# Viswanath Nagarajan's research while affiliated with University of Michigan and other places

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## Publications (101)

We consider a general online network design problem in which a sequence of N requests arrive over time, each of which needs to use some subset of the available resources E. The cost incurred by a resource [Formula: see text] is some function f e of the total load [Formula: see text] on that resource. The objective is to minimize the total cost [For...

Adaptivity in Stochastic Submodular Cover
Solutions to stochastic optimization problems are typically sequential decision processes that make decisions one by one, waiting for (and using) the feedback from each decision. Whereas such “adaptive” solutions achieve the best objective, they can be very time-consuming because of the need to wait for fee...

We study the $K$-armed dueling bandit problem, a variation of the traditional multi-armed bandit problem in which feedback is obtained in the form of pairwise comparisons. Previous learning algorithms have focused on the $\textit{fully adaptive}$ setting, where the algorithm can make updates after every comparison. The "batched" dueling bandit prob...

We consider the stochastic score classification problem. There are several binary tests, where each test i is associated with a probability pi of being positive, a cost ci, and a weight ai. The score of an outcome is a weighted sum of all positive tests, and the range of possible scores is partitioned into intervals corresponding to different class...

The $K$-armed dueling bandit problem, where the feedback is in the form of noisy pairwise comparisons, has been widely studied. Previous works have only focused on the sequential setting where the policy adapts after every comparison. However, in many applications such as search ranking and recommendation systems, it is preferable to perform compar...

The k-Supplier problem is an important location problem that has been actively studied in both general and Euclidean metrics. Many of its variants have also been studied, primarily on general metrics. We study two variants of k-Supplier, namely Priority k-Supplier and k-Supplier with Outliers, in Euclidean metrics. We obtain (1+3)-approximation alg...

A fundamental task in active learning involves performing a sequence of tests to identify an unknown hypothesis that is drawn from a known distribution. This problem, known as optimal decision tree induction, has been widely studied for decades and the asymptotically best-possible approximation algorithm has been devised for it. We study a generali...

Assortment optimization involves selecting a subset of products to offer to customers in order to maximize revenue. Often, the selected subset must also satisfy some constraints, such as capacity or space usage. Two key aspects in assortment optimization are (1) modeling customer behavior and (2) computing optimal or near-optimal assortments effici...

The $k$-Supplier problem is an important location problem that has been actively studied in both general and Euclidean metrics. Many of its variants have also been studied, primarily on general metrics. We study two variants of $k$-Supplier, namely Priority $k$-Supplier and $k$-Supplier with Outliers, in Euclidean metrics. We obtain $(1+\sqrt{3})$-...

We study stochastic combinatorial optimization problems where the objective is to minimize the expected maximum load (a.k.a. the makespan). In this framework, we have a set of n tasks and m resources, where each task j uses some subset of the resources. Tasks have random sizes \(X_j\), and our goal is to non-adaptively select t tasks to minimize th...

We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasi-polynomial time [Formula: see text]-appr...

We consider the stochastic score classification problem. There are several binary tests, where each test $i$ is associated with a probability $p_i$ of being positive and a cost $c_i$. The score of an outcome is a weighted sum of all positive tests, and the range of possible scores is partitioned into intervals corresponding to different classes. Th...

In the stochastic submodular cover problem, the goal is to select a subset of stochastic items of minimum expected cost to cover a submodular function. Solutions in this setting correspond to sequential decision processes that select items one by one "adaptively" (depending on prior observations). While such adaptive solutions achieve the best obje...

We consider the problem of makespan minimization on unrelated machines when job sizes are stochastic. The goal is to find a fixed assignment of jobs to machines, to minimize the expected value of the maximum load over all the machines. For the identical-machines special case when the size of a job is the same across all machines, a constant-factor...

We consider the a priori traveling repairman problem, which is a stochastic version of the classic traveling repairman problem. Given a metric (V,d) with a root r∈V, the traveling repairman problem (TRP) involves finding a tour originating from r that minimizes the sum of arrival-times at all vertices. In its a priori version, we are also given ind...

We study stochastic combinatorial optimization problems where the objective is to minimize the expected maximum load (a.k.a. the makespan). In this framework, we have a set of n tasks and m resources, where each task j uses some subset of the resources. Tasks have random sizes \(X_j\), and our goal is to non-adaptively select t tasks to minimize th...

We study stochastic combinatorial optimization problems where the objective is to minimize the expected maximum load (a.k.a. the makespan). In this framework, we have a set of n tasks and m resources, where each task j uses some subset of the resources. Tasks have random sizes X_j, and our goal is to non-adaptively select t tasks to minimize the ex...

The [Formula: see text]-supplier problem is a fundamental location problem that involves opening [Formula: see text] facilities to minimize the maximum distance of any client to an open facility. We consider the [Formula: see text]-supplier problem in Euclidean metrics (of arbitrary dimension) and present an algorithm with approximation ratio [Form...

This paper provides a unified family of algorithms with performance guarantees for malleable scheduling problems on flows. A flow represents a set of jobs with precedence constraints. Each job has a speedup function that governs the rate at which work is done on the job as a function of the number of processors allocated to it. In our setting, each...

We consider fractional online covering problems with \(\ell _q\)-norm objectives as well as its dual packing problems. The problem of interest is of the form \(\min \{ f(x) \,:\, Ax\ge 1, x\ge 0\}\) where \(f(x)=\sum _{e} c_e \Vert x(S_e)\Vert _{q_e} \) is the weighted sum of \(\ell _q\)-norms and A is a non-negative matrix. The rows of A (i.e. cov...

We consider the problem of makespan minimization on unrelated machines when job sizes are stochastic. The goal is to find a fixed assignment of jobs to machines, to minimize the expected value of the maximum load over all the machines. For the identical machines special case when the size of a job is the same across all machines, a constant-factor...

We consider the a priori traveling repairman problem, which is a stochastic version of the classic traveling repairman problem (also called the traveling deliveryman or minimum latency problem). Given a metric $(V,d)$ with a root $r\in V$, the traveling repairman problem (TRP) involves finding a tour originating from $r$ that minimizes the sum of a...

We consider the following general network design problem on directed graphs. The input is an asymmetric metric $(V,c)$, root $r\in V$, monotone submodular function $f:2^V \xrightarrow{} \mathbb{R}_+$ and budget $B$. The goal is to find an $r$-rooted arborescence $T$ of cost at most $B$ that maximizes $f(T)$. Our main result is a quasi-polynomial ti...

We consider the max-cut and max-$k$-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a $\frac{1}{2}$-approximation algorithm for this class of problems.

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) combin...

We study a general stochastic ranking problem where an algorithm needs to adaptively select a sequence of elements so as to “cover” a random scenario (drawn from a known distribution) at minimum expected cost. The coverage of each scenario is captured by an individual submodular function, where the scenario is said to be covered when its function v...

An instance of the graph-constrained max-cut (\(\mathsf {GCMC}\)) problem consists of (i) an undirected graph \(G=(V,E)\) and (ii) edge-weights \(c:{V\atopwithdelims ()2} \rightarrow \mathbb {R}_+\) on a complete undirected graph. The objective is to find a subset \(S \subseteq V\) of vertices satisfying some graph-based constraint in G that maximi...

We consider fractional online covering problems with $\ell_q$-norm objectives. The problem of interest is of the form $\min\{ f(x) \,:\, Ax\ge 1, x\ge 0\}$ where $f(x)=\sum_{e} c_e \|x(S_e)\|_{q_e} $ is the weighted sum of $\ell_q$-norms and $A$ is a non-negative matrix. The rows of $A$ (i.e. covering constraints) arrive online over time. We provid...

