Computational geometry

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions—the Nash equilibria—may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context—it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widely studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is $O(\log k)$, where $k$ is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-89te3x19ow.pdf

The Price of Stability for Network Design with Fair Cost Allocation

In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Flows in Networks

Journal of the ACM

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions - the Nash equilibria - may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context - it is the optimal solution that can be proposed from which no user will "defect". We consider the price of stability for network design with respect to one of the most widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-4b3bbv50mq.pdf

The price of stability for network design with fair cost allocation

We present an approximate distance oracle for a point set S with n points and doubling dimension {\lambda}. For every {\epsilon}>0, the oracle supports (1+{\epsilon})-approximate distance queries in (universal) constant time, occupies space [{\epsilon}^{-O({\lambda})} + 2^{O({\lambda} log {\lambda})}]n, and can be constructed in [2^{O({\lambda})} log3 n + {\epsilon}^{-O({\lambda})} + 2^{O({\lambda} log {\lambda})}]n expected time. This improves upon the best previously known constructions, presented by Har-Peled and Mendel. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2^{O({\lambda})} log n + {\epsilon}^{-O({\lambda})} + 2^{O({\lambda} log {\lambda})} update time. This is the first fully dynamic (1+{\epsilon})-distance oracle.

Fast, precise and dynamic distance queries

Let (G = (V(G),E(G))) be a directed graph with nonnegative edge lengths and let P be a shortest path from s to t in G. In the replacement paths problem we are required to compute for every edge e in P, the length of a shortest path from s to t that avoids e. The fastest known algorithm for solving the problem in weighted directed graphs is the trivial one: each edge in P is removed from the graph in its turn and the distance from s to t in the modified graph is computed. The running time of this algorithm is O(m n + n2 log n), where n = vV(G)v and m = vE(G)v.The replacement paths problem is strongly motivated by two different applications. First, the fastest algorithm to compute the k simple shortest paths from s to t in directed graphs [Yen 1971; Lawler 1972] repeatedly computes the replacement paths from s to t. Its running time is O(kn (m + n log n)). Second, the computation of Vickrey pricing of edges in distributed networks can be reduced to the replacement paths problem. An open question raised by Nisan and Ronen [2001] asks whether it is possible to compute the Vickrey pricing faster than the trivial algorithm described in the previous paragraph.In this article we present a near-linear time algorithm for computing replacement paths in weighted planar directed graphs. In particular, the algorithm computes the lengths of the replacement paths in O (n log3n) time (recall that in planar graphs m = O (n)). This result immediately improves the running time of the two applications mentioned before by almost a linear factor. Our algorithm is obtained by combining several new ideas with a data structure of Klein [2005] that supports multisource shortest paths queries in planar directed graphs in logarithmic time.Our algorithm can be adapted to address the variant of the problem in which one is interested in the replacement path itself (rather than the length of the path). In that case the algorithm is executed in a preprocessing stage constructing a data structure that supports replacement path queries in time O (h), where h is the number of hops in the replacement path. In addition, we can handle the variant in which vertices should be avoided instead of edges.

A near-linear-time algorithm for computing replacement paths in planar directed graphs

Thorup and Zwick (J. ACM 52(1):1–24, 2005 and STOC’01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer
 k≥1 it is possible to preprocess the graph in $\tilde {O}(mn^{1/k})$
 time and generate a compact data structure of size O(kn
 1+1/k
 ). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k−1 in O(k) time. For k=2 this gives an oracle of O(n
 1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtrascu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n
 5/3) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs. We present a data structure that produces estimated distances with 1+e stretch. The size of the data structure is O(nm
 1−e′) and the query time is $\tilde {O}(m^{1-\varepsilon '})$
 . Using it for sparse unweighted graphs we can get a data structure of size O(n
 1.87) that can supply in O(n
 0.87) time estimated distances with multiplicative stretch 1.75.

Preprocess, Set, Query!

Let V be a set of points in a d-dimensional l p-metric space. Let s,t∈V and let L be a real number. An L-bounded leg path from s to t is an ordered set of points which connects s to t such that the leg between any two consecutive points in the set has length of at most L. The minimal path among all these paths is the L-bounded leg shortest path from s to t. In the s–t Bounded Leg Shortest Path (stBLSP) problem we are given two points s and t and a real number L, and are required to compute an L-bounded leg shortest path from s to t. In the All-Pairs Bounded Leg Shortest Path (apBLSP) problem we are required to build a data structure that, given any two query points from V and a real number L, outputs the length of the L-bounded leg shortest path (a distance query) or the path itself (a path query). In this paper we obtain the following results:


An algorithm for the apBLSP problem in any l p-metric which, for any fixed e>0, computes in O(n3(log 3n+log 2n⋅e−d)) time a data structure which approximates any bounded leg shortest path within a multiplicative error of (1+e). It requires O(n2log n) space and distance queries are answered in O(log log n) time.
An algorithm for the stBLSP problem that, given s,t∈V and a real number L, computes in O(n⋅polylog(n)) the exact L-bounded shortest path from s to t. This algorithm works in l1 and l∞ metrics. In the Euclidean metric we also obtain an exact algorithm but with a running time of O(n4/3+e), for any e>0.
For any weighted directed graph we give a data structure of size O(n2.5log n) which is capable of answering path queries with a multiplicative error of (1+e) in O(log log n+l) time, where l is the length of the reported path.

Our results improve upon the results given by Bose et al. (Comput. Geom. Theory Appl. 29:233–249, 2004). Our algorithms incorporate several new ideas along with an interesting observation made on geometric spanners, which is of independent interest.

On Bounded Leg Shortest Paths Problems

Let $$G=V,E$$ be an n-vertices m-edges directed graph. Let $$s\in V$$ be any designated source vertex, and let T be an arbitrary reachability tree rooted at s. We address the problem of finding a set of edges $$\mathcal{E}\subseteq E\backslash T$$ of minimum size such that on a failure of any vertex $$w\in V$$, the set of vertices reachable from s in $$T\cup \mathcal{E} \backslash \{w\}$$ is the same as the set of vertices reachable from s in $$G\backslash \{w\}$$. We obtain the following results:The optimal set $$\mathcal E$$ for any arbitrary reachability tree T has at most $$n-1$$ edges.There exists an $$Om \log n$$-time algorithm that computes the optimal set $$\mathcal{E}$$ for any given reachability tree T.

For the restricted case when the reachability tree T is a Depth-First-Search DFS tree it is straightforward to bound the size of the optimal set $$\mathcal{E}$$ by $$n-1$$ using semidominators with respect to DFS trees from the celebrated work of Lengauer and Tarjan [13]. Such a set $$\mathcal E$$ can be computed in Om time using the algorithm of Buchsbaum et. ali¾?[4].

To bound the size of the optimal set in the general case we define semidominators with respect to arbitrary trees. We also present a simple $$Om \log n$$ time algorithm for computing such semidominators. As a byproduct, we get an alternative algorithm for computing dominators in $$Om \log n$$ time.

/pdf/fault-tolerant-reachability-for-directed-graphs-18qws9b2qe.pdf

Liam Roditty

Papers

Fast, precise and dynamic distance queries

A near-linear-time algorithm for computing replacement paths in planar directed graphs

Preprocess, Set, Query!

On Bounded Leg Shortest Paths Problems

Fault Tolerant Reachability for Directed Graphs