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The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive quantitative bounds on the expected suprema and the degree of continuity of sample paths for many processes.

One of the crowning achievements of the theory is Talagrand’s tight alternative characterization of the suprema of Gaussian processes in terms of majorizing measures. The proof of this theorem was difficult, and thus considerable effort was put into the task of developing both shorter and easier to understand proofs. A major reason for this difficulty was considered to be theory of majorizing measures itself, which had the reputation of being opaque and

mysterious. As a consequence, most recent treatments of the theory (including by Talagrand himself) have eschewed the use of majorizing measures in favor of a purely combinatorial approach (the *generic chaining*) where objects based on sequences of partitions provide roughly matching upper and lower bounds on the desired expected supremum.

In this paper, we return to majorizing measures as a primary object of study, and give a viewpoint that we think is natural and clarifying from an optimization perspective. As our main contribution, we give an algorithmic proof

of the majorizing measures theorem based on two parts:

- We make the simple (but apparently new) observation that finding the best majorizing measure can be cast as a convex program.

This also allows for efficiently computing the measure using off-the-shelf methods from convex optimization. - We obtain tree-based upper and lower bound certificates by
*rounding*, in a series of steps, the primal and dual solutions to this convex program.

While duality has conceptually been part of the theory since its beginnings, as far as we are aware no explicit link to convex optimization has been previously made.

A classical problem in appointment scheduling, with applications in health care, concerns the determination of the patients’ arrival times that minimize a cost function that is a weighted sum of mean waiting times and mean idle times. Part of this problem is the sequencing problem, which focuses on ordering the patients. We assess the performance of the smallest-variance-first (SVF) rule, which sequences patients in order of increasing variance of their service durations. While it was known that SVF is not always optimal, many papers have found that it performs well in practice and simulation. We give theoretical justification for these observations by proving quantitative worst-case bounds on the ratio between the cost incurred by the SVF rule and the minimum attainable cost, in a number of settings. We also show that under quite general conditions, this ratio approaches 1 as the number of patients grows large, showing that the SVF rule is asymptotically optimal. While this viewpoint in terms of approximation ratio is a standard approach in many algorithmic settings, our results appear to be the first of this type in the appointment scheduling literature.

Hypercongestion—the phenomenon that higher traffic densities can reduce throughput—is well understood at the link level, but has also been observed in a macroscopic form at the level of traffic networks; for instance, in morning rush-hour traffic into a downtown core. In this paper, we show that macroscopic hypercongestion can occur as a purely emergent effect of dynamic equilibrium behaviour on a network, even if the underlying link dynamics (we consider Vickrey bottlenecks with spaceless vertical queues) do not exhibit hypercongestion.

We introduce a new iterative rounding technique to round a point in a matroid polytope subject to further matroid constraints. This technique returns an independent set in one matroid with limited violations of the other ones. On top of the classical steps of iterative relaxation approaches, we iteratively refine/split involved matroid constraints to obtain a more restrictive constraint system, that is amenable to iterative relaxation techniques. Hence, throughout the iterations, we both tighten constraints and later relax them by dropping constrains under certain conditions. Due to the refinement step, we can deal with considerably more general constraint classes than existing iterative relaxation/rounding methods, which typically round on one matroid polytope with additional simple cardinality constraints that do not overlap too much.

We show how our rounding method, combined with an application of a matroid intersection algorithm, yields the first 2-approximation for finding a maximum-weight common independent set in 3 matroids. Moreover, our 2-approximation is LP-based, and settles the integrality gap for the natural relaxation of the problem. Prior to our work, no better upper bound than 3 was known for the integrality gap, which followed from the greedy algorithm. We also discuss various other applications of our techniques, including an extension that allows us to handle a mixture of matroid and knapsack constraints.

Flows over time have received substantial attention from both an optimization and (more recently) a game-theoretic perspective. In this model, each arc has an associated delay for traversing the arc, and a bound on the rate of flow entering the arc; flows are time-varying. We consider a setting which is very standard within the transportation economic literature, but has received little attention from an algorithmic perspective. The flow consists of users who are able to choose their route but also their departure time, and who desire to arrive at their destination at a particular time, incurring a *scheduling cost* if they arrive earlier or later. The total cost of a user is then a combination of the time they spend commuting, and the scheduling cost they incur. We present a combinatorial algorithm for the natural optimization problem, that of minimizing the average total cost of all users (i.e., maximizing the social welfare). Based on this, we also show how to set tolls so that this optimal flow is induced as an equilibrium of the underlying game.

We give new approximation algorithms for the submodular joint replenishment problem and the inventory routing problem, using an iterative rounding approach. In both problems, we are given a set of *N* items and a discrete time horizon of *T* days in which given demands for the items must be satisfied. Ordering a set of items incurs a cost according to a set function, with properties depending on the problem under consideration. Demand for an item at time *t* can be satisfied by an order on any day prior to *t*, but a holding

cost is charged for storing the items during the intermediate period; the goal is to minimize the sum of the ordering and holding cost. Our approximation factor for both problems is *O*(log log min(*N,T*)); this

improves exponentially on the previous best results.

We consider the following natural scheduling problem: Given a sequence of jobs with weights and processing times, one needs to assign each job to one of *m* identical machines in order to minimize the sum of weighted completion times. The twist is that for machine the jobs assigned to it must obey the order of the input sequence, as is the case in multi-server queuing systems. We establish a constant factor approximation algorithm for this (strongly NP-hard) problem. Our approach is necessarily very different from what has been used for similar scheduling problems without the fixed-order assumption. We also give a QPTAS for the special case of unit processing times

We present a new strongly polynomial algorithm for generalized flow maximization. The first strongly polynomial algorithm for this problem was given in [Végh 2016]; our new algorithm is much simpler, and much faster. The complexity bound $O((m+n\log n)mn\log (n^2/m))$ improves on the previous estimate in [ Végh 2016] by almost a factor $O(n^2)$. Even for small numerical parameter values, our algorithm is essentially as fast as the best weakly polynomial algorithms. The key new technical idea is relaxing primal feasibility conditions. This allows us to work almost exclusively with integral flows, in contrast to all previous algorithms for the problem.

SODA 2018

We consider the problem of finding a spanning tree satisfying a family of additional constraints. Several settings have been considered previously, the most famous being the problem of finding a spanning tree with degree constraints. Since the problem is hard, the goal is typically to find a spanning tree that violates the constraints as little as possible.

Iterative rounding became the tool of choice for constrained spanning tree problems. However, iterative rounding approaches are very hard to adapt to settings where an edge can be part of a super-constant number of constraints. We consider a natural constrained spanning tree problem of this type, namely where upper bounds are imposed on a family of cuts forming a chain. Our approach reduces the problem to a family of independent matroid intersection problems, leading to a spanning tree that violates each constraint by a factor of at most 9.

We also present strong hardness results: among other implications, these are the first to show, in the setting of a basic constrained spanning tree problem, a qualitative difference between what can be achieved when allowing multiplicative as opposed to additive constraint violations.

The bottleneck of the currently best (ln(4) + epsilon)-approximation algorithm for the NP-hard Steiner tree problem is the solution of its large, so called hypergraphic, linear programming relaxation (HYP). Hypergraphic LPs are NP-hard to solve exactly, and it is a formidable computational task to even approximate them sufficiently well. We focus on another well-studied but poorly understood LP relaxation of the problem: the bidirected cut relaxation (BCR). This LP is compact, and can therefore be solved efficiently. Its integrality gap is known to be greater than 1.16, and while this is widely conjectured to be close to the real answer, only a (trivial) upper bound of 2 is known. In this paper, we give an efficient constructive proof that BCR and HYP are polyhedrally equivalent in instances that do not have an (edge-induced) claw on Steiner vertices, i.e., they do not contain a Steiner vertex with 3 Steiner neighbors. This implies faster ln(4)-approximations for these graphs, and is a significant step forward from the previously known equivalence for (so called quasi-bipartite) instances in which Steiner vertices form an independent set. We complement our results by showing that even restricting to instances where Steiner vertices induce one single star, determining whether the two relaxations are equivalent is NP-hard.

Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the two following cases: we say $T$ is explosive if \[ \lim_{n\to \infty}m_n(T) < \infty, \] and say that $T$ exhibits linear growth if \[ \liminf_{n\to \infty} \frac{m_n(T)}{n} > 0. \]

We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres. As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.

Robust network design refers to a class of optimization problems that occur when designing networks to efficiently handle variable demands. The notion of “hierarchical hubbing” was introduced (in the narrow context of a specific robust network design question), by Olver and Shepherd [2010]. Hierarchical hubbing allows for routings with a multiplicity of “hubs” which are connected to the terminals and to each other in a treelike fashion. Recently, Fréchette et al. [2013] explored this notion much more generally, focusing on its applicability to an extension of the well-studied hose model that allows for upper bounds on individual point-to-point demands. In this paper, we consider hierarchical hubbing in the context of a previously studied (and extremely natural) generalization of the hose model, and prove that the optimal hierarchical hubbing solution can be found efficiently. This result is relevant to a recently proposed generalization of the “VPN Conjecture”.