We consider the problem of constructing optimal decision trees: given a collection of tests that can disambiguate between a set of m possible diseases, each test having a cost, and the a priori likelihood of any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? This pr...

We study the Minimum Latency Submodular Cover (MLSC) problem, which consists of a metric (V, d) with source r ∈ V and m monotone submodular functions f1, f2, …, fm: 2V → [0, 1]. The goal is to find a path originating at r that minimizes the total “cover time” of all functions. This generalizes well-studied problems, such as Submodular Ranking...

We consider the hub label optimization problem, which arises in designing fast preprocessing-based shortest-path algorithms. We give O(log n)-approximation algorithms for the objectives of minimizing the maximum label size (ℓ∞-norm) and simultaneously minimizing a constant number of ℓp-norms. Prior to this, an O(log n)-approximation algorithm was k...

We study a general adaptive ranking problem where an algorithm needs to perform a sequence of actions on a random user, drawn from a known distribution, so as to "satisfy" the user as early as possible. The satisfaction of each user is captured by an individual submodular function, where the user is said to be satisfied when the function value goes...

We study a location-routing problem in the context of capacitated vehicle routing. The input to the k-location capacitated vehicle routing problem (k-LocVRP) consists of a set of demand locations in a metric space and a fleet of k identical vehicles, each of capacity Q. The objective is to locate k depots, one for each vehicle, and compute routes f...

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) combin...

An instance of the graph-constrained max-cut (\(\mathsf {GCMC}\)) problem consists of (i) an undirected graph \(G=(V,E)\) and (ii) edge-weights \(c:{V\atopwithdelims ()2} \rightarrow \mathbb {R}_+\) on a complete undirected graph. The objective is to find a subset \(S \subseteq V\) of vertices satisfying some graph-based constraint in G that maximi...

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) combin...

An instance of the graph-constrained max-cut (GCMC) problem consists of (i)
an undirected graph G and (ii) edge-weights on a complete undirected graph on
the same vertex set. The objective is to find a subset of vertices satisfying
some graph-based constraint in G that maximizes the total weight of edges in
the cut. The types of graph constraints w...

In the classical k-median problem, we are given a metric space and want to open k centers so as to minimize the sum (over all the vertices) of the distance of each vertex to its nearest open center. In this paper we present the first constant-factor approximation algorithms for two natural generalizations of this problem that handle matroid or knap...

We consider the following two deterministic inventory optimization problems
over a finite planning horizon $T$ with non-stationary demands.
(a) Submodular Joint Replenishment Problem: This involves multiple item types
and a single retailer who faces demands. In each time step, any subset of
item-types can be ordered incurring a joint ordering cost...

In the stochastic orienteering problem, we are given a finite metric space, where each node contains a job with some deterministic reward and a random processing time. The processing time distributions are known and independent across nodes. However the actual processing time of a job is not known until it is completely processed. The objective is...

We introduce the X-Flex cross-platform scheduler. X-Flex is intended as an alternative to the Dominant Resource Fairness (DRF) scheduler currently employed by both YARN and Mesos. There are multiple design differences between X-Flex and DRF. For one thing, DRF is based on an instantaneous notion of fairness, while X-Flex monitors instantaneous fair...

We consider circuit routing with an objective of minimizing energy, in a
network of routers that are speed scalable and that may be shutdown when idle.
We consider both multicast routing and unicast routing. It is known that this
energy minimization problem can be reduced to a capacitated flow network design
problem, where vertices have a common ca...

We consider virtual circuit routing protocols, with an objective of minimizing energy, in a network of components that are speed scalable, and that may be shutdown when idle. We assume that the speed s of a link is proportional to its load, and assume the standard model for component power, namely that the power is some constant static power σ plus...

We introduce FlowFlex, a highly generic and effective scheduler for flows of MapReduce jobs connected by precedence constraints. Such a flow can result, for example, from a single user-level Pig, Hive or Jaql query. Each flow is associated with an arbitrary function describing the cost incurred in completing the flow at a particular time. The overa...

