1,917 research outputs found

    Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP

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    Given a metric space on n points, an {\alpha}-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of the induced subtree is within an {\alpha} factor of the optimal Steiner tree cost for X. An {\alpha}-approximate differentially private algorithm for the Steiner tree problem takes as input a subset X of vertices, and outputs a tree distribution that induces a solution within an {\alpha} factor of the optimal as before, and satisfies the additional property that for any set X' that differs in a single vertex from X, the tree distributions for X and X' are "close" to each other. Universal and differentially private algorithms for TSP are defined similarly. An {\alpha}-approximate universal algorithm for the Steiner tree problem or TSP is also an {\alpha}-approximate differentially private algorithm. It is known that both problems admit O(logn)-approximate universal algorithms, and hence O(log n)-approximate differentially private algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation ratio achievable for the universal Steiner tree problem and the universal TSP, matching the known upper bounds. Our lower bound for the Steiner tree problem holds even when the algorithm is allowed to output a more general solution of a distribution on paths to the root.Comment: 14 page

    Designing Networks with Good Equilibria under Uncertainty

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    We consider the problem of designing network cost-sharing protocols with good equilibria under uncertainty. The underlying game is a multicast game in a rooted undirected graph with nonnegative edge costs. A set of k terminal vertices or players need to establish connectivity with the root. The social optimum is the Minimum Steiner Tree. We are interested in situations where the designer has incomplete information about the input. We propose two different models, the adversarial and the stochastic. In both models, the designer has prior knowledge of the underlying metric but the requested subset of the players is not known and is activated either in an adversarial manner (adversarial model) or is drawn from a known probability distribution (stochastic model). In the adversarial model, the designer's goal is to choose a single, universal protocol that has low Price of Anarchy (PoA) for all possible requested subsets of players. The main question we address is: to what extent can prior knowledge of the underlying metric help in the design? We first demonstrate that there exist graphs (outerplanar) where knowledge of the underlying metric can dramatically improve the performance of good network design. Then, in our main technical result, we show that there exist graph metrics, for which knowing the underlying metric does not help and any universal protocol has PoA of Ω(logk)\Omega(\log k), which is tight. We attack this problem by developing new techniques that employ powerful tools from extremal combinatorics, and more specifically Ramsey Theory in high dimensional hypercubes. Then we switch to the stochastic model, where each player is independently activated. We show that there exists a randomized ordered protocol that achieves constant PoA. By using standard derandomization techniques, we produce a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu

    Algorithms and Adaptivity Gaps for Stochastic k-TSP

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    Given a metric (V,d)(V,d) and a rootV\textsf{root} \in V, the classic \textsf{k-TSP} problem is to find a tour originating at the root\textsf{root} of minimum length that visits at least kk nodes in VV. In this work, motivated by applications where the input to an optimization problem is uncertain, we study two stochastic versions of \textsf{k-TSP}. In Stoch-Reward kk-TSP, originally defined by Ene-Nagarajan-Saket [ENS17], each vertex vv in the given metric (V,d)(V,d) contains a stochastic reward RvR_v. The goal is to adaptively find a tour of minimum expected length that collects at least reward kk; here "adaptively" means our next decision may depend on previous outcomes. Ene et al. give an O(logk)O(\log k)-approximation adaptive algorithm for this problem, and left open if there is an O(1)O(1)-approximation algorithm. We totally resolve their open question and even give an O(1)O(1)-approximation \emph{non-adaptive} algorithm for this problem. We also introduce and obtain similar results for the Stoch-Cost kk-TSP problem. In this problem each vertex vv has a stochastic cost CvC_v, and the goal is to visit and select at least kk vertices to minimize the expected \emph{sum} of tour length and cost of selected vertices. This problem generalizes the Price of Information framework [Singla18] from deterministic probing costs to metric probing costs. Our techniques are based on two crucial ideas: "repetitions" and "critical scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for our problems, if we truncate the random variables at an ideal threshold and repeat, then their expected values form a good surrogate. Unfortunately, this ideal threshold is adaptive as it depends on how far we are from achieving our target kk, so we truncate at various different scales and identify a "critical" scale.Comment: ITCS 202

    Probabilistic bounds on the kk-Traveling Salesman Problem and the Traveling Repairman Problem

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    The kk-traveling salesman problem (kk-TSP) seeks a tour of minimal length that visits a subset of knk\leq n points. The traveling repairman problem (TRP) seeks a complete tour with minimal latency. This paper provides constant-factor probabilistic approximations of both problems. We first show that the optimal length of the kk-TSP path grows at a rate of Θ(k/n12(1+1k1))\Theta\left(k/n^{\frac{1}{2}\left(1+\frac{1}{k-1}\right)}\right). The proof provides a constant-factor approximation scheme, which solves a TSP in a high-concentration zone -- leveraging large deviations of local concentrations. Then, we show that the optimal TRP latency grows at a rate of Θ(nn)\Theta(n\sqrt n). This result extends the classical Beardwood-Halton-Hammersley theorem to the TRP. Again, the proof provides a constant-factor approximation scheme, which visits zones by decreasing order of probability density. We discuss practical implications of this result in the design of transportation and logistics systems. Finally, we propose dedicated notions of fairness -- randomized population-based fairness for the kk-TSP and geographical fairness for the TRP -- and give algorithms to balance efficiency and fairness

    Improved guarantees for the a priori TSP

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    We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space (V,c)(V,c) and an activation probability p(v)p(v) for each customer vVv\in V. We ask for a TSP tour TT for VV that minimizes the expected length after cutting TT short by skipping the inactive customers. All known approximation algorithms select a nonempty subset SS of the customers and construct a master route solution, consisting of a TSP tour for SS and two edges connecting every customer vVSv\in V\setminus S to a nearest customer in SS. We address the following questions. If we randomly sample the subset SS, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a deterministic polynomial-time algorithm.Comment: 39 pages, 6 figures, extended abstract to appear in the proceedings of ISAAC 202
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