Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses

We present a deterministic polynomial-time algorithm for computing $d^{d+o(d)}$-approximate (pure) Nash equilibria in weighted congestion games with polynomial cost functions of degree at most $d$. This is an exponential improvement of the approximation factor with respect to the previously best deterministic algorithm. An appealing additional feature of our algorithm is that it uses only best-improvement steps in the actual game, as opposed to earlier approaches that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by Caragiannis et al. [FOCS'11, TEAC 2015], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of $[d/\mathcal{W}(d/\rho)]^{d+1}$ for the Price of Anarchy (PoA) of $\rho$-approximate equilibria in weighted congestion games, where $\mathcal{W}$ is the Lambert-W function. More specifically, we show that this PoA is exactly equal to $\Phi_{d,\rho}^{d+1}$, where $\Phi_{d,\rho}$ is the unique positive solution of the equation $\rho (x+1)^d=x^{d+1}$. Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, while our lower bound is simple enough to apply even to singleton congestion games

Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time

In the decremental single-source shortest paths (SSSP) problem we want to maintain the distances between a given source node $s$ and every other node in an $n$-node $m$-edge graph $G$ undergoing edge deletions. While its static counterpart can be solved in near-linear time, this decremental problem is much more challenging even in the undirected unweighted case. In this case, the classic $O(mn)$ total update time of Even and Shiloach [JACM 1981] has been the fastest known algorithm for three decades. At the cost of a $(1+\epsilon)$-approximation factor, the running time was recently improved to $n^{2+o(1)}$ by Bernstein and Roditty [SODA 2011]. In this paper, we bring the running time down to near-linear: We give a $(1+\epsilon)$-approximation algorithm with $m^{1+o(1)}$ expected total update time, thus obtaining near-linear time. Moreover, we obtain $m^{1+o(1)} \log W$ time for the weighted case, where the edge weights are integers from $1$ to $W$. The only prior work on weighted graphs in $o(m n)$ time is the $m n^{0.9 + o(1)}$-time algorithm by Henzinger et al. [STOC 2014, ICALP 2015] which works for directed graphs with quasi-polynomial edge weights. The expected running time bound of our algorithm holds against an oblivious adversary. In contrast to the previous results which rely on maintaining a sparse emulator, our algorithm relies on maintaining a so-called sparse $(h, \epsilon)$-hop set introduced by Cohen [JACM 2000] in the PRAM literature. An $(h, \epsilon)$-hop set of a graph $G=(V, E)$ is a set $F$ of weighted edges such that the distance between any pair of nodes in $G$ can be $(1+\epsilon)$-approximated by their $h$-hop distance (given by a path containing at most $h$ edges) on $G'=(V, E\cup F)$. Our algorithm can maintain an $(n^{o(1)}, \epsilon)$-hop set of near-linear size in near-linear time under edge deletions.Comment: Accepted to Journal of the ACM. A preliminary version of this paper was presented at the 55th IEEE Symposium on Foundations of Computer Science (FOCS 2014). Abstract shortened to respect the arXiv limit of 1920 character

Efficient Algorithms and Hardness Results for the Weighted kk-Server Problem

In this paper, we study the weighted $k$-server problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) $k$-server problem which has a polynomial-time solution using min-cost flows, there are strong computational lower bounds for the weighted $k$-server problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomial-time algorithms with a sub-polynomial approximation factor, even if we use $c$-resource augmentation for $c < 2$. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least $\ell$ resource augmentation, where $\ell$ is the number of distinct server weights. We complement these results by obtaining a constant-approximation algorithm via LP rounding, with a resource augmentation of $(2+\epsilon)\ell$ for any constant $\epsilon > 0$. In the online setting, an $\exp(k)$ lower bound is known for the competitive ratio of any randomized algorithm for the weighted $k$-server problem on the uniform metric. In contrast, we show that $2\ell$-resource augmentation can bring the competitive ratio down by an exponential factor to only $O(\ell^2 \log \ell)$. Our online algorithm uses the two-stage approach of first obtaining a fractional solution using the online primal-dual framework, and then rounding it online.Comment: This paper will appear in the proceedings of APPROX 202

A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits

We give a nontrivial algorithm for the satisfiability problem for cn-wire threshold circuits of depth two which is better than exhaustive search by a factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first nontrivial satisfiability algorithm for cn-wire threshold circuits of depth two. The independently interesting problem of the feasibility of sparse 0-1 integer linear programs is a special case. To our knowledge, our algorithm is the first to achieve constant savings even for the special case of Integer Linear Programming. The key idea is to reduce the satisfiability problem to the Vector Domination Problem, the problem of checking whether there are two vectors in a given collection of vectors such that one dominates the other component-wise. We also provide a satisfiability algorithm with constant savings for depth two circuits with symmetric gates where the total weighted fan-in is at most cn. One of our motivations is proving strong lower bounds for TC^0 circuits, exploiting the connection (established by Williams) between satisfiability algorithms and lower bounds. Our second motivation is to explore the connection between the expressive power of the circuits and the complexity of the corresponding circuit satisfiability problem

Constant Factor Approximation for Capacitated k-Center with Outliers

The $k$-center problem is a classic facility location problem, where given an edge-weighted graph $G = (V,E)$ one is to find a subset of $k$ vertices $S$, such that each vertex in $V$ is "close" to some vertex in $S$. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied. In the capacitated version of the problem each vertex is assigned a capacity, which is a strict upper bound on the number of clients a facility can serve, when located at this vertex. A constant factor approximation for the capacitated $k$-center was obtained last year by Cygan, Hajiaghayi and Khuller [FOCS'12], which was recently improved to a 9-approximation by An, Bhaskara and Svensson [arXiv'13]. In a different generalization of the problem some clients (denoted as outliers) may be disregarded. Here we are additionally given an integer $p$ and the goal is to serve exactly $p$ clients, which the algorithm is free to choose. In 2001 Charikar et al. [SODA'01] presented a 3-approximation for the $k$-center problem with outliers. In this paper we consider a common generalization of the two extensions previously studied separately, i.e. we work with the capacitated $k$-center with outliers. We present the first constant factor approximation algorithm with approximation ratio of 25 even for the case of non-uniform hard capacities.Comment: 15 pages, 3 figures, accepted to STACS 201

QoS Constrained Optimal Sink and Relay Placement in Planned Wireless Sensor Networks

We are given a set of sensors at given locations, a set of potential locations for placing base stations (BSs, or sinks), and another set of potential locations for placing wireless relay nodes. There is a cost for placing a BS and a cost for placing a relay. The problem we consider is to select a set of BS locations, a set of relay locations, and an association of sensor nodes with the selected BS locations, so that number of hops in the path from each sensor to its BS is bounded by hmax, and among all such feasible networks, the cost of the selected network is the minimum. The hop count bound suffices to ensure a certain probability of the data being delivered to the BS within a given maximum delay under a light traffic model. We observe that the problem is NP-Hard, and is hard to even approximate within a constant factor. For this problem, we propose a polynomial time approximation algorithm (SmartSelect) based on a relay placement algorithm proposed in our earlier work, along with a modification of the greedy algorithm for weighted set cover. We have analyzed the worst case approximation guarantee for this algorithm. We have also proposed a polynomial time heuristic to improve upon the solution provided by SmartSelect. Our numerical results demonstrate that the algorithms provide good quality solutions using very little computation time in various randomly generated network scenarios