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Register a new userA tight lower bound for an online hypercube packing problem and bounds for prices of anarchy of a related game
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Added by
Fl\\'avio Miyazawa
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We prove a tight lower bound on the asymptotic performance ratio $rho$ of the bounded space online $d$-hypercube bin packing problem, solving an open question raised in 2005. In the classic $d$-hypercube bin packing problem, we are given a sequence of $d$-dimensional hypercubes and we have an unlimited number of bins, each of which is a $d$-dimensional unit hypercube. The goal is to pack (orthogonally) the given hypercubes into the minimum possible number of bins, in such a way that no two hypercubes in the same bin overlap. The bounded space online $d$-hypercube bin packing problem is a variant of the $d$-hypercube bin packing problem, in which the hypercubes arrive online and each one must be packed in an open bin without the knowledge of the next hypercubes. Moreover, at each moment, only a constant number of open bins are allowed (whenever a new bin is used, it is considered open, and it remains so until it is considered closed, in which case, it is not allowed to accept new hypercubes). Epstein and van Stee [SIAM J. Comput. 35 (2005), no. 2, 431-448] showed that $rho$ is $Omega(log d)$ and $O(d/log d)$, and conjectured that it is $Theta(log d)$. We show that $rho$ is in fact $Theta(d/log d)$. To obtain this result, we elaborate on some ideas presented by those authors, and go one step further showing how to obtain better (offline) packings of certain special instances for which one knows how many bins any bounded space algorithm has to use. Our main contribution establishes the existence of such packings, for large enough $d$, using probabilistic arguments. Such packings also lead to lower bounds for the prices of anarchy of the selfish $d$-hypercube bin packing game. We present a lower bound of $Omega(d/log d)$ for the pure price of anarchy of this game, and we also give a lower bound of $Omega(log d)$ for its strong price of anarchy.
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In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $rho$ of the online bounded space variant is $Omega(log d)$ and $O(d/log d)$, and conjectured that it is $Theta(log d)$. We show that $rho$ is in fact $Theta(d/log d)$, using probabilistic arguments.
We consider the file maintenance problem (also called the online labeling problem) in which n integer items from the set {1,...,r} are to be stored in an array of size m >= n. The items are presented sequentially in an arbitrary order, and must be stored in the array in sorted order (but not necessarily in consecutive locations in the array). Each new item must be stored in the array before the next item is received. If r<=m then we can simply store item j in location j but if r>m then we may have to shift the location of stored items to make space for a newly arrived item. The algorithm is charged each time an item is stored in the array, or moved to a new location. The goal is to minimize the total number of such moves done by the algorithm. This problem is non-trivial when n=<m<r. In the case that m=Cn for some C>1, algorithms for this problem with cost O(log(n)^2) per item have been given [IKR81, Wil92, BCD+02]. When m=n, algorithms with cost O(log(n)^3) per item were given [Zha93, BS07]. In this paper we prove lower bounds that show that these algorithms are optimal, up to constant factors. Previously, the only lower bound known for this range of parameters was a lower bound of Omega(log(n)^2) for the restricted class of smooth algorithms [DSZ05a, Zha93]. We also provide an algorithm for the sparse case: If the number of items is polylogarithmic in the array size then the problem can be solved in amortized constant time per item.
Pool block withholding attack is performed among mining pools in digital cryptocurrencies, such as Bitcoin. Instead of mining honestly, pools can be incentivized to infiltrate their own miners into other pools. These infiltrators report partial solutions but withhold full solutions, share block rewards but make no contribution to block mining. The block withholding attack among mining pools can be modeled as a non-cooperative game called the miners dilemm, which reduces effective mining power in the system and leads to potential systemic instability in the blockchain. However, existing literature on the game-theoretic properties of this attack only gives a preliminary analysis, e.g., an upper bound of 3 for the pure price of anarchy (PPoA) in this game, with two pools involved and no miner betraying. Pure price of anarchy is a measurement of how much mining power is wasted in the miners dilemma game. Further tightening its upper bound will bring us more insight into the structure of this game, so as to design mechanisms to reduce the systemic loss caused by mutual attacks. In this paper, we give a tight bound of (1, 2] for the pure price of anarchy. Moreover, we show the tight bound holds in a more general setting, in which infiltrators may betray.We also prove the existence and uniqueness of pure Nash equilibrium in this setting. Inspired by experiments on the game among three mining pools, we conjecture that similar results hold in the N-player miners dilemma game (N>=2).
Vizings celebrated theorem asserts that any graph of maximum degree $Delta$ admits an edge coloring using at most $Delta+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses $2Delta-1$ colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with $Delta=O(log n)$, and they conjectured the existence of online algorithms using $Delta(1+o(1))$ colors for $Delta=omega(log n)$. Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS03 and Bahmani et al., SODA10). We resolve the above conjecture for emph{adversarial} vertex arrivals in bipartite graphs, for which we present a $(1+o(1))Delta$-edge-coloring algorithm for $Delta=omega(log n)$ known a priori. Surprisingly, if $Delta$ is not known ahead of time, we show that no $big(frac{e}{e-1} - Omega(1) big) Delta$-edge-coloring algorithm exists. We then provide an optimal, $big(frac{e}{e-1}+o(1)big)Delta$-edge-coloring algorithm for unknown $Delta=omega(log n)$. Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.
We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $ell$ servers, where servers have capacity $k$ and we assume that the graph has $ell cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $varepsilon k$ for constant $varepsilon>0$. Whenever $operatorname{ONL}$ learns about a new edge, the algorithm is allowed to move vertices from one server to another. Its objective is to minimize the number of vertex moves. More specifically, $operatorname{ONL}$ should minimize the competitive ratio: the total cost $operatorname{ONL}$ incurs compared to an optimal offline algorithm $operatorname{OPT}$. Our main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of $O(log ell + log k)$ on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than $Omega(log ell + log k)$. We also settle the open problem of the achievable competitive ratio by deterministic online algorithms, by deriving a competitive ratio of $Theta(ell lg k)$; to this end, we present an improved lower bound as well as a deterministic polynomial-time online algorithm. Our algorithms rely on a novel technique which combines efficient integer programming with a combinatorial approach for maintaining ILP solutions. We believe this technique is of independent interest and will find further applications in the future.
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