Motivated by the recent emergence of the so-called opportunistic communication networks, we consider the issue of adaptivity in the most basic continuous time (asynchronous) rumor spreading process. In our setting a rumor has to be spread to a population; the service provider can push it at any time to any node in the network and has unit cost for doing this. On the other hand, as usual in rumor spreading, nodes share the rumor upon meeting and this imposes no cost on the service provider. Rather than fixing a budget on the number of pushes, we consider the cost version of the problem with a fixed deadline and ask for a minimum cost strategy that spreads the rumor to every node. A non-adaptive strategy can only intervene at the beginning and at the end, while an adaptive strategy has full knowledge and intervention capabilities. Our main result is that in the homogeneous case (where every pair of nodes randomly meet at the same rate) the benefit of adaptivity is bounded by a constant. This requires a subtle analysis of the underlying random process that is of interest in its own right.

Game Theory and Mechanism Design are by now standard tools for studying and designing massive decentralized systems. Unfortunately, designing mechanisms that induce socially efficient outcomes often requires full information and prohibitively large computational resources. In this work we study simple mechanisms that require only *local* information. Specifically, in the setting of a classic scheduling problem, we demonstrate local mechanisms that induce outcomes with social cost close to that of the socially optimal solution. Somewhat counter-intuitively, we find that mechanisms yielding Pareto dominated outcomes may in fact enhance the overall performance of the system, and we provide a justification of these results by interpreting these inefficiencies as externalities being internalized. We also show how to employ randomization to obtain yet further improvements. Lastly, we use the game-theoretic insights gained to obtain a new combinatorial approximation algorithm for the underlying optimization problem.

Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications.

One property that has been useful in applications is negative correlation of the resulting vector. There are some further properties that would be interesting to derive, but do not seem to follow from negative correlation. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting.

We introduce a simple but useful technique called *concavity of pessimistic estimators*. This technique allows us to show concentration of submodular functions and concentration of matrix sums under pipage rounding. The former result answers a question of Chekuri et al. (2009). To prove the latter result, we derive a new variant of Lieb’s celebrated concavity theorem in matrix analysis.

We provide numerous applications of these results. One is to *spectrally-thin trees*, a spectral analog of the *thin trees* that played a crucial role in the recent breakthrough on the asymmetric traveling salesman problem. We show a polynomial time algorithm that, given a graph where every edge has effective conductance at least $\kappa$, returns an $O(\kappa^{-1} \cdot \log n / \log \log n)$-spectrally-thin tree. There are further applications to rounding of semidefinite programs and to a geometric question of extracting a nearly-orthonormal basis from an isotropic distribution.

We consider robust network design problems where the set of feasible demands may be given by an arbitrary polytope or convex body more generally. This model, introduced by Ben-Ameur and Kerivin (2003), generalizes the well studied virtual private network (VPN) problem. Most research in this area has focused on ﬁnding constant factor approximations for speciﬁc polytope of demands, such as the class of hose matrices used in the deﬁnition of VPN. As pointed out in Chekuri (2007), however, the general problem was only known to be APX-hard (based on a reduction from the Steiner tree problem). We show that the general robust design is hard to approximate to within logarithmic factors. We establish this by showing a general reduction of buy-at-bulk network design to the robust network design problem. In the second part of the paper, we introduce a natural generalization of the VPN problem. In this model, the set of feasible demands is determined by a tree with edge capacities; a demand matrix is feasible if it can be routed on the tree. We give a constant factor approximation algorithm for this problem that achieves factor 8 in general, and 2 for the case where the tree has unit capacities.

We consider the following network design problem. We are given an undirected graph $G=(V,E)$ with edges costs $c(e)$ and a set of terminal nodes $W$. A *hose* demand matrix for $W$ is any symmetric matrix $[D_{ij}]$ such that for each $i$, $\sum_{j \neq i} D_{ij} \leq 1$. We must compute the minimum cost edge capacities that are able to support the oblivious routing of every hose matrix in the network. An *oblivious routing* template, in this context, is a simple path $P_{ij}$ for each pair $i,j \in W$. Given such a template, if we are to route a demand matrix $D$, then for each $i,j$ we send $D_{ij}$ units of flow along each $P_{ij}$. Fingerhut et al. (1997) and Gupta et al. (2001) obtained a $2$-approximation for this problem, using a solution template in the form of a tree. It has been widely asked and subsequently conjectured that this solution actually results in the optimal capacity for the single path VPN design problem; this has become known as the *VPN conjecture*.

