Improved Approximation Results for Stochastic Knapsack Problems

We consider a stochastic variant of the NP-hard 0/1 knapsack problem, in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. Our goal is to compute a solution “policy” that maximizes the expected value of items successfully placed in the knapsack, where the final overflowing item contributes no value. We consider both nonadaptive policies (that designate a priori a fixed sequence of items to insert) and adaptive policies (that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far). An important facet of our work lies in characterizing the benefit of adaptivity. For this purpose we advocate the use of a measure called the adaptivity gap: the ratio of the expected value obtained by an optimal adaptive policy to that obtained by an optimal nonadaptive policy. We bound the adaptivity gap of the stochastic knapsack problem by demonstrating a polynomial-time algorithm that computes a nonadaptive policy whose expected value approximates that of an optimal adaptive policy to within a factor of four. We also devise a polynomial-time adaptive policy that approximates the optimal adaptive policy to within a factor of 3 + ε for any constant ε > 0.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight Guarantees for Multi-unit Prophet Inequalities and Online Stochastic KnapsackJiashuo Jiang, Will Ma, and Jiawei ZhangJiashuo Jiang, Will Ma, and Jiawei Zhangpp.1221 - 1246Chapter DOI:https://doi.org/10.1137/1.9781611977073.51PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Prophet inequalities are a useful tool for designing online allocation procedures and comparing their performance to the optimal offline allocation. In the basic setting of k-unit prophet inequalities, the procedure of Alaei [2] with its celebrated performance guarantee of has found widespread adoption in mechanism design and general online allocation problems in online advertising, healthcare scheduling, and revenue management. Despite being commonly used for implementing a fractional allocation in an online fashion, the tightness of Alaei's procedure for a given k has remained unknown. In this paper we resolve this question, characterizing the tight bound by identifying the structure of the optimal online implementation, and consequently improving the best-known guarantee for k-unit prophet inequalities for all k > 1. We also consider the more general online stochastic knapsack problem where each individual allocation can consume an arbitrary fraction of the initial capacity. Here we introduce a new “best-fit” procedure for implementing a fractionally-feasible knapsack solution online, with a performance guarantee of ≈ 0.319, which we also show is tight with respect to the standard LP relaxation. This improves the previously best-known guarantee of 0.2 for online knapsack. Our analysis differs from existing ones by eschewing the need to split items into “large” or “small” based on capacity consumption, using instead an invariant for the overall utilization on different sample paths. All in all, our results imply tight (non-greedy) Online Contention Resolution Schemes for k-uniform matroids and the knapsack polytope, respectively, which has further implications. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

In this paper we study and solve two different variants of static knapsack problems with random weights: The stochastic knapsack problem with simple recourse as well as the stochastic knapsack problem with probabilistic constraint. Special interest is given to the corresponding continuous problems and three different problem solving methods are presented. The resolution of the continuous problems allows to provide upper bounds in a branch-and-bound framework in order to solve the original problems. Numerical results on a dataset from the literature as well as a set of randomly generated instances are given.

