Following previous theoretical work by Srinivasan (FOCS 2001) and the first author (STACS 2006) and a first experimental evaluation on random instances (ALENEX 2009), we investigate how the recently developed different approaches to generate randomized roundings satisfying disjoint cardinality constraints behave when used in two classical algorithmic problems, namely low-congestion routing in networks and max-coverage problems in hypergraphs.
We generally find that all randomized rounding algorithms work well, much better than what is guaranteed by existing theoretical work. The derandomized versions produce again significantly better rounding errors, with running times still negligible compared to the one for solving the corresponding LP. It thus seems worth preferring them over the randomized variants.
The data created in these experiments lets us propose and investigate the following new ideas. For the low-congestion routing problems, we suggest to solve a second LP, which yields the same congestion, but aims at producing a solution that is easier to round. Experiments show that this reduces the rounding errors considerably, both in combination with randomized and derandomized rounding.
For the max-coverage instances, we generally observe that the greedy heuristics also performs very good. We develop a strengthened method of derandomized rounding, and a simple greedy/rounding hybrid approach using greedy and LP-based rounding elements, and observe that both these improvements yield again better solutions than both earlier approaches on their own.
For unit disk max-domination, we also develop a PTAS. Contrary to all other algorithms investigated, it performs not much better in experiments than in theory; thus, unless extremely good solutions are to be obtained with huge computational resources, greedy, LP-based rounding or hybrid approaches are preferable.
We present an algorithm for the asymmetric traveling salesman problem on instances which satisfy the triangle inequality. Like several existing algorithms, it achieves approximation ratio O(log n). Unlike previous algorithms, it uses randomized rounding.
We describe a novel algorithm for rounding packing integer programs based on multidimensional Brownian motion in $\mathbb{R}^n$. Starting from an optimal fractional feasible solution $\bar{x}$, the procedure converges in polynomial time to a distribution over (possibly infeasible) point set $P \subset {\{0,1 \}}^n$ such that the expected value of any linear objective function over $P$ equals the value at $\bar{x}$. This is an alternate approach to the classical randomized rounding method of Raghavan and Thompson \cite{RT:87}. Our procedure is very general and in conjunction with discrepancy ba…
We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems where iterated rounding has been useful.
Randomized rounding is a standard method, based on the probabilistic method, for designing combinatorial approximation algorithms. In Raghavan's seminal paper introducing the method (1988), he writes: "The time taken to solve the linear program relaxations of the integer programs dominates the net running time theoretically (and, most likely, in practice as well)." This paper explores how this bottleneck can be avoided for randomized rounding algorithms for packing and covering problems (linear programs, or mixed integer linear programs, having no negative coefficients). The resulting algorith…
We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speed-scaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same qua…
Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative rounding. We provide an extension of the discrepancy-based rounding algorithm due to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such a…
We give a new randomized LP-rounding 1.725-approximation algorithm for the metric Fault-Tolerant Uncapacitated Facility Location problem. This improves on the previously best known 2.076-approximation algorithm of Swamy & Shmoys. To the best of our knowledge, our work provides the first application of a dependent-rounding technique in the domain of facility location. The analysis of our algorithm benefits from, and extends, methods developed for Uncapacitated Facility Location; it also helps uncover new properties of the dependent-rounding approach. An important concept that we develop is a no…
Many network design problems deal with the design of low-cost networks that are resilient to the failure of their elements, such as nodes or links. One such problem is Connectivity Augmentation, where the goal is to cheaply increase the connectivity of a network from a value k to k+1. The most studied setting focuses on edge-connectivity, which reduces to k=2, called Cactus Augmentation. From an approximation perspective, Byrka, Grandoni, and Jabal Ameli (2020) were the first to break the 2-approximation barrier for this problem, by exploiting a connection to the Steiner Tree problem, and by t…
International Computer Science Institute（国际计算机科学研究所）；University of California Berkeley（加州大学伯克利分校）；Xerox Research Centre India（施乐印度研究中心）；Purdue University（普渡大学）
We present and analyze a simple, two-step algorithm to approximate the optimal solution of the sparse PCA problem. Our approach first solves a L1 penalized version of the NP-hard sparse PCA optimization problem and then uses a randomized rounding strategy to sparsify the resulting dense solution. Our main theoretical result guarantees an additive error approximation and provides a tradeoff between sparsity and accuracy. Our experimental evaluation indicates that our approach is competitive in practice, even compared to state-of-the-art toolboxes such as Spasm.
