## Generalized Knapsack Solvers for Multi-Unit Combinatorial Auctions: Analysis and Application to Computational Resource Allocation (2004)

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Venue: | In Workshop on Agent Mediated Electronic Commerce VI: Theories for and Engineering of Distributed Mechanisms and Systems |

Citations: | 17 - 3 self |

### BibTeX

@INPROCEEDINGS{Kelly04generalizedknapsack,

author = {Terence Kelly},

title = {Generalized Knapsack Solvers for Multi-Unit Combinatorial Auctions: Analysis and Application to Computational Resource Allocation},

booktitle = {In Workshop on Agent Mediated Electronic Commerce VI: Theories for and Engineering of Distributed Mechanisms and Systems},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

The problem of allocating discrete computational resources motivates interest in general multi-unit combinatorial exchanges. This paper considers the problem of computing optimal (surplus-maximizing) allocations, assuming unrestricted quasi-linear preferences. We present a solver whose pseudo-polynomial time and memory requirements are linear in three of four natural measures of problem size: number of agents, length of bids, and units of each resource. In applications where the number of resource types is inherently a small constant, e.g., computational resource allocation, such a solver offers advantages over more elaborate approaches developed for high-dimensional problems.

### Citations

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Citation Context ...t § defines utility over all NR possible resource bundles (the case of rational preferences) then the time requirement becomes O ¦ RTN 2R § . The classic 0-1 and integer knapsack problems are NP-ha=-=rd [32, 33]-=-. MDMCK includes these as special cases, and therefore it too is NP-hard. However knapsack problems are not NP-hard in the strong sense, i.e., they admit pseudo-polynomial solution if dimensionality i... |

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Citation Context ...allocation (integral quantities of goods). Our results apply to the allocator of proper economic mechanisms such as the Generalized Vickrey Auction (GVA) [8] or Vickrey-Clarke-Groves (VCG) mechanisms =-=[9]-=-, but we do not consider incentive issues surrounding auctions. Finally, although a wide range of approximation schemes for KPs have been proposed, we restrict attention to exact methods. This is appr... |

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Citation Context ...t § defines utility over all NR possible resource bundles (the case of rational preferences) then the time requirement becomes O ¦ RTN 2R § . The classic 0-1 and integer knapsack problems are NP-ha=-=rd [32, 33]-=-. MDMCK includes these as special cases, and therefore it too is NP-hard. However knapsack problems are not NP-hard in the strong sense, i.e., they admit pseudo-polynomial solution if dimensionality i... |

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Citation Context ...ce literature in this regard. 4 Dynamic Programming Solver This section presents a simple dynamic programming (DP) algorithm for MDMCK; it generalizes multi-dimensional and multiple-choice KP solvers =-=[20, 30].sLets¦ N1¡¤-=-¢£¢¤¢¤¡ NR§ N denote the multi-dimensional “size” of our resource pool, and let 0 denote the R-vector consisting entirely of zeros. We say thatsa b if every component of vector a is not le... |

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Citation Context ...tant measure of problem size in a single-unit CA is the number of good types, and for this measure the winner determination problem (WDP) is NP-hard by reduction from the weighted set packing problem =-=[3]-=-. An unfortunate consequence of excessive attention to single-unit CAs has been excessive pessimism regarding efficient and exact winner determination in more general problems. The few papers that hav... |

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Citation Context ...ith user needs. Data centers are partitioned so that an application’s performance depends only upon the resources it receives; in auction contexts this property is sometimes called “no externaliti=-=es” [17]-=-. Multi-tiered applications for large computing environments are horizontally scalable by design, i.e., they exploit variable quantities of resources at each tier. Application performance exhibits bot... |

243 |
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Citation Context ... literature exists on these special cases, but relatively little on MDMCK itself. Kellerer et al. devote roughly three pages to MDMCK and identify approximate heuristic algorithms dating back to 1997 =-=[20]-=-. They report that to the best of their knowledge no exact algorithm for MDMCK has ever been published. In fact, Tennenholtz briefly sketched an exact solver suitable for low-dimensional MDMCK instanc... |