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2011 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Bidimensionality and EPTASFedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket SaurabhFedor V. FominDepartment of Informatics, University of Bergen, Norway.Department of Computer Science and Engineering, University of California, San DiegoThe Institute of Mathematical Sciences, Chennai, India.The Institute of Mathematical Sciences, Chennai, India.Search for more papers by this author, Daniel LokshtanovDepartment of Informatics, University of Bergen, Norway.Department of Computer Science and Engineering, University of California, San DiegoThe Institute of Mathematical Sciences, Chennai, India.The Institute of Mathematical Sciences, Chennai, India.Search for more papers by this author, Venkatesh RamanDepartment of Informatics, University of Bergen, Norway.Department of Computer Science and Engineering, University of California, San DiegoThe Institute of Mathematical Sciences, Chennai, India.The Institute of Mathematical Sciences, Chennai, India.Search for more papers by this author, and Saket SaurabhDepartment of Informatics, University of Bergen, Norway.Department of Computer Science and Engineering, University of California, San DiegoThe Institute of Mathematical Sciences, Chennai, India.The Institute of Mathematical Sciences, Chennai, India.Search for more papers by this authorpp.748 - 759Chapter DOI:https://doi.org/10.1137/1.9781611973082.59PDFBibTexSections ToolsAdd to favoritesDownload CitationsTrack CitationsEmail SectionsAboutAbstract Bidimensionality theory appears to be a powerful framework for the development of meta-algorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al [SODA 2010] established a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. One of the important conditions required in the framework developed by Demaine and Hajiaghayi [SODA 2005] is that to obtain an EPTAS for a graph optimization problem П, we have to know a constant-factor approximation algorithm for П. Our approach eliminates this strong requirement, which makes it amenable to more problems. At the heart of our framework is a decomposition lemma which states that for “most” bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an ε > 0 outputs a vertex set X of size ε · OPT such that the treewidth of G\X is O(1/ε). Here, OPT is the objective function value of the problem in question This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including Cycle Packing, Vertex-H-Packing, Maximum Leaf Spanning Tree, and Partial r-Dominating Set no EPTASs on planar graphs were previously known. Previous chapter Next chapter RelatedDetails Published:2011ISBN:978-0-89871-993-2eISBN:978-1-61197-308-2 https://doi.org/10.1137/1.9781611973082Book Series Name:ProceedingsBook Code:PR138Book Pages:xviii-1788

We study stochastic variants of Packing Integer Programs (PIP) --- the problems of finding a maximum-value 0/1 vector x satisfying Ax ≤ b, with A and b nonnegative. Many combinatorial problems belong to this broad class, including the knapsack problem, maximum clique, stable set, matching, hypergraph matching (a.k.a. set packing), b-matching, and others. PIP can also be seen as a multidimensional knapsack problem where we wish to pack a maximum-value collection of items with vector-valued sizes. In our stochastic setting, the vector-valued sizes of each item is known to us apriori only as a probability distribution, and the size of an item is instantiated once we commit to including the item in our solution.Following the framework of [3], we consider both adaptive and non-adaptive policies for solving such problems, adaptive policies having the flexibility of being able to make decisions based on the instantiated sizes of items already included in the solution. We investigate the adaptivity gap for these problems: the maximum ratio between the expected values achieved by optimal adaptive and non-adaptive policies. We show tight bounds on the adaptivity gap for set packing and b-matching, and we also show how to find efficiently non-adaptive policies approximating the adaptive optimum. For instance, we can approximate the adaptive optimum for stochastic set packing to within O(d1/2), which is not only optimal with respect to the adaptivity gap, but it is also the best known approximation factor in the deterministic case. It is known that there is no polynomial-time d1/2-e approximation for set packing, unless NP = ZPP. Similarly, for b-matching, we obtain algorithmically a tight bound on the adaptivity gap of O(λ) where λ satisfies Σ λbj+1 = 1.For general Stochastic Packing, we prove that a simple greedy algorithm provides an O(d)-approximation to the adaptive optimum. For A ∈ [0, 1]dxn, we provide an O(λ) approximation where Σ 1/λbj = 1. (For b = (B, B,..., B), we get λ = d1/B.) We also improve the hardness results for deterministic PIP: in the general case, we prove that a polynomial-time d1-e-approximation algorithm would imply NP = ZPP. In the special case when A ∈ [0,1]dxn and b = (B,B,...,B), we show that a d1/B-∈-approximation would imply NP = ZPP. Finally, we prove that it is PSPACE-hard to find the optimal adaptive policy for Stochastic Packing in any fixed dimension d ≥ 2.

A faster FPTAS for knapsack problem with cardinality constraint

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA)Faster p-norm minimizing flows, via smoothed q-norm problemsDeeksha Adil and Sushant SachdevaDeeksha Adil and Sushant Sachdevapp.892 - 910Chapter DOI:https://doi.org/10.1137/1.9781611975994.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We present faster high-accuracy algorithms for computing ℓp-norm minimizing flows. On a graph with m edges, our algorithm can compute a (1 + 1/poly(m))-approximate unweighted ℓp-norm minimizing flow with operations, for any p ≥ 2, giving the best bound for all p ≳ 5.24. Combined with the algorithm from the work of Adil et al. (SODA '19), we can now compute such flows for any 2 ≤ p ≤ mo(1) in time at most O(m1.24). In comparison, the previous best running time was Ω(m1.33) for large constant p. For p ∼ σ−1 log m, our algorithm computes a (1 + σ)-approximate maximum flow on undirected graphs using m1+o(1)σ−1 operations, matching the current best bound, albeit only for unit-capacity graphs. We also give an algorithm for solving general ℓp-norm regression problems for large p. Our algorithm makes calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted ℓp-norm minimizing flows that runs in time o(m1.5) for some p = mΩ(1). Our key technical contribution is to show that smoothed ℓp-norm problems introduced by Adil et al., are interreducible for different values of p. No such reduction is known for standard ℓp-norm problems. Previous chapter Next chapter RelatedDetails Published:2020eISBN:978-1-61197-599-4 https://doi.org/10.1137/1.9781611975994Book Series Name:ProceedingsBook Code:PRDA20Book Pages:xxii + 3011

Discrete-order admission ATP model with joint effect of margin and order size in a MTO environment

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)The Complexity of Optimal Multidimensional PricingXi Chen, Ilias Diakonikolas, Dimitris Paparas, Xiaorui Sun, and Mihalis YannakakisXi Chen, Ilias Diakonikolas, Dimitris Paparas, Xiaorui Sun, and Mihalis Yannakakispp.1319 - 1328Chapter DOI:https://doi.org/10.1137/1.9781611973402.97PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing a revenue-optimal pricing can be solved in polynomial time for distributions of support size 2 and its decision version is NP-complete for distributions of support size 3. We also show that the problem remains NP-complete for the case of identical distributions. Previous chapter Next chapter RelatedDetails Published:2014ISBN:978-1-61197-338-9eISBN:978-1-61197-340-2 https://doi.org/10.1137/1.9781611973402Book Series Name:ProceedingsBook Code:PRDA14Book Pages:viii + 1885

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Partitioning into ExpandersShayan Oveis Gharan and Luca TrevisanShayan Oveis Gharan and Luca Trevisanpp.1256 - 1266Chapter DOI:https://doi.org/10.1137/1.9781611973402.93PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Let G = (V, E) be an undirected graph, λk be the kth smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that λk > 0 if and only if G has at most k – 1 connected components. We prove a robust version of this fact. If λk > 0, then for some 1 ≤ ℓ ≤ k – 1, V can be partitioned into ℓ sets P1, …, Pℓ such that each Pi is a low-conductance set in G and induces a high conductance induced subgraph. In particular, and G[Pi] ≳ λk/k2-expander. We make our results algorithmic by designing a simple polynomial time spectral algorithm to find such partitioning of G with a quadratic loss in the inside conductance of Pi's. Unlike the recent results on higher order Cheeger's inequality [6, 9], our results does not use higher order eigenfunctions of G. If there is a sufficiently large gap between λk and λk+1, more precisely if then our algorithm finds a k partitioning of V into sets P1, …, Pk such that the induced subgraph G[Pi] has a singnificantly larger conductance than the conductance of Pi in G. Such a partitioning may represent the best k clusterings of G. Our algorithm is a simple local search that only uses the Spectral Partitioning algorithm as a subroutine. We expect to see further applications of this simple algorithm in clustering applications. Let ρ(k) = mindisjoint max1≤i≤k φ(Ai) be the order k conductance constant of G, in words, ρ(k) is the smallest value of the maximum conductance of any k disjoint subsets of V. Our main technical lemma shows that if (1 + ∊)ρ(k) < ρ(k+1), then V can be partitioned into k sets P1, …, Pk such that for each 1 ≤ i ≤ k, φ(G[Pi]) ≳ ∊ · ρ(k + 1)/k and φ(Pi) ≤ k · ρ(k). This significantly improves a recent result of Tanaka [13] who assumed an exponential (in k) gap between ρ(k) and ρ(k + 1). Previous chapter Next chapter RelatedDetails Published:2014ISBN:978-1-61197-338-9eISBN:978-1-61197-340-2 https://doi.org/10.1137/1.9781611973402Book Series Name:ProceedingsBook Code:PRDA14Book Pages:viii + 1885