The input to the stochastic orienteering problem [14] consists of a budget B and metric (V,d) where each vertex v ∈ V has a job with a deterministic reward and a random processing time (drawn from a known distribution). The processing times are independent across vertices. The goal is to obtain a non-anticipatory policy (originating from a given ro...

We consider the problem of approximating optimal hub labelings in the context of labeling algorithms for the shortest path problem. A previous result was a O(logn) approximating for minimizing the total label size. We give an O(logn)-approximation algorithm for the maximum label size. We also give O(logn)-approximation algorithms for natural genera...

In the k-supplier problem, we are given a set of clients C and set of facilities F located in a metric (C ∪ F, d), along with a bound k. The goal is to open a subset of k facilities so as to minimize the maximum distance of a client to an open facility, i.e., min S ⊆ F: |S| = k
max v ∈ C
d(v,S), where d(v,S) = min u ∈ S
d(v,u) is the minimum distan...

We study a general stochastic probing problem defined on a universe V, where each element e ∈ V is “active” independently with probability p
e
. Elements have weights {w
e
:e ∈ V} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the p
e
values—to determine whether or not an element e is acti...

We consider a class of multi-stage robust covering problems, where additional information is revealed about the problem instance in each stage, but the cost of taking actions increases. The dilemma for the decision-maker is whether to wait for additional information and risk the inflation, or to take early actions to hedge against rising costs. We...

In the binary paintshop problem, there are m cars appearing in a sequence of length 2m, with each car occurring twice. Each car needs to be colored with two colors. The goal is to choose for each car, which of its occurrences receives either color, so as to minimize the total number of color changes in the sequence. We show that the binary paintsho...

We consider virtual circuit multicast routing in a network of links that are speed scalable. We assume that a link with load f uses power σ + f
α
, where σ is the static power, and α > 1 is some constant. We assume that a link may be shutdown if not in use. In response to the arrival of client i at vertex t
i
a routing path (the virtual circuit) P...

A covering integer program (CIP) is a mathematical program of the form:
$$\begin{aligned} \min \{ c^\top \mathbf{x} \mid A\mathbf{x} \geq \mathbf{1},\; \mathbf{0} \leq \mathbf{x} \leq \mathbf{u},\; \mathbf{x} \in {\ensuremath{\mathbb{Z}}}^n\},\nonumber \end{aligned}$$ where \(A \in R_{\geq 0}^{m \times n}, c,u \in {\ensuremath{\mathbb{R}}}_{\geq 0}...

In the Stochastic Orienteering problem, we are given a metric, where each node also has a job located there with some deterministic reward and a random size. (Think of the jobs as being chores one needs to run, and the sizes as the amount of time it takes to do the chore.) The goal is to adaptively decide which nodes to visit to maximize total expe...

We study the distance constrained vehicle routing problem (DVRP) (Laporte et al., Networks 14 (1984), 47–61, Li et al., Oper Res 40 (1992), 790–799): given a set of vertices in a metric space, a specified depot, and a distance bound D, find a minimum cardinality set of tours originating at the depot that covers all vertices, such that each tour has...

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed – but costs...

We consider the vehicle routing problem with stochastic demands (VRPSD). We give randomized approximation algorithms achieving approximation guarantees of 1 + α for split-delivery VRPSD, and 2 + α for unsplit-delivery VRPSD; here a is the best approximation guarantee for the traveling salesman problem. These bounds match the best known for even the...

In the classical k-median problem, we are given a metric space and would like to open k centers so as to minimize the sum (over all the vertices) of the distance of each vertex to its nearest open center. In this paper, we consider the following generalization of the problem: instead of opening at most k centers, what if each center belongs to one...

