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Computationally Manageable Combinatorial Auctions
, 1998
"... There is interest in designing simultaneous auctions for situations in which the value of assets to a bidder depends upon which other assets he or she wins. In such cases, bidders may well wish to submit bids for combinations of assets. When this is allowed, the problem of determining the revenue ma ..."
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Cited by 345 (1 self)
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There is interest in designing simultaneous auctions for situations in which the value of assets to a bidder depends upon which other assets he or she wins. In such cases, bidders may well wish to submit bids for combinations of assets. When this is allowed, the problem of determining the revenue maximizing set of nonconflicting bids can be a difficult one. We analyze this problem, identifying several different structures of combinatorial bids for which computational tractability is constructively demonstrated and some structures for which computational tractability 1 Introduction Some auctions sell many assets simultaneously. Often these assets, like U.S. treasury bills, are interchangeable. However, sometimes the assets and the bids for them are distinct. This happens frequently, as in the U.S. Department of the Interior's simultaneous sales of offshore oil leases, in some private farm land auctions, and in the Federal Communications Commission's recent multibillion dollar sales...
Computationally Manageable Combinational Auctions
, 1998
"... There is interest in designing simultaneous auctions for situations such as the recent FCC radio spectrum auctions, in which the value of assets to a bidder depends on which other assets he or she wins. In such auctions, bidders may wish to submit bids for combinations of assets. When this is allowe ..."
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Cited by 96 (6 self)
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There is interest in designing simultaneous auctions for situations such as the recent FCC radio spectrum auctions, in which the value of assets to a bidder depends on which other assets he or she wins. In such auctions, bidders may wish to submit bids for combinations of assets. When this is allowed, the problem of determining the revenue maximizing set of nonconflicting bids can be difficult. We analyze this problem, identifying several different structures of permitted combinational bids for which computational tractability is constructively demonstrated and some structures for which computational tractability cannot be guaranteed.
Tiling a Polygon with Rectangles
 Proc. 33rd Symp. Foundations of Computer Science
, 1992
"... We study the problem of tiling a simple polygon of surface n with rectangles of given types (tiles). We present a linear time algorithm for deciding if a polygon can be tiled with 1 \Theta m and k \Theta 1 tiles (and giving a tiling when it exists), and a quadratic algorithm for the same problem whe ..."
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Cited by 35 (4 self)
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We study the problem of tiling a simple polygon of surface n with rectangles of given types (tiles). We present a linear time algorithm for deciding if a polygon can be tiled with 1 \Theta m and k \Theta 1 tiles (and giving a tiling when it exists), and a quadratic algorithm for the same problem when the tile types are m \Theta k and k \Theta m. 1 Introduction We present algorithms for tiling a simple region of Z 2 (i.e. polygonal region without holes) with rectangular tiles. In the first part, we present a linear time algorithm (in the surface area n of the region) when there are two tile types, the 1 \Theta m and l \Theta 1 rectangles. Previously the only known algorithms other than exhaustive search were for the case of dominoes (m = 2; l = 2), where several combinatorial methods were known (using matchings, or maxflowmincut algorithms) . Our algorithm generalizes a domino tiling algorithm of W. P. Thurston [1] based on ideas of J. H. Conway and Lagarias which rely on geome...
Critical chromatic number and the complexity of perfect packings in graphs
 17th ACMSIAM Symposium on Discrete Algorithms (SODA), (2006), 851– 859. Oliver Cooley, Daniela Kühn & Deryk Osthus School of Mathematics Birmingham University Edgbaston Birmingham B15 2TT UK Email addresses: {cooleyo,kuehn,osthus}@maths.bham.ac.uk
"... Abstract. Let H be any nonbipartite graph. We determine asymptotically the minimum degree of a graph G which ensures that G has a perfect Hpacking. More precisely, we determine the smallest number τ having the following property: For every positive constant γ there exists an integer n0 = n0(γ, H) ..."
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Cited by 11 (7 self)
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Abstract. Let H be any nonbipartite graph. We determine asymptotically the minimum degree of a graph G which ensures that G has a perfect Hpacking. More precisely, we determine the smallest number τ having the following property: For every positive constant γ there exists an integer n0 = n0(γ, H) such that every graph G whose order n ≥ n0 is divisible by H  and whose minimum degree is at least (τ + γ) n contains a perfect Hpacking. The value of τ depends on the relative sizes of the colour classes in the optimal colourings of H. The proof is algorithmic, which shows that the problem of finding a maximum Hpacking is polynomially solvable for graphs G whose minimum degree is at least (τ + γ)n. On the other hand, given any positive constant γ, we show that for infinitely many (nonbipartite) graphs H the corresponding decision problem becomes NPcomplete if one considers input graphs G of minimum degree at least (τ − γ)n. 1.
Finding nonoverlapping substructures of a sparse matrix
 Electronic Transaction on Numerical Analysis, 21:107–124, 2005. Raghu Ramakrishnan and Johannes
"... Abstract. Many applications of scientific computing rely on sparse matrix computations, thus efficient implementations of sparse matrix kernels are crucial for the overall efficiency of these applications. Due to the low computetomemory ratio and irregular memory access patterns, the performance o ..."
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Abstract. Many applications of scientific computing rely on sparse matrix computations, thus efficient implementations of sparse matrix kernels are crucial for the overall efficiency of these applications. Due to the low computetomemory ratio and irregular memory access patterns, the performance of sparse matrix kernels is often far away from the peak performance on modern processors. Alternative matrix representations have been proposed, where the matrix is split into and, so that contains all dense blocks of a specified form in the matrix, and contains the remaining entries. This facilitates using dense matrix kernels on the entries of, producing better memory performance. We study the problem of finding a maximum number of nonoverlapping rectangular dense blocks in a sparse matrix. We show that the maximum nonoverlapping dense blocks problem is NPcomplete by a reduction from the maximum independent set problem on cubic planar graphs. We also propose aapproximation algorithm for blocks that runs in linear time in the number of nonzeros in the matrix. We discuss alternatives to rectangular blocks such as diagonal blocks and cross blocks and present complexity analysis and approximation algorithms. Key words. Memory performance, memoryefficient data structures, highperformance computing, sparse matrices, independent sets, NPcompleteness, approximation algorithms AMS subject classifications. 65F50, 65Y20, 05C50, 05C69, 68Q17,68W25 1. Introduction. Sparse
Tile Invariants: New Horizons
 THEOR. COMP. SCI
, 2000
"... Let T be a finite set of tiles. The group of invariants G(T), introduced by the author [P], is a group of linear relations between the number of copies of tiles in tilings of the same region. We survey known results about G , the height function approach, the local move property, various applica ..."
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Cited by 3 (2 self)
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Let T be a finite set of tiles. The group of invariants G(T), introduced by the author [P], is a group of linear relations between the number of copies of tiles in tilings of the same region. We survey known results about G , the height function approach, the local move property, various applications and special cases.
Fast TileBased Adaptive Sampling with UserSpecified Fourier Spectra Supplementary Material
"... In this supplementary material, we provide a more comprehensive treatment of the various technical aspects involved in our approach, along with some extensions. In particular, we show how to extend our construction (proposed in the original paper) to arbitrary subdivision factors λ and polyhex sizes ..."
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Cited by 2 (1 self)
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In this supplementary material, we provide a more comprehensive treatment of the various technical aspects involved in our approach, along with some extensions. In particular, we show how to extend our construction (proposed in the original paper) to arbitrary subdivision factors λ and polyhex sizes m, and discuss in greater detail why we settled on our particular choices of λ = 37 and m = 3 based on considerations including memory size and spectral quality. 1 Tiling Construction With Randomized Boundaries Summary: In this section, we provide formal details on the boundary randomization process. We first generalize the construction presented in our original paper to a large set of subdivision factors λ. Then, we demonstrate conditions on edge modifiers sufficient to ensure topologically consistent perturbations (Lemma 1). Finally, we detail the combinatorics of edge modifiers with respect to the desired subdivision factor (Lemma 2). a0 a1 a2 a3 c0 c1c2 b0 b1 b2 a0 a0 a0 a1 a1 a2 c0 c0 c0c1c1 c2 c3 b1 b1 b2 b0 b0 b0 b3 (a) (b) hi hj hk (c) Figure 1: Construction of the regular tiling system: (a) hexagon sequences A, B and C starting from a given origin point (in gray) for n = {1, 2, 3, 4} (from left to right). (b) tile construction via the joining of three triplets of paths for n = {1, 2, 3}. (c) overall tiling construction for n = 4. From a partition of , we define a recursive structure into multihexagon regions we call tiles. Tile boundaries are comprised of hexagon sequences of equal length n aligned along one of three given directions (Fig. 1(a)): • A:={a0,..., an−1}, following the
The 2×2 Simple Packing Problem
"... We significantly extend the class of polygons for which the 2×2 simple packing problem can be solved in polynomial time. 1 ..."
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We significantly extend the class of polygons for which the 2×2 simple packing problem can be solved in polynomial time. 1