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2013 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Near Optimal Leader Election in Multi-Hop Radio NetworksMohsen Ghaffari and Bernhard HaeuplerMohsen GhaffariMITMITSearch for more papers by this author and Bernhard HaeuplerMITMITSearch for more papers by this authorpp.748 - 766Chapter DOI:https://doi.org/10.1137/1.9781611973105.54PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We design leader election protocols for multi-hop radio networks that elect a leader in almost the same time TBC that it takes for broadcasting one message (one ID). For the setting without collision detection our algorithm runs whp. in rounds on any n-node network with diameter D. Since is a lower bound, our upper bound is optimal up to a factor of at most log log n and the extra log n factor on the additive term. Our algorithm is furthermore the first O(n) time algorithm for this setting. Our algorithm improves over a 23 year old simulation approach of Bar-Yehuda, Goldreich and Itai with a O(TBC log n) running time: In 1987 they designed a fast broadcast protocol and subsequently in 1989 they showed how it can be used to simulate one round of a single-hop network that has collision detection in TBC time. The prime application of this simulation was to simulate Willards single-hop leader election protocol, which elects a leader in O(log n) rounds whp. and O(log log n) rounds in expectation. While it was subsequently shown that Willards bounds are tight, it was unclear whether the simulation approach is optimal. Our results break this barrier and essentially remove the logarithmic slowdown over the broadcast time TBC. This is achieved by going away from the simulation approach. We also give an leader election algorithm for the setting with collision detection (even with single-bit messages). This is optimal up to log log n factors and improves over a deterministic algorithm that requires Θ(n) rounds independently of D. Our almost optimal leader election protocols are especially important because countless communication protocols in radio networks use leader election as a crucial first step to solve various, seemingly unrelated, communication primitives such as gathering, multiple unicasts or multiple broadcasts. Even though leader election seems easier than these tasks, its best-known O(TBC log n) running time had become a bottleneck, preventing optimal algorithms. Breaking the simulation barrier for leader election in this paper has subsequently led to the development of near optimal protocols for these communication primitives. Previous chapter Next chapter RelatedDetails Published:2013ISBN:978-1-61197-251-1eISBN:978-1-61197-310-5 https://doi.org/10.1137/1.9781611973105Book Series Name:ProceedingsBook Code:PR143Book Pages:xix + 1915

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Inapproximability of the Multi-level Uncapacitated Facility Location ProblemRavishankar Krishnaswamy and Maxim SviridenkoRavishankar Krishnaswamy and Maxim Sviridenkopp.718 - 734Chapter DOI:https://doi.org/10.1137/1.9781611973099.59PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract In this paper, we present improved inapproximability results for the k-level uncapacitated facility location problem. In particular, we show that there is no polynomial time approximation algorithm with performance guarantee better than 1.539 unless NP is contained in DTIME(nO(log log n)) for the case when k = 2. For the case of general k (tendining to infinity) we obtain a better hardness factor of 1.61. Interestingly, our results show that the two-level problem is computationally harder than the well known uncapacitated facility location problem (k = 1) since the best known approximation guarantee for the latter problem is 1.488 due to Li [22], and our inapproximability is a factor of 1.539 for the two-level problem. The only inapproximability result known before for this class of metric facility location problems is the bound of 1.463 due to Guha and Khuller [17], which holds even for the case of k = 1. Previous chapter Next chapter RelatedDetails Published:2012ISBN:978-1-61197-210-8eISBN:978-1-61197-309-9 https://doi.org/10.1137/1.9781611973099Book Series Name:ProceedingsBook Code:PR141Book Pages:xiii + 1757