We study graph partitioning problems from a min-max perspective, in which an
input graph on n vertices should be partitioned into k parts, and the objective
is to minimize the maximum number of edges leaving a single part. The two main
versions we consider are where the k parts need to be of equal-size, and where
they must separate a set of k given...

We study the Minimum Latency Submodular Cover problem (MLSC), which consists
of a metric $(V,d)$ with source $r\in V$ and $m$ monotone submodular functions
$f_1, f_2, ..., f_m: 2^V \rightarrow [0,1]$. The goal is to find a path
originating at $r$ that minimizes the total cover time of all functions. This
generalizes well-studied problems, such as S...

This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed
k-TSP problem: given an asymmetric metric (V,d), a root r∈V and a target k≤|V|, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time -approximation algorithm for this problem. We use this algor...

We study a basic resource allocation problem that arises in cloud computing environments. The physical network of the cloud is represented as a graph with vertices denoting servers and edges corresponding to communication links. A workload is a set of processes with processing requirements and mutual communication requirements. The workloads arrive...

Dial a ride problems consist of a metric space (denoting travel time between
vertices) and a set of m objects represented as source-destination pairs, where
each object requires to be moved from its source to destination vertex. We
consider the multi-vehicle Dial a ride problem, with each vehicle having
capacity k and its own depot-vertex, where th...

We study a location-routing problem in the context of capacitated vehicle routing. The input to k-LocVRP is a set of demand locations in a metric space and a fleet of k vehicles each of capacity Q. The objective is to locate
k depots, one for each vehicle, and compute routes for the vehicles so that all demands are satisfied and the total cost is m...

Consider the following problem: given a set system (U, Ω) and an edge-weighted graph G = (U, E) on the same universe U, find the set A ∈ Ω such that the Steiner tree cost with terminals A is as large as possible—“which set in Ω is the most difficult to connect up?” This is an example of a max-min problem: find the set A ∈ Ω such that the val...

The capacitated vehicle routing problem (CVRP) [21] involves distributing (identical) items from a depot to a set of demand locations in the shortest possible time, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present...

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained
network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling
more complex degree constraints in the minimum spanning tree problem (namely laminar crossing spann...

Consider a random graph model where each possible edge e is present independently with some probability p
e
. We are given these numbers p
e
, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the grap...

Consider a random graph model where each possible edge $e$ is present independently with some probability $p_e$. Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, w...

Consider a random graph model where each possible edge e is present independently with some probability pe. We are given these numbers pe, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, w...

We consider the problem of constructing optimal decision trees: given a collection of tests which can disambiguate between
a set of m possible diseases, each test having a cost, and the a-priori likelihood of the patient having any particular disease, what
is a good adaptive strategy to perform these tests to minimize the expected cost to identify...

In the classic broadcast scheduling problem, there are n pages stored at a server, and requests for these pages arrive over time. Whenever a page is broadcast, it satisfies all outstanding requests for that page. The objective is to minimize the average flowtime of the requests. In this paper, for any ε> 0, we give a (1 + ε)-speed O(1/ε
3)-competit...

This note presents improved approximation guarantees for the requirement cut problem: given an n-vertex edge-weighted graph G=(V,E), and g groups of vertices X1,…,Xg⊆V with each group Xi having a requirement ri between 0 and |Xi|, the goal is to find a minimum cost set of edges whose removal separates each group Xi into at least ri disconnected com...

We consider the Survivable Network Design Problem (SNDP) and the Symmetric Traveling Salesman Problem (STSP). We give simpler proofs of the existence of a -edge and 1-edge in any extreme point of the natural LP relaxations for the SNDP and STSP, respectively. We formulate a common generalization of both problems and show our results by a new counti...

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing span...

This results in this paper have been merged with the result in arXiv:1002.3763v1 The authors would like to withdraw this version. Please see arXiv:1008.5356v1 for the merged version. Comment: This paper has been withdrawn due to new merged paper arXiv:1008.5356v1

where for any $F \subseteq V,d(j,F) = \min _{feF} d(j,f)$ . This is a "min-max" or "robust" version of the k-median problem. Note that in contrast to the recent papers on robust and stochastic problems, we have only one stage of decision-making where we select a set of k facilities to open. Once a set of open facilities is fixed, each client in the...

Submodular function maximization is a central problem in combinatorial optimization, generalizing many im- portant problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maxi- m...

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow and require coverage, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case covering cost (summed over both days) is minimized? We consider the k-robust model [6,15] where...

Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n × n symmetric nonnegative matrices [Formula: see text] and [Formula: see text...

Dial-a-Ride problems consist of a set V of n vertices in a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs \(\{(s_i,t_i)\}^m_{i=1}\), where each object requires to be moved from its source to destination vertex. In the multi-vehicle Dial-a-Ride problem, there are q vehicles each ha...

We present polynomial-time approximation algorithms for some degree-bounded directed network design problems. Our main result is for intersecting supermodular connectivity requirements with degree bounds: given a directed graph $G=(V,E)$ with nonnegative edge-costs, a connectivity requirement specified by an intersecting supermodular function $f$,...

We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and
$O(k^2)$. We also show that the integrality gap...

In flow shop scheduling there are m machines and n jobs, such that every job has to be processed on the machines in the fixed order 1,…,m. In the permutation flow shop problem, it is also required that each machine process the set of all jobs in the same order. Formally, given n jobs along with their processing times on each machine, the goal is to...

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important prob- lems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, max- imum entropy sampling, and maximum facility location prob- lems. Unlike submodular minimization, submodular max...

Suppose we are given a schedule of train movements over a rail network into which a new train is to be included. The origin and the destination are specified for the new train; it is required that a schedule (including the path) be determined for it so as to minimize the time taken without affecting the schedules for the old trains. In the standard...

We present polynomial-time approximation algorithms for some degree-bounded directed network design problems. Our main result is for intersecting supermodular connectivity with degree bounds: given a directed graph G=(V,E) with non-negative edge-costs, a connectivity requirement specified by an intersecting supermodular function f, and upper bounds...

This paper studies an extension of the k-median prob- lem where we are given a metric space (V;d) and not just one but m client sets fSi µ V gm i=1, and the goal is to open k facilities F to minimize: maxi2(m) 'P j2Si d(j;F) " ; i.e., the worst-case cost over all the client sets. This is a \min-max" or \robust" version of the k-median prob- lem; ho...

We study the directed minimum latency problem: given an n-vertex asymmetric metric (V,d) with a root vertex r ∈ V, find a spanning path originating at r that minimizes the sum of latencies at all vertices (the latency of any vertex v ∈ V is the distance from r to v along the path). This problem has been well-studied on symmetric metrics, and the be...

The k-forest problem is a common generalization of both the k-MST and the dense-k-subgraph problems. Formally, given a metric space on n vertices V , with m demand pairs ⊆ V × V and a “target” k ≤ m , the goal is to find a minimum cost subgraph that connects at least k pairs. In this paper, we give an O (min{√ n ⋅log k ,√ k })-approximation algorit...

This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed k
-TSP problem: given an asymmetric metric (V,d), a root r ∈ V and a target k ≤ |V|, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time O(log2
n·logk)-approximation algorithm for this problem...

The Dial-a-Ride problem is the following: given an n point metric space, m objects each with its own source and destination, and a vehicle of capacity k, find the minimum length tour that uses this vehicle to move each object from its source to destination subject to there being at most k objects in the vehicle at any time. We want that the tour be...

In this paper, we study the following vehicle routing problem: given n vertices in a metric space, a specified root vertex r (the depot), and a length bound D, find a minimum cardinality set of r-paths that covers all vertices, such that each path has length at most D. This problem is \(\mathcal{NP}\)-complete, even when the underlying metric is in...

