## Improved Hardness of Approximation for Stackelberg Shortest-Path Pricing

Author: Bundit Laekhanukit, Danupon Nanongkai
(Alphabetical order)

Conference: WINE 2010: 6th Workshop on Internet & Network Economics [wiki].  Published in LNCS Vol. 6484.

Abstract:

We consider the Stackelberg shortest-path pricing problem, which is deﬁned as follows. Given a graph G with ﬁxed-cost and pricable edges and two distinct vertices s and t, we may assign
prices to the pricable edges. Based on the predeﬁned ﬁxed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable
edges belonging to the path. Our goal is to ﬁnd prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide
the ﬁrst explicit approximation threshold and prove hardness of approximation within 2 − o(1). We also prove that for the nicely structured type of instance resulting from our reduction, the
gap between the revenue of an optimal pricing and the only known general upper bound can still be logarithmically large.

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[v1] October 1, 2010 (Conference version)

## Regret-Minimizing Representative Databases

Author: Danupon Nanongkai, Atish Das Sarma, Ashwin Lall, Richard J. Lipton, Jun Xu

Conference: VLDB 2010: 36th International Conference on Very Large Databases [wiki]

Abstract:

We propose the k-representative regret minimization query (k-regret) as an operation to support multi-criteria decision making. Like top-k, the k-regret query assumes that users have some utility or scoring functions; however, it never asks the users to provide such functions. Like skyline, it filters out a set of interesting points from a potentially large database based on the users’ criteria; however, it never overwhelms the users by outputting too many tuples.

In particular, for any number k and any class of utility functions, the k-regret query outputs k tuples from the database and tries to minimize the {\em maximum regret ratio}. This captures how disappointed a user could be had she seen k-representative tuples instead of the whole database. We focus on the class of linear utility functions, which is widely applicable.

The first challenge of this approach is that it is not clear if the maximum regret ratio can be small, or even bounded. We answer this question affirmatively. Theoretically, we prove that the maximum regret ratio can be bounded and this bound is independent of the database size. Moreover, our extensive experiments on real and synthetic datasets suggest that in practice the maximum regret ratio is reasonably small. Additionally, algorithms developed in this paper are practical as they run in linear time in the size of the database and the experiments show that their running time is small when they run on top of the skyline operation which means that these algorithm could be integrated into current database systems.

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[v1] June 28, 2010 (Conference version)

## Efficient Distributed Random Walks with Applications

Author: Atish Das Sarma, Danupon Nanongkai, Gopal Pandurangan, Prasad Tetali

Conference: PODC 2010

Journal: Journal of the ACM 2013

Abstract:

We  focus on  the problem of performing random walks efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain a random walk sample. We first present a fast sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk. Our algorithm performs a random walk of length $\ell$  in $\tilde{O}(\sqrt{\ell D})$  rounds (with high probability) on an undirected  network, where $D$ is the diameter of the network. This improves over the previous best algorithm that ran in $\tilde{O}(\ell^{2/3}D^{1/3})$ rounds (Das Sarma et al., PODC 2009). We further extend our algorithms to efficiently perform $k$ independent random walks in   $\tilde{O}(\sqrt{k\ell D} + k)$ rounds. We then show that there is a fundamental difficulty in improving the dependence on $\ell$ any further by proving a lower bound of $\Omega(\sqrt{\frac{\ell}{\log \ell}} + D)$ under a general model of distributed random walk algorithms. Our random walk algorithms are useful in speeding up distributed algorithms for a variety of applications that use random walks as a subroutine. We present two main applications. First, we give a fast distributed algorithm for computing a random spanning tree (RST) in an arbitrary (undirected) network which runs in $\tilde{O}(\sqrt{m}D)$ rounds (with high probability; here $m$ is the number of edges). Our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network. Our algorithm is fully decentralized and can serve as a building block in the design of topologically-aware networks.

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Mar 03, 2009 (New version posted on ArXiv)
Nov 06, 2009 (Link to arXiv posted)
Feb 18, 2010 (New version posted)
Feb 18, 2013 (Journal version posted)

## Stackelberg Pricing is Hard to Approximate within 2−ε

Author Parinya Chalermsook, Bundit Lekhanukit, Danupon Nanongkai

Conference:

Abstract:

Stackelberg Pricing Games is a two-level combinatorial pricing problem studied in the Economics, Operation Research, and Computer Science communities. In this paper, we consider the decade-old shortest path version of this problem which is the first and most studied problem in this family. The game is played on a graph (representing a network) consisting of fixed cost edges and pricable or variable cost edges. The fixed cost edges already have some fixed price (representing the competitor’s prices). Our task is to choose prices for the variable cost edges. After that, a client will buy the cheapest path from a node $s$ to a node $t$, using any combination of fixed cost and variable cost edges. The goal is to maximize the revenue on variable cost edges.

