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Register a new userHow to Catch Marathon Cheaters: New Approximation Algorithms for Tracking Paths
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Pedro Matias
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Given an undirected graph, $G$, and vertices, $s$ and $t$ in $G$, the tracking paths problem is that of finding the smallest subset of vertices in $G$ whose intersection with any $s$-$t$ path results in a unique sequence. This problem is known to be NP-complete and has applications to animal migration tracking and detecting marathon course-cutting, but its approximability is largely unknown. In this paper, we address this latter issue, giving novel algorithms having approximation ratios of $(1+epsilon)$, $O(lg OPT)$ and $O(lg n)$, for $H$-minor-free, general, and weighted graphs, respectively. We also give a linear kernel for $H$-minor-free graphs and make improvements to the quadratic kernel for general graphs.
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We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t, Tracking Shortest Paths asks if there exists a k-sized subset of vertices (referred to as tracking set) that intersects each shortest s-t path in a distinct set of vertices. We first generalize this problem for set systems, namely Tracking Set System, where given a family of subsets of a universe, we are required to find a subset of elements from the universe that has a unique intersection with each set in the family. Tracking Set System is shown to be fixed-parameter tractable due to its relation with a known problem, Test Cover. By a reduction to the well-studied d-hitting set problem, we give a polynomial (with respect to k) kernel for the case when the set sizes are bounded by d. This also helps solving Tracking Shortest Paths when the input graph diameter is bounded by d. While the results for Tracking Set System help to show that Tracking Shortest Paths is fixed-parameter tractable, we also give an independent algorithm by using some preprocessing rules, resulting in an improved running time.
We study the classical NP-hard problems of finding maximum-size subsets from given sets of $k$ terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is $Omega(log^{1/2-epsilon}{n})$, assuming NP$~ otsubseteq~$ZPTIME$(n^{mathrm{poly}log n})$. This constitutes a significant gap to the best known approximation upper bound of $O(sqrt{n})$ due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an $O(1)$-approximation when edges (or nodes) may be used by $O(frac{log n}{loglog n})$ paths. In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number $r$ of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following. * For MaxEDP, we give an $O(sqrt{r}cdot log^{1.5}{kr})$-approximation algorithm. As $rleq n$, up to logarithmic factors, our result strengthens the best known ratio $O(sqrt{n})$ due to Chekuri et al. * Further, we show how to route $Omega(mathrm{OPT})$ pairs with congestion $O(frac{log{kr}}{loglog{kr}})$, strengthening the bound obtained by the classic approach of Raghavan and Thompson. * For MaxNDP, we give an algorithm that gives the optimal answer in time $(k+r)^{O(r)}cdot n$. If $r$ is at most triple-exponential in $k$, this improves the best known algorithm for MaxNDP with parameter $k$, by Kawarabayashi and Wollan (STOC 2010). We complement these positive results by proving that MaxEDP is NP-hard even for $r=1$, and MaxNDP is W$[1]$-hard for parameter $r$.
The general adwords problem has remained largely unresolved. We define a subcase called {em $k$-TYPICAL}, $k in Zplus$, as follows: the total budget of all the bidders is sufficient to buy $k$ bids for each bidder. This seems a reasonable assumption for a typical instance, at least for moderate values of $k$. We give a randomized online algorithm, achieving a competitive ratio of $left(1 - {1 over e} - {1 over k} right)$, for this problem. We also give randomized online algorithms for other special cases of adwords. Another subcase, when bids are small compared to budgets, has been of considerable practical significance in ad auctions cite{MSVV}. For this case, we give an optimal randomized online algorithm achieving a competitive ratio of $left(1 - {1 over e} right)$. Previous algorithms for this case were based on LP-duality; the impact of our new approach remains to be seen. The key to these results is a simplification of the proof for RANKING, the optimal algorithm for online bipartite matching, given in cite{KVV}. Our algorithms for adwords can be seen as natural extensions of RANKING.
Approximating the length of the longest increasing sequence (LIS) of an array is a well-studied problem. We study this problem in the data stream model, where the algorithm is allowed to make a single left-to-right pass through the array and the key resource to be minimized is the amount of additional memory used. We present an algorithm which, for any $delta > 0$, given streaming access to an array of length $n$ provides a $(1+delta)$-multiplicative approximation to the emph{distance to monotonicity} ($n$ minus the length of the LIS), and uses only $O((log^2 n)/delta)$ space. The previous best known approximation using polylogarithmic space was a multiplicative 2-factor. Our algorithm can be used to estimate the length of the LIS to within an additive $delta n$ for any $delta >0$ while previous algorithms could only achieve additive error $n(1/2-o(1))$. Our algorithm is very simple, being just 3 lines of pseudocode, and has a small update time. It is essentially a polylogarithmic space approximate implementation of a classic dynamic program that computes the LIS. We also give a streaming algorithm for approximating $LCS(x,y)$, the length of the longest common subsequence between strings $x$ and $y$, each of length $n$. Our algorithm works in the asymmetric setting (inspired by cite{AKO10}), in which we have random access to $y$ and streaming access to $x$, and runs in small space provided that no single symbol appears very often in $y$. More precisely, it gives an additive-$delta n$ approximation to $LCS(x,y)$ (and hence also to $E(x,y) = n-LCS(x,y)$, the edit distance between $x$ and $y$ when insertions and deletions, but not substitutions, are allowed), with space complexity $O(k(log^2 n)/delta)$, where $k$ is the maximum number of times any one symbol appears in $y$.
The restless bandit problem is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit problem in decision theory. In its ultimate generality, the restless bandit problem is known to be PSPACE-Hard to approximate to any non-trivial factor, and little progress has been made despite its importance in modeling activity allocation under uncertainty. We consider a special case that we call Feedback MAB, where the reward obtained by playing each of n independent arms varies according to an underlying on/off Markov process whose exact state is only revealed when the arm is played. The goal is to design a policy for playing the arms in order to maximize the infinite horizon time average expected reward. This problem is also an instance of a Partially Observable Markov Decision Process (POMDP), and is widely studied in wireless scheduling and unmanned aerial vehicle (UAV) routing. Unlike the stochastic MAB problem, the Feedback MAB problem does not admit to greedy index-based optimal policies. We develop a novel and general duality-based algorithmic technique that yields a surprisingly simple and intuitive 2+epsilon-approximate greedy policy to this problem. We then define a general sub-class of restless bandit problems that we term Monotone bandits, for which our policy is a 2-approximation. Our technique is robust enough to handle generalizations of these problems to incorporate various side-constraints such as blocking plays and switching costs. This technique is also of independent interest for other restless bandit problems. By presenting the first (and efficient) O(1) approximations for non-trivial instances of restless bandits as well as of POMDPs, our work initiates the study of approximation algorithms in both these contexts.
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