Designing Networks with Good Equilibria under Uncertainty

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 $\Omega(\log k)$, 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

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20

Capacitated Network Design on Outerplanar Graphs

Network design problems model the efficient allocation of resources like routers, optical fibres, roads, canals etc. to effectively construct and operate critical infrastructures. In this thesis, we consider the capacitated network design problem (CapNDP), which finds applications in supply-chain logistics problems and network security. Here, we are given a network and for each edge in the network, several security reinforcement options. In addition, for each pair of nodes in the network, there is a specified level of protection demanded. The objective is to select a minimum-cost set of reinforcements for all the edges so that an adversary with strength less than the protection level of a particular pair of nodes cannot disconnect these nodes. The optimal solution to this problem cannot, in general, be found in reasonable time. One way to tackle such hard problems is to develop approximation algorithms, which are fast algorithms that are guaranteed to find near-optimal solutions; the worst-case ratio between the cost of the solution output by the algorithm and the optimum cost is called the approximation ratio of the algorithm. In this thesis, we investigate CapNDP when the network structure is constrained to belong to a class of graphs called outerplanar graphs. This particular special case was first considered by Carr, Fleischer, Leung and Philips; while they claimed to obtain an approximation ratio arbitrarily close to 1, their algorithm has certain fatal flaws. We build upon some of the ideas they use to approximate CapNDP on general networks to develop a new algorithm for CapNDP on outerplanar graphs. The approximation ratio achieved by our algorithm improves the state-of-the-art by a doubly exponential factor. We also notice that our methods can be applied to a more general class of problems called column-restricted covering integers programs, and be adapted to improve the approximation ratio on more instances of CapNDP if the structure of the network is known. Furthermore, our techniques also yield interesting results for a completely unrelated problem in the area of data structures

Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions)

(see paper for full abstract) Given a vertex-weighted directed graph $G=(V,E)$ and a set $T=\{t_1, t_2, \ldots t_k\}$ of $k$ terminals, the objective of the SCSS problem is to find a vertex set $H\subseteq V$ of minimum weight such that $G[H]$ contains a $t_{i}\rightarrow t_j$ path for each $i\neq j$. The problem is NP-hard, but Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel $n^{O(k)}$ algorithm for the SCSS problem, where $n$ is the number of vertices in the graph and $k$ is the number of terminals. We explore how much easier the problem becomes on planar directed graphs: - Our main algorithmic result is a $2^{O(k)}\cdot n^{O(\sqrt{k})}$ algorithm for planar SCSS, which is an improvement of a factor of $O(\sqrt{k})$ in the exponent over the algorithm of Feldman and Ruhl. - Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an $f(k)\cdot n^{o(\sqrt{k})}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. The following additional results put our upper and lower bounds in context: - In general graphs, we cannot hope for such a dramatic improvement over the $n^{O(k)}$ algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs does not have an $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. - Feldman and Ruhl generalized their $n^{O(k)}$ algorithm to the more general Directed Steiner Network (DSN) problem; here the task is to find a subgraph of minimum weight such that for every source $s_i$ there is a path to the corresponding terminal $t_i$. We show that, assuming ETH, there is no $f(k)\cdot n^{o(k)}$ time algorithm for DSN on acyclic planar graphs.Comment: To appear in SICOMP. Extended abstract in SODA 2014. This version has a new author (Andreas Emil Feldmann), and the algorithm is faster and considerably simplified as compared to conference versio