In this paper, we show that the problem is hard to approximate within $2-\epsilon$, improving the previous APX-hardness result by Joret [to appear in Networks]. Our technique combines the existing ideas with a new insight into the price structure and its relation to the hardness of the instances.

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[v1] Oct 2, 2009 (Manuscript posted on arXiv)

## Faster Algorithms for Semi-Matching Problems

Author Jittat Fakcharoenphol, Bundit Lekhanukit, Danupon Nanongkai

Conference: ICALP 2010

Abstract:

We consider the problem of finding semi-matching in bipartite graphs, a problem also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted case.

For the weighted case, we give an $O(nm\log n)$-time algorithm, where $n$ is the number of vertices and $m$ is the number of edges, by exploiting geometric structure of the problem. This improves the classical $O(n^3)$ algorithms by Horn [Operations Research 1973] and Brono, Coffman and Sethi [Communications of the ACM 1974].

For the unweighted case, the bound could be improved even further. We give a simple divide-and-conquer algorithm which runs in time $O(\sqrt{n}m\log n)$, improving two previous $O(nm)$-time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lovasz and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the Balance Edge Cover problem in time $O(\sqrt{n}m\log n)$, improving the previous $O(nm)$-time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008].

## Randomized Multi-pass Streaming Skyline Algorithms (VLDB’09)

Author (ordered alphabetically): Atish Das Sarma, Ashwin Lall, Danupon Nanongkai, Jun Xu

Journal: Soon

Conference: VLDB 2009: 35th International Conference on Very Large Databases [wiki]

Abstract:

We consider external algorithms for skyline computation without pre-processing. Our goal is to develop an algorithm with a good worst case guarantee while performing well on average. Due to the nature of disks, it is desirable that such algorithms access the input as a stream (even if in multiple passes). Using the tools of randomness, proved to be useful in many applications, we present an efficient multi-pass streaming algorithm, RAND, for skyline computation. As far as we are aware, RAND is the first randomized skyline algorithm in the literature.

RAND is near-optimal for the streaming model, which we prove via a simple lower bound. Additionally, our algorithm is distributable and can handle partially ordered domains on each attribute. Finally, we demonstrate the robustness of RAND via extensive experiments on both real and synthetic datasets. RAND is comparable to the existing algorithms in average case and additionally tolerant to simple modifications of the data, while other algorithms degrade considerably with such variation.

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[v1] August 20, 2009 (Conference version)

## Fast Distributed Random Walks (PODC’09)

Authors: Atish Das Sarma, Danupon Nanongkai, Gopal Pandurangan

Journal: Journal of the ACM 2013

Conference: PODC 2009: 28th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computin[wiki]

Abstract:

Performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science, including distributed computing. In this paper, we focus on the problem of performing random walks efficiently in a distributed network. Given bandwidth constraints, the goal is to minimize the number of rounds required to obtain a random walk sample.

All previous algorithms that compute a random walk sample of length $\ell$ as a subroutine always do so naively, i.e., in $O(\ell)$ rounds. The main contribution of this paper is a fast distributed
algorithm for performing random walks. We show that  a random walk sample of length $\ell$ can be computed in $\tilde{O}(\ell^{2/3}D^{1/3})$ rounds on an undirected unweighted network, where $D$ is the diameter of the network. ($\tilde{O}$ hides $\frac{log{n}}{\delta}$ factors where $n$ is the number of nodes in the network and $\delta$ is the minimum degree.) For small diameter graphs, this is a significant improvement over the naive $O(\ell)$ bound. We also show that our algorithm  can be applied to speedup the more general Metropolis-Hastings sampling.

We extend our algorithms to perform a large number, $k$, of random walks efficiently. We show how $k$ destinations can be sampled in $\tilde{O}((k\ell)^{2/3}D^{1/3})$ rounds if $k\leq \ell^2$ and $\tilde{O}((k\ell)^{1/2})$ rounds otherwise. We  also present faster algorithms for performing random walks of length larger than (or equal to) the mixing time of the underlying graph. Our techniques can be useful in speeding up distributed algorithms for a variety of applications that use random walks as a subroutine.

Keywords: Random walks, Random sampling, Distributed algorithm, Metropolis-Hastings sampling.

Update History

[v1] May 31, 2009 (Conference version)
[v2] Feb 18, 2013 (Journal version posted)