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167 | Combinatorial auctions: A survey
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Citation Context ...of goods. Although CAs are applicable to a wide range of allocation problems, the U.S. Federal Communications Commission’s spectrum allocation problem largely motivated the 1990s surge of CA researc=-=h [1,2]. Sp-=-ecial properties of spectrum auctions—particularly the restriction that only a single unit of each type of good is available—received much attention in E-commerce research literature. An important... |

159 | Winner Determination in Combinatorial Auction Generalizations
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Citation Context ...express willingness to engage in complex atomic (all-or-nothing) transactions. Thus the MDMCK formulation supports very general combinatorial exchanges, e.g., the dozen CA variants considered in Ref. =-=[26]-=-. 3.2 Auction and KP Taxonomies CA WDPs are often linked to set packing, even in the multi-unit case [5]. Connections with generalized knapsack problems, however, seem more natural and more useful for... |

138 | Combinatorial auctions with decreasing marginal utilities
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Citation Context ...ctions has restricted the form of bids, e.g., demand must be monotonic in per-unit price [22] or atomic bids are forbidden [23]; monotonicity restrictions have also appeared in multi-good CA analyses =-=[24]-=-. In the single-good-type case, divisibility is required for existence of a uniform price that maximizes surplus according to restricted-form bids (which might not represent actual agent preferences).... |

138 | Towards a universal test suite for combinatorial auction algorithms
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Citation Context ...le for solver benchmarking, so we must rely on synthetic benchmarks. A thorough evaluation of any WDP solver should include instances intended to mimic typical inputs, such as those generated by CATS =-=[34]-=-, as well as hard instances to expose worst-case behavior. The connection between WDPs and KPs allows us to exploit many years of research on hard KP instances for WDP solver evaluation. There are two... |

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116 | A Unified Approach to Approximating Resource Allocation and Scheduling
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Citation Context ..., and a handful casually state that the connection is well known, without saying by whom [3,24,75]. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits =-=[4,15,29,33,44,50,55,65,66,77]-=-; twelve CiteSeer hits [3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]; and eleven Science Direct hits [13, 23, 25, 39,60, 61, 64,67, 70,72, 75]. Somewhat ironically, the only detailed discussion of the c... |

109 | Towards a characterization of truthful combinatorial auctions - Lavi, Mu’alem, et al. - 2003 |

96 | A Parametrization of the Auction Design Space
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Citation Context ...rature on hard KP instances. The boundaries of the present investigation are as follows: We consider only oneshot sealed-bid auctions, an important subset of auction types in a comprehensive taxonomy =-=[7]-=-. We consider only discrete allocation (integral quantities of goods). Our results apply to the allocator of proper economic mechanisms such as the Generalized Vickrey Auction (GVA) [8] or Vickrey-Cla... |

95 | Truthful approximation mechanisms for restricted combinatorial auctions: extended abstract
- MU’ALEM, NISAN
(Show Context)
Citation Context ..., and a handful casually state that the connection is well known, without saying by whom [3,24,75]. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits =-=[4,15,29,33,44,50,55,65,66,77]-=-; twelve CiteSeer hits [3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]; and eleven Science Direct hits [13, 23, 25, 39,60, 61, 64,67, 70,72, 75]. Somewhat ironically, the only detailed discussion of the c... |

93 | Vickrey prices and shortest paths: What is an edge worth - Hershberger, Suri - 2001 |

91 | Integer programming for combinatorial auction winner determination. ICMAS
- Andersson, Tenhunen, et al.
- 2000
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Citation Context ...ally no restrictions on the expression of agent utility functions. The connection between CA WDPs and generalized KPs allows us to retain much of the flexibility and generality of integer programming =-=[6]-=- while exploiting the special structure of KPs to obtain simple and efficient exact solvers. In special cases such as single-good multi-unit auctions, textbook uni-dimensional KP solvers compare rathe... |

69 | Optimal solutions for multi-unit combinatorial auctions: Branch and bound heuristics
- Gonen, Lehmann
- 2000
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Citation Context ... Furthermore, benchmarks for multi-unit CAs could draw upon extensive research on hard KP instances. Empirical evaluations of MUCA WDP solvers to date have employed similar input synthesis procedures =-=[4, 5, 26]-=-, which produce multidimensional variants of the uncorrelated and weakly correlated cases of Figure 2; for uni-dimensional KPs, these are not hard instances. Finally, awareness of the WDP-KP connectio... |

66 | Knapsack problems
- Pisinger, Toