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 consider the fundamental algorithmic problem of finding a cycle of minimum weight in a weighted graph. In particular, we show that the minimum weight cycle problem in an undirected n-node graph with edge weights in {1,...,M} or in a directed n-node graph with edge weights in {-M,..., M} and no negative cycles can be efficiently reduced to finding a minimum weight triangle in an Theta(n)-node undirected graph with weights in {1,...,O(M)}. Roughly speaking, our reductions imply the following surprising phenomenon: a minimum cycle with an arbitrary number of weighted edges can be "encoded" using only three edges within roughly the same weight interval! This resolves a longstanding open problem posed by Itai and Rodeh [SIAM J. Computing 1978 and STOC'77]. 
A direct consequence of our efficient reductions are O (Mn^{omega})-time algorithms using fast matrix multiplication (FMM) for finding a minimum weight cycle in both undirected graphs with integral weights from the interval [1,M] and directed graphs with integral weights from the interval [-M,M]. The latter seems to reveal a strong separation between the all pairs shortest paths (APSP) problem and the minimum weight cycle problem in directed graphs as the fastest known APSP algorithm has a running time of O(M^{0.681}n^{2.575}) by Zwick [J. ACM 2002]. 
In contrast, when only combinatorial algorithms are allowed (that is, without FMM) the only known solution to minimum weight cycle is by computing APSP. Interestingly, any separation between the two problems in this case would be an amazing breakthrough as by a recent paper by Vassilevska W. and Williams [FOCS'10], any O(n^{3-eps})-time algorithm (eps>0) for minimum weight cycle immediately implies a O(n^{3-delta})-time algorithm (delta>0) for APSP.

Minimum Weight Cycles and Triangles: Equivalences and Algorithms

Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time $\tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k M)$. If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of $\tilde{O}(2^k \poly(k) n^\omega(\log\log M + 1/\varepsilon))$. For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time $\tilde{O}(2^k \poly(k) M n^3) $ and a $(1+\varepsilon)$-approximate solution of running time $\tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon))$. All of the above algorithms are randomized with a polynomially-small error probability.

Finding the Minimum-Weight k-Path

Thorup and Zwick [19] introduced the notion of approximate distance oracles, a data structure that produces for an n-vertices, m-edges weighted undirected graph $G=(V,E)$, distance estimations in constant query time. They presented a distance oracle of size $O(kn^{1+1/k})$ that given a pair of vertices $u,v \in V$ at distance d(u, v) produces in O(k) time an estimation that is bounded by $(2k-1)d(u,v)$, i.e., a $(2k-1)$-multiplicative approximation (stretch). Thorup and Zwick [19] presented also a lower bound based on the girth conjecture of Erdős.

Approximate Distance Oracles with Improved Stretch for Sparse Graphs.

In the $\{-1,0,1\}$-APSP problem the goal is to compute all-pairs shortest paths (APSP) on a directed graph whose edge weights are all from $\{-1,0,1\}$. In the (min,max)-product problem the input is two $n\times n$ matrices $A$ and $B$, and the goal is to output the (min,max)-product of $A$ and $B$. 
This paper provides a new algorithm for the $\{-1,0,1\}$-APSP problem via a simple reduction to the target-(min,max)-product problem where the input is three $n\times n$ matrices $A,B$, and $T$, and the goal is to output a Boolean $n\times n$ matrix $C$ such that the $(i,j)$ entry of $C$ is 1 if and only if the $(i,j)$ entry of the (min,max)-product of $A$ and $B$ is exactly the $(i,j)$ entry of the target matrix $T$. If (min,max)-product can be solved in $T_{MM}(n) = \Omega(n^2)$ time then it is straightforward to solve target-(min,max)-product in $O(T_{MM}(n))$ time. Thus, given the recent result of Bringmann, K\"unnemann, and Wegrzycki [STOC 2019], the $\{-1,0,1\}$-APSP problem can be solved in the same time needed for solving approximate APSP on graphs with positive weights. 
Moreover, we design a simple algorithm for target-(min,max)-product when the inputs are restricted to the family of inputs generated by our reduction. Using fast rectangular matrix multiplication, the new algorithm is faster than the current best known algorithm for (min,max)-product.

{-1, 0, 1}-APSP and (min, max)-Product Problems.

The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted $m$-edge and $n$-node graphs require $\Omega(\min\{n^{\omega}, mn\})$ time (for $2\leq\omega<2.373$). In this paper, we drastically improve these runtimes as follows: 
* Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in $\widetilde{O}(m)$ time computes an $\widetilde{O}(1)$-multiplicative approximation of the girth as well as an $\widetilde{O}(1)$-multiplicative roundtrip spanner with $\widetilde{O}(n)$ edges with high probability (w.h.p). 
* Nearly Tight Additive Approximations: For unweighted graphs and any $\alpha \in (0,1)$ we give an algorithm that in $\widetilde{O}(mn^{1 - \alpha})$ time computes an $O(n^\alpha)$-additive approximation of the girth w.h.p, and partially derandomize it. We show that the runtime of our algorithm cannot be significantly improved without a breakthrough in combinatorial Boolean matrix multiplication. 
Our main technical contribution to achieve these results is the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous roundtrip spanner constructions. Previously it was not known how to compute these significantly faster than $\Omega(\min\{n^\omega, mn\})$ time. Given the traditional difficulty in efficiently processing directed graphs, we hope our techniques may find further applications.

Liam Roditty

Papers

Minimum Weight Cycles and Triangles: Equivalences and Algorithms

Finding the Minimum-Weight k-Path

Approximate Distance Oracles with Improved Stretch for Sparse Graphs.

{-1, 0, 1}-APSP and (min, max)-Product Problems.

Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners