We consider the one-to-one Pickup and Delivery Problem (PDP) in Euclidean Space with arbitrary dimension @math where @math transportation requests are picked i.i.d. with a separate origin-destination pair for each object to be moved. First, we consider the problem from the customer perspective where the objective is to compute a plan for transporting the objects such that the Euclidean distance traveled by the vehicles when carrying objects is minimized. We develop a polynomial time asymptotically optimal algorithm for vehicles with capacity @math for this case. This result also holds imposing LIFO constraints for loading and unloading objects. Secondly, we extend our algorithm to the classical single-vehicle PDP where the objective is to minimize the total distance traveled by the vehicle and present results indicating that the extended algorithm is asymptotically optimal for a fixed vehicle capacity if the origins and destinations are picked i.i.d. using the same distribution.
Stein @cite_6 also conducts a probabilistic analysis but he looks at the variant where @math and @math and where @math origins and @math destinations are picked independently using a uniform distribution on some planar region. Stein shows that the value of the optimal solution divided by @math converges almost surely to a constant times the square root of the area of the region. Stein also shows how to solve the problem he considers by concatenating two TSP tours on the origins and destinations respectively. This way of solving the problem yields a solution which is roughly @math O(n^2) @math 4 @math d=2 @math c=+ @math d=2 @math c = O( n) @math d @math c$ @cite_13 .
@cite_13 @cite_6
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In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward traveling salesman problem (TSP), motivated by robot navigation. In both problems, we are given a graph with lengths on edges and rewards on nodes, and a start node @math . In the Orienteering problem, the goal is to find a path starting at @math that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor @math , and the goal is to maximize the total discounted reward collected, where the reward for a node reached at time @math is discounted by @math . This problem is motivated by an approximation to a planning problem in the Markov decision process (MDP) framework under the commonly employed infinite horizon discounted reward optimality criterion. The approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one-time events and nonrepeatable rewards (such as for package deliveries). We also consider tree and multiple-path variants of these problems and provide approximations for those as well. Although the unrooted Orienteering problem, where there is no fixed start node @math , has been known to be approximable using algorithms for related problems such as @math -TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem in [E. M. Arkin, J. S. B. Mitchell, and G. Narasimhan, Proceedings of the @math th ACM Symposium on Computational Geometry, 1998, pp. 307-316] and [B. Awerbuch, Y. Azar, A. Blum, and S. Vempala, SIAM J. Comput., 28 (1998), pp. 254-262]. We complement our approximation result for Orienteering by showing that the problem is APX-hard. We consider the problem of selecting the best subset of exactly @math columns from an @math matrix @math . We present and analyze a novel two-stage algorithm that runs in @math time and returns as output an @math matrix @math consisting of exactly @math columns of @math . In the first (randomized) stage, the algorithm randomly selects @math columns according to a judiciously-chosen probability distribution that depends on information in the top- @math right singular subspace of @math . In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly @math columns from the set of columns selected in the first stage. Let @math be the @math matrix containing those @math columns, let @math denote the projection matrix onto the span of those columns, and let @math denote the best rank- @math approximation to the matrix @math . Then, we prove that, with probability at least 0.8, @math This Frobenius norm bound is only a factor of @math worse than the best previously existing existential result and is roughly @math better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, @math This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on @math , whereas previous results depend on @math ; if these two quantities are comparable, then our bound is asymptotically worse by a @math factor. Abstract Recently Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503], using probabilistic methods alone, has given new proofs for the existence, asymptotics and uniqueness of travelling wave solutions to the K-P-P equation. Following in this vein we outline alternative probabilistic proofs. Specifically the techniques are confined to the study of additive and multiplicative martingales and spinal path decompositions along the lines of [B. Chauvin, A. Rouault, Probab. Theory Related Fields 80 (1988) 299], [R. Lyons, in: K.B. Athreya, P. Jagers (eds.), Classical and Modern Branching Processes, Vol. 84, Springer-Verlag, New York, 1997, pp. 217–222] and [R. , Ann. Probab. 23 (1995) 1125]. We also make use of a new decomposition where the spine is a conditioned process. Some new results concerning martingale convergence are obtained as a by-product of the analysis. The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Academie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7-8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper). It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.--A. Charnes, Northwestern Technological Institute and The Transportation Center. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. We consider the problem of provably optimal exploration in reinforcement learning for finite horizon MDPs. We show that an optimistic modification to value iteration achieves a regret bound of @math where @math is the time horizon, @math the number of states, @math the number of actions and @math the number of time-steps. This result improves over the best previous known bound @math achieved by the UCRL2 algorithm of , 2010. The key significance of our new results is that when @math and @math , it leads to a regret of @math that matches the established lower bound of @math up to a logarithmic factor. Our analysis contains two key insights. We use careful application of concentration inequalities to the optimal value function as a whole, rather than to the transitions probabilities (to improve scaling in @math ), and we define Bernstein-based "exploration bonuses" that use the empirical variance of the estimated values at the next states (to improve scaling in @math ). Consider a uniform random rooted tree on vertices labelled by @math , with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number @math of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider @math and let @math denote the event that all @math cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if @math , we have @math , whereas if @math we have @math . We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson( @math ) number of cars arrive at each vertex. Let @math be the number of cars which visit the root of the tree. Then for @math , we have @math , whereas for @math , we have @math . This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. Given a graph @math and a subset @math of terminals, a of @math is a tree that spans @math . In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of @math . We study a natural generalization of the VST problem motivated by multi-level graph construction, the (V-GSST), which can be stated as follows: given a graph @math and terminals @math , where each terminal @math requires a facility of a minimum grade of service @math , compute a Steiner tree @math by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in @math with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on @math , the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a @math -approximation, where @math is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. Given a directed graph @math and a list @math of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of @math that contains a directed @math path for every @math . The special case Directed Steiner Tree (when we ask for paths from a root @math to terminals @math ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every @math to every other @math ) is known to be W[1]-hard. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if @math is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list @math of requests form a directed graph that is a member of @math . Our main result is a complete characterization of the classes @math resulting in fixed-parameter tractable special cases: we show that if every pattern in @math has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable @math not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], @math -Root Steiner Tree is FPT for constant @math [Such 'y, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [, SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before. Abstract At the heart of this article will be the study of a branching Brownian motion (BBM) with killing , where individual particles move as Brownian motions with drift − ρ , perform dyadic branching at rate β and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the ‘one-sided’ FKPP travelling-wave equation of speed − ρ with solutions f : [ 0 , ∞ ) → [ 0 , 1 ] satisfying f ( 0 ) = 1 and f ( ∞ ) = 0 has a unique solution with a particular asymptotic when ρ 2 β , and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503–517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53–72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is 2 β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ ( 0 , 2 β − ρ ) . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > 2 β − ρ . This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299–314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon–Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future. * Recasts topics in random fields by following a completely new way of handling both geometry and probability * Significant exposition of the work of others in the field * Presentation is clear and pedagogical * Excellent reference work as well as excellent work for self study @PARASPLIT This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. @PARASPLIT The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities. @PARASPLIT "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 The traveling salesman problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem an algorithm that for any fixed @math computes in randomized polynomial time a @math -approximation to the optimal tour in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora [J. ACM, 45 (1998), pp. 753--782] and Mitchell [SIAM J. Comput., 28 (1999), pp. 1298--1309] prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar [Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pp. 281--290]. Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. A central theme in distributed network algorithms concerns understanding and coping with the issue of locality . Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for . In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard @math model of computation and define @math (for local decision ) as the class of decision problems that can be solved in @math communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class @math , containing all languages for which there exists a randomized algorithm that runs in @math rounds, accepts correct instances with probability at least @math and rejects incorrect ones with probability at least @math . We show that @math is a threshold for the containment of @math in @math . More precisely, we show that there exists a language that does not belong to @math for any @math but does belong to @math for any @math such that @math . On the other hand, we show that, restricted to hereditary languages, @math , for any function @math and any @math such that @math . In addition, we investigate the impact of non-determinism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that non-determinism does help, but that this help is limited, as there exist languages that cannot be decided non-deterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with non-determinism that enables to decide languages . Finally, we introduce the notion of local reduction, and establish some completeness results. Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. Indyk and Sidiropoulos (2007) proved that any orientable graph of genus @math can be probabilistically embedded into a graph of genus @math with constant distortion. Viewing a graph of genus @math as embedded on the surface of a sphere with @math handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion @math , by iteratively removing the handles. By removing all @math handles at once, we present a probabilistic embedding with distortion @math for both orientable and non-orientable graphs. Our result is obtained by showing that the nimum-cut graph of Erickson and Har Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009). We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice ( Z^d ) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions (d>2 )) and prove new optimal estimates for regularized versions of the corrector (in dimensions (d 2 )). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients. In the highway problem, we are given an @math -edge path graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is to choose weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly @math -hard only recently [K. M. , in Proceedings of the International Symposium on Algorithmic Game Theory (SAGT), 2009, pp. 275--286]. The best-known approximation is @math [I. Gamzu and D. Segev, in Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP), 2010, pp. 582--593], which improves on the previous best @math approximation [M.-F. Balcan and A. Blum, in Proceedings of the ACM Conference on Electronic Commerce,...