The conjecture has previously been proven for some restricted classes of graphs (Hurkens et al. 2005, Grandoni et al. 2007, Fiorini et al. 2007). Our main theorem establishes that this conjecture is true in general graphs. This proves that the single path VPN problem is solvable in polynomial time. We also show that the multipath version of the conjecture is false.

Consider a branching random walk on $\mathbb{R}$, with offspring distribution $Z$ and nonnegative displacement distribution W. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution $Z$ is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs $(Z, W)$ for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur. We also investigate, under additional smoothness assumptions, the behavior of $M_n$, the position of the particle in generation $n$ closest to the origin, when explosion does not occur (and hence $\lim_{n\rightarrow\infty}M_n=\infty$).

Until recently, LP relaxations have played a limited role in the design of approximation algorithms for the Steiner tree problem. In 2010, Byrka et al. presented a ln(4)+epsilon approximation based on a hypergraphic LP relaxation, but surprisingly, their analysis does not provide a matching bound on the integrality gap.

We take a fresh look at hypergraphic LP relaxations for the Steiner tree problem – one that heavily exploits methods and results from the theory of matroids and submodular functions – which leads to stronger integrality gaps, faster algorithms, and a variety of structural insights of independent interest. More precisely, we present a deterministic ln(4)+epsilon approximation that compares against the LP value and therefore proves a matching ln(4) upper bound on the integrality gap.

Similarly to Byrka et al., we iteratively fix one component and update the LP solution. However, whereas they solve an LP at every iteration after contracting a component, we show how feasibility can be maintained by a greedy procedure on a well-chosen matroid. Apart from avoiding the expensive step of solving a hypergraphic LP at each iteration, our algorithm can be analyzed using a simple potential function. This gives an easy means to determine stronger approximation guarantees and integrality gaps when considering restricted graph topologies. In particular, this readily leads to a 73/60 bound on the integrality gap for quasi-bipartite graphs.

For the case of quasi-bipartite graphs, we present a simple algorithm to transform an optimal solution to the bidirected cut relaxation to an optimal solution of the hypergraphic relaxation, leading to a fast 73/60 approximation for quasi-bipartite graphs. Furthermore, we show how the separation problem of the hypergraphic relaxation can be solved by computing maximum flows, providing a fast independence oracle for our matroids.

Consider the robust network design problem of finding a minimum cost network with enough capacity to route all traffic demand matrices in a given polytope. We investigate the impact of different routing models in this robust setting: in particular, we compare *oblivious* routing, where the routing between each terminal pair must be fixed in advance, to *dynamic* routing, where routings may depend arbitrarily on the current demand. Our main result is a construction that shows that the optimal cost of such a network based on oblivious routing (fractional or integral) may be a factor of $\Omega(\log{n})$ more than the cost required when using dynamic routing. This is true even in the important special case of the asymmetric hose model. This answers a question in Chekuri (2007), and is tight up to constant factors. Our proof technique builds on a connection between expander graphs and robust design for single-sink traffic patterns (Chekuri et al. 2007).

We consider a priority-based selﬁsh routing model, where agents may have diﬀerent priorities on a link. An agent with a higher priority on a link can traverse it with a smaller delay or cost than an agent with lower priority. This general framework can be used to model a number of diﬀerent problems. The structural properties that lead to ineﬃciencies in routing choices appear diﬀerent in this priority-based model compared to the classical model. In particular, in parallel link networks with nonatomic agents, the price of anarchy is exactly one in the priority-based model; that is, selﬁsh behaviour leads to optimal routings. In contrast, in the standard model the worst possible price of anarchy can be achieved in a simple two-link network. For multi-commodity networks, selﬁsh routing does lead to ineﬃciencies in the priority-based model. We present tight bounds on the price of anarchy for such networks. Speciﬁcally, in the nonatomic case the worst-case price of anarchy is exactly $(d + 1)^{d+1}$ for polynomial latency functions of degree $d$ (hence 4 for linear cost functions). For atomic games, the worst-case price of anarchy is exactly $3+2\sqrt{2}$ in the weighted case, and exactly 17/3 in the unweighted case. An upper bound of $O(2^dd^d)$ is also shown for polynomial cost functions in the atomic case, although this is not tight. Our framework (and results) also generalise to include models similar to congestion games.