Motivated by several applications, we consider the problem of randomly rounding a fractional solution in a matroid (base) polytope to an integral one. We consider the pipage rounding technique and also present a new technique, randomized swap rounding. Our main technical results are concentration bounds for functions of random variables arising from these rounding techniques. We prove Chernoff-type concentration bounds for linear functions of random variables arising from both techniques, and also a lower-tail exponential bound for monotone submodular functions of variables arising from random…
The Virtual Network Embedding Problem (VNEP) captures the essence of many resource allocation problems of today's infrastructure providers, which offer their physical computation and networking resources to customers. Customers request resources in the form of Virtual Networks, i.e. as a directed graph which specifies computational requirements at the nodes and communication requirements on the edges. An embedding of a Virtual Network on the shared physical infrastructure is the joint mapping of (virtual) nodes to physical servers together with the mapping of (virtual) edges onto paths in the …
Unsplittable flow problems cover a wide range of telecommunication and transportation problems and their efficient resolution is key to a number of applications. In this work, we study algorithms that can scale up to large graphs and important numbers of commodities. We present and analyze in detail a heuristic based on the linear relaxation of the problem and randomized rounding. We provide empirical evidence that this approach is competitive with state-of-the-art resolution methods either by its scaling performance or by the quality of its solutions. We provide a variation of the heuristic w…
The maximum volume $j$-simplex problem asks to compute the $j$-dimensional simplex of maximum volume inside the convex hull of a given set of $n$ points in $\mathbb{Q}^d$. We give a deterministic approximation algorithm for this problem which achieves an approximation ratio of $e^{j/2 + o(j)}$. The problem is known to be $\mathrm{NP}$-hard to approximate within a factor of $c^{j}$ for some constant $c > 1$. Our algorithm also gives a factor $e^{j + o(j)}$ approximation for the problem of finding the principal $j\times j$ submatrix of a rank $d$ positive semidefinite matrix with the largest det…
The multiway-cut problem is, given a weighted graph and k >= 2 terminal nodes, to find a minimum-weight set of edges whose removal separates all the terminals. The problem is NP-hard, and even NP-hard to approximate within 1+delta for some small delta > 0. Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this paper, we give improved randomized rounding schemes for their relaxation, yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation algorithm in general. Our approach hinges on …
Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations wh…
We design an expected polynomial-time, truthful-in-expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are m matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions, matroid weighted-rank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P != NP. Ours is the first truthful-in-expectation …
We develop polynomial-size LP-relaxations for {\em orienteering} and the {\em regret-bounded vehicle routing problem} (\rvrp) and devise suitable LP-rounding algorithms that lead to various new insights and approximation results for these problems. In orienteering, the goal is to find a maximum-reward $r$-rooted path, possibly ending at a specified node, of length at most some given budget $B$. In \rvrp, the goal is to find the minimum number of $r$-rooted paths of {\em regret} at most a given bound $R$ that cover all nodes, where the regret of an $r$-$v$ path is its length $-$ $c_{rv}$. For {…
The request-trip-vehicle assignment problem is at the heart of a popular decomposition strategy for online vehicle routing. In this framework, assignments are done in batches in order to exploit any shareability among vehicles and incoming travel requests. We study a natural ILP formulation and its LP relaxation. Our main result is an LP-based randomized rounding algorithm that, whenever the instance is feasible, leverages mild assumptions to return an assignment whose: i) expected cost is at most that of an optimal solution, and ii) expected fraction of unassigned requests is at most $1/e$. I…
We present a spectral approach to design approximation algorithms for network design problems. We observe that the underlying mathematical questions are the spectral rounding problems, which were studied in spectral sparsification and in discrepancy theory. We extend these results to incorporate additional non-negative linear constraints, and show that they can be used to significantly extend the scope of network design problems that can be solved. Our algorithm for spectral rounding is an iterative randomized rounding algorithm based on the regret minimization framework. In some settings, thi…
We study the problem of allocating $m$ items to $n$ agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a $1/e$ approximation factor of the objective. Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree $m$-homogeneous stable polynomial on $m$ variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our r…
We study problems that integrate depot location decisions along with the inventory routing problem of serving clients from these depots over time balancing the costs of routing vehicles from the depots with the holding costs of demand delivered before they are due. Since the inventory routing problem is already complex, we study the version that assumes that the daily vehicle routes are direct connections from the depot thus forming stars as solutions, and call this problem the Star Inventory Routing Problem with Facility Location (SIRPFL). As a stepping stone to solving SIRPFL, we first study…
We consider the vehicle routing problem with stochastic demands (VRPSD) on tree structured networks with a single depot. The problem we are concerned with in this paper is to find a set of tours for the vehicle with minimum expected length. Every tour begins at the depot, visits a subset of customers and returns to the depot without violating the capacity constraint. Randomized approximation algorithm achieving approximation guarantees of 2 for split-delivery VRPSD, and 3 for un-split delivery VRPSD are obtained.
Let A be a matrix, c be any linear objective function and x be a fractional vector, say an LP solution to some discrete optimization problem. Then a recurring task in theoretical computer science (and in approximation algorithms in particular) is to obtain an integral vector y such that Ax is roughly Ay and c*y exceeds c*x by only a moderate factor. We give a new randomized rounding procedure for this task, provided that A has bounded Delta-approximate entropy. This property means that for uniformly chosen random signs chi(j) in {-1,+1} on any subset of the columns, the outcome A*chi can be ap…
Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay $δ$ is incurred during which no data can be transmitted. For the offline version of the problem we present a $(1-\frac{1}{e}-ε)$ approximation ratio (for any c…
Given an undirected graph G=(V,E), a collection (s_1,t_1),...,(s_k,t_k) of k source-sink pairs, and an integer c, the goal in the Edge Disjoint Paths with Congestion problem is to connect maximum possible number of the source-sink pairs by paths, so that the maximum load on any edge (called edge congestion) does not exceed c. We show an efficient randomized algorithm to route $Ω(OPT/\poly\log k)$ source-sink pairs with congestion at most 14, where OPT is the maximum number of pairs that can be simultaneously routed on edge-disjoint paths. The best previous algorithm that routed $Ω(OPT/\poly\lo…
Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of $k$ independent sets (that is, $k$-fold matroid union), computing $k$ disjoint bases (a.k.a. matroid base packing), and covering the elements by $k$ bases (a.k.a. matroid base covering). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement a…
Cut games are among the most fundamental strategic games in algorithmic game theory. It is well-known that computing an exact pure Nash equilibrium in these games is PLS-hard, so research has focused on computing approximate equilibria. We present a polynomial-time algorithm that computes $2.7371$-approximate pure Nash equilibria in cut games. This is the first improvement to the previously best-known bound of $3$, due to the work of Bhalgat, Chakraborty, and Khanna from EC 2010. Our algorithm is based on a general recipe proposed by Caragiannis, Fanelli, Gravin, and Skopalik from FOCS 2011 an…
We introduce a novel method for clustering using a semidefinite programming (SDP) relaxation of the Max k-Cut problem. The approach is based on a new methodology for rounding the solution of an SDP relaxation using iterated linear optimization. We show the vertices of the Max k-Cut relaxation correspond to partitions of the data into at most k sets. We also show the vertices are attractive fixed points of iterated linear optimization. Each step of this iterative process solves a relaxation of the closest vertex problem and leads to a new clustering problem where the underlying clusters are mor…
We consider the following fundamental routing problem. An adversary inputs packets arbitrarily at sources, each packet with an arbitrary destination. Traffic is constrained by link capacities and buffer sizes, and packets may be dropped at any time. The goal of the routing algorithm is to maximize throughput, i.e., route as many packets as possible to their destination. Our main result is an $O\left(\log n\right)$-competitive deterministic algorithm for an $n$-node line network (i.e., $1$-dimensional grid), requiring only that buffers can store at least $5$ packets, and that links can deliver …
Recently a number of randomized 3/4-approximation algorithms for MAX SAT have been proposed that all work in the same way: given a fixed ordering of the variables, the algorithm makes a random assignment to each variable in sequence, in which the probability of assigning each variable true or false depends on the current set of satisfied (or unsatisfied) clauses. To our knowledge, the first such algorithm was proposed by Poloczek and Schnitger; Van Zuylen subsequently gave an algorithm that set the probabilities differently and had a simpler analysis. She also set up a framework for deriving s…
We consider vehicle-routing problems (VRPs) that incorporate the notion of {\em regret} of a client, which is a measure of the waiting time of a client relative to its shortest-path distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the {\em regret-bounded vehicle routing problem} (RVRP). In these problems, we are given an undirected complete graph $G=(\{r\}\cup V,E)$ on $n$ nodes with a distinguished root (depot) node $r$, edge costs $\{c_{uv}\}$ that form a metric, and a regret bound $R$. Given a path $P$ rooted at $r$ and a node $v…
We study the NP-hard problem of approximating a Minimum Routing Cost Spanning Tree in the message passing model with limited bandwidth (CONGEST model). In this problem one tries to find a spanning tree of a graph $G$ over $n$ nodes that minimizes the sum of distances between all pairs of nodes. In the considered model every node can transmit a different (but short) message to each of its neighbors in each synchronous round. We provide a randomized $(2+ε)$-approximation with runtime $O(D+\frac{\log n}ε)$ for unweighted graphs. Here, $D$ is the diameter of $G$. This improves over both, the (expe…
We study a submodular maximization problem motivated by applications in online retail. A platform displays a list of products to a user in response to a search query. The user inspects the first $k$ items in the list for a $k$ chosen at random from a given distribution, and decides whether to purchase an item from that set based on a choice model. The goal of the platform is to maximize the engagement of the shopper defined as the probability of purchase. This problem gives rise to a less-studied variation of submodular maximization in which we are asked to choose an $\textit{ordering}$ of a s…