In this paper, we unify several graph partitioning problems including multicut, multiway cut, and k-cut, into a single problem. The input to a requirement cut problem is an undirected edge-weighted graph G=(V,E), and g groups of vertices X
1, ⋯ ,X
g
⊆ V, each with a requirement r
i
between 0 and |X
i
|. The goal is to find a minimum cost set of edg...

## Citations

... This is because, as shown by [19], any B-round algorithm for batched MAB in the adversarial setting has regret Ω(T /B). Adaptivity and batch processing has been recently studied for stochastic submodular cover [1,20,22,23], and for various stochastic "maximization" problems such as knapsack [12,16], matching [9,11], probing [26] and orienteering [10,24,25]. Recently, there have also been several results examining the role of adaptivity in (deterministic) submodular optimization; e.g. ...

... All the bounds in the table have a dependence on n, meaning that none of the listed SBFE problems has a constant adaptivity gap. This contrasts with recent work of Ghuge et al. [14], which shows that the adaptivity gaps for the SBFE problem for symmetric Boolean functions and linear threshold functions are O (1). ...

... Note. An independent recent work of Lee, Nagarajan and Wang [12] obtained a (1 + √ 3)approximation algorithm for Robust Euclidean k-Supplier. Our approach is similar to theirs, and at a high level essentially identical, but there are technical differences in both the polytope used for the round-or-cut algorithm as well as in the algorithm that solves the constraint version of Edge Cover that appears as an intermediate problem during the iterations of the ellipsoid algorithm. ...

... We prove that there does not exist a constant-factor approximation for these extensions assuming the widely-used Exponential Time Hypothesis (ETH) in theoretical computer science community. These results further illustrate the hardness of our problem under the base model in view of the existence of constant-factor approximation for extensions of some closely related problems in the literature (e.g., the constant-factor approximation for the cardinality-constrained assortment optimization problems under the PCL model studied in Ghuge et al. (2022)). ...

... There are non-adaptive O(1)-approximations known for identical machines [KRT00], unrelated machines [GKNS21] and the ℓ q -norm objective [Mol19]. [GKNS22] give non-adaptive poly(log log m)-approximations for more general load balancing problems, where at least t stochastic sets out of a structured set system of size n have to be selected. ...

... Their algorithm also yields an O(log 2 n log k)approximation in general graphs via embeddings into tree metrics [6,13]. Better approximation in quasi-polynomial time are known [11,17,15]. Set Connectivity is a generalization of group Steiner tree problem. ...

... These results are asymptotically tight as shown by the lower bound of Ω log m log log m on the adaptivity gap [GKNS21] and the lower bound of Ω(log m) on the competitive ratio of any deterministic online algorithm, even for deterministic requests [ANR95]. In particular, the theorem implies that the adaptivity gap for stochastic load balancing is Θ log m log log m . ...

... Recent work has focused on the a priori TRP [45,31]. Following earlier work on the a priori TSP [11,25,27], this problem seeks a master tour under demand uncertainty, where each vertex is present with some probability. ...

... A major challenge in designing algorithms for such cost-functions is that one needs the balance two opposing goals (1) aggregating demands in the concave regime and (2) separating demands in the convex regime. Prior work [6,7,31] has mainly focused on the special case of uniform (or related) cost functions where the α e s and σ e ξ e s are uniform across all resources e. ...

... The best known polynomial-time approximation ratio for the problem is only O(k ) with a running time of n O(1/ ) for any constant > 0 [Zel97, CCC + 99]. With quasi-polynomial time algorithms, one can achieve an GN20]. It is known from the work of Halperin and Krauthgamer [HK03] that unless NP ⊆ ZPTIME(n poly log(n) ), there is no polynomial time O(log 2− k)-approximation for the problem for any constant > 0. The long-standing open question on DST is whether a poly-logarithmic approximation for the problem can be achieved in polynomial time. ...