time bound holds not only in expectation but also with high probability.

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arXiv2019

Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+eps)-approximations in f(k,eps)n^{O(1)} time where k is some parameter of the input. We obtain the following results on parameterized approximability: 1) In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(loglog n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant eps>0 and integer k>0, in time f(k,eps)n^{g(eps)}, either outputs a solution of size at least k/(1+eps), or declares that the optimum solution has size less than k. 2) In the (2-dimensional) geometric knapsack problem (TDK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of TDK with rotations (TDKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factors are 558/325+eps and 4/3+eps, resp. [Galvez et al., FOCS'17]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for TDKR. For all considered problems, getting time f(k,eps)n^{O(1)}, rather than f(k,eps)n^{g(eps)}, would give FPT time f'(k)n^{O(1)} exact algorithms using eps=1/(k+1), contradicting W[1]-hardness.

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arXiv2019

Dynamic Set Cover: Improved Algorithms & Lower Bounds

We give new upper and lower bounds for the {\em dynamic} set cover problem. First, we give a $(1+ε) f$-approximation for fully dynamic set cover in $O(f^2\log n /ε^5)$ (amortized) update time, for any $ε> 0$, where $f$ is the maximum number of sets that an element belongs to. In the decremental setting, the update time can be improved to $O(f^2/ε^5)$, while still obtaining an $(1+ε) f$-approximation. These are the first algorithms that obtain an approximation factor linear in $f$ for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of $O(f^2)$ in the dynamic setting. To complement our upper bounds, we also show that a linear dependence of the update time on $f$ is necessary unless we can tolerate much worse approximation factors. Using the recent distributed PCP-framework, we show that any dynamic set cover algorithm that has an amortized update time of $O(f^{1-ε})$ must have an approximation factor that is $Ω(n^δ)$ for some constant $δ>0$ under the Strong Exponential Time Hypothesis.

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arXiv2018

The Matching Augmentation Problem: A $\frac74$-Approximation Algorithm

We present a $\frac74$ approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. We first present a reduction of any given MAP instance to a collection of well-structured MAP instances such that the approximation guarantee is preserved. Then we present a $\frac74$ approximation algorithm for a well-structured MAP instance. The algorithm starts with a min-cost 2-edge cover and then applies ear-augmentation steps. We analyze the cost of the ear-augmentations using an approach similar to the one proposed by Vempala and Vetta for the (unweighted) min-size 2-ECSS problem (`Factor 4/3 approximations for minimum 2-connected subgraphs,' APPROX 2000, LNCS 1913, pp.262-273).

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arXiv2018

$O(\log^2k/\log\log{k})$-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm

In the Directed Steiner Tree (DST) problem we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost arborescence that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial-time. By adjusting the parameters in the hardness result of Halperin and Krauthgamer, we show the matching lower bound of $Ω(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial-time algorithms. This is the first improvement on the DST problem since the classical quasi-polynomial-time $O(\log^3 k)$ approximation algorithm by Charikar et al. (The paper erroneously claims an $O(\log^2k)$ approximation due to a mistake in prior work.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Label-Consistent Subtree (LCST) problem. The LCST instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve LCST instances, which is inspired by the framework developed by Rothvoß. We consider a Sherali-Adams lifting of a proper LP relaxation of LCST. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. A small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves.

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arXiv2018

Improved Approximation for Tree Augmentation: Saving by Rewiring

The Tree Augmentation Problem (TAP) is a fundamental network design problem in which we are given a tree and a set of additional edges, also called \emph{links}. The task is to find a set of links, of minimum size, whose addition to the tree leads to a $2$-edge-connected graph. A long line of results on TAP culminated in the previously best known approximation guarantee of $1.5$ achieved by a combinatorial approach due to Kortsarz and Nutov [ACM Transactions on Algorithms 2016], and also by an SDP-based approach by Cheriyan and Gao [Algorithmica 2017]. Moreover, an elegant LP-based $(1.5+ε)$-approximation has also been found very recently by Fiorini, Groß, Könemann, and Sanitá [SODA 2018]. In this paper, we show that an approximation factor below $1.5$ can be achieved, by presenting a $1.458$-approximation that is based on several new techniques.

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arXiv2018

Improved Pseudo-Polynomial-Time Approximation for Strip Packing

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of a fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P$\neq$NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a $(\frac{7}{5}+ε)$-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm. In this paper, we make further progress on the PPT approximability of strip packing, by presenting a $(\frac43+ε)$-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the \emph{empty space} left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by $90^\circ$, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well.

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arXiv2017

Approximating Geometric Knapsack via L-packings

We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + ε) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17/9 + ε) < 1.89 approximation, which improves to (558/325 + ε) < 1.72 in the cardinality case. Essentially all prior work on 2DK approximation packs items inside a constant number of rectangular containers, where items inside each container are packed using a simple greedy strategy. We deviate for the first time from this setting: we show that there exists a large profit solution where items are packed inside a constant number of containers plus one L-shaped region at the boundary of the knapsack which contains items that are high and narrow and items that are wide and thin. As a second major and the main algorithmic contribution of this paper, we present a PTAS for this case. We believe that this will turn out to be useful in future work in geometric packing problems. We also consider the variant of the problem with rotations (2DKR), where items can be rotated by 90 degrees. Also, in this case, the best-known polynomial-time approximation factor (even for the cardinality case) is (2 + ε) [Jansen and Zhang, SODA 2004]. Exploiting part of the machinery developed for 2DK plus a few additional ideas, we obtain a polynomial-time (3/2 + ε)-approximation for 2DKR, which improves to (4/3 + ε) in the cardinality case.

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