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2009 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Exact Algorithms for Partial Curve Matching via the Fréchet DistanceKevin Buchin, Maike Buchin, and Yusu WangKevin Buchin, Maike Buchin, and Yusu Wangpp.645 - 654Chapter DOI:https://doi.org/10.1137/1.9781611973068.71PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Curve matching is a fundamental problem that occurs in many applications. In this paper, we study the problem of measuring partial similarity between curves. Specifically, given two curves, we wish to maximize the total length of subcurves that are close to each other, where closeness is measured by the Fréchet distance, a common distance measure for curves. The resulting maximal length is called the partial Fréchet similarity between the two input curves. Given two polygonal curves P and Q in IRd of size m and n, respectively, we present the first exact algorithm that runs in polynomial time to compute ℱδ(P, Q), the partial Fréchet similarity between P and Q, under the L1 and L∞ norms. Specifically, we formulate the problem of computing ℱδ(P, Q) as a longest path problem, and solve it in O(mn(m + n) log(mn)) time, under the L1 or L∞ norm, using a “shortest-path map” type decomposition. To the best of our knowledge, this is the first paper to study this natural definition of partial curve similarity in the continuous setting (with all points in the curve considered), and present a polynomial-time exact algorithm for it. Previous chapter Next chapter RelatedDetails Published:2009ISBN:978-0-89871-680-1eISBN:978-1-61197-306-8 https://doi.org/10.1137/1.9781611973068Book Series Name:ProceedingsBook Code:PR132Book Pages:xviii + 1288

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2012 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Using Hashing to Solve the Dictionary Problem (In External Memory)John Iacono and Mihai PătraşcuJohn IaconoPolytechnic Institute of NYUAT&T Labs*Research supported by NSF grants CCF-1018370, CCF-0430849 and by an Alfred P. Sloan fellowship. Research partially completed as a visiting professor at MADALGO (Center for Massive Data Algorithmics, a Center of the Danish National Research Foundation), Department of Computer Science, Aarhus University, IT Parken, Åbogade 34, DK-8200 Århus N, Denmark.Search for more papers by this author and Mihai PătraşcuPolytechnic Institute of NYUAT&T LabsSearch for more papers by this authorpp.570 - 582Chapter DOI:https://doi.org/10.1137/1.9781611973099.48PDFBibTexSections ToolsAdd to favoritesDownload CitationsTrack CitationsEmail SectionsAboutAbstract We consider the dictionary problem in external memory and improve the update time of the well-known buffer tree by roughly a logarithmic factor. For any λ > max(lg lg n, logM/B (n/B)}, we can support updates in time O(λ/B) and queries in sublogarithmic time, O(logλ n). We also present a lower bound in the cell-probe model showing that our data structure is optimal. In the RAM, hash tables have been use to solve the dictionary problem faster than binary search for more than half a century. By contrast, our data structure is the first to beat the comparison barrier in external memory. Ours is also the first data structure to depart convincingly from the indivisibility paradigm. Previous chapter Next chapter RelatedDetails Published:2012ISBN:978-1-61197-210-8eISBN:978-1-61197-309-9 https://doi.org/10.1137/1.9781611973099Book Series Name:ProceedingsBook Code:PR141Book Pages:xiii + 1757

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2014 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Half-integrality, LP-branching and FPT AlgorithmsMagnus WahlströmMagnus WahlströmRoyal Holloway, University of London.Search for more papers by this authorpp.1762 - 1781Chapter DOI:https://doi.org/10.1137/1.9781611973402.128PDFBibTexSections ToolsAdd to favoritesDownload CitationsTrack CitationsEmail SectionsAboutAbstract A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser-Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994). Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0,1}V to {0,½,1}V such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and Živný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties. Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an O*(|Σ|2k)-time algorithm for node-deletion Unique Label Cover with label set Σ (improving the previous bound of due to Chitnis et al., 2012) and an O*(4k)-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access (improving on the previous bound of O*(2O(klogk)) due to Cygan et al., 2012). The latter bound is optimal under the Exponential Time Hypothesis. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Interestingly, despite the half-integrality, our result do not imply any approximation results (as may be expected, given the Unique Games-hardness of the covered problems). Previous chapter Next chapter RelatedDetails Published:2014ISBN:978-1-61197-338-9eISBN:978-1-61197-340-2 https://doi.org/10.1137/1.9781611973402Book Series Name:ProceedingsBook Code:PRDA14Book Pages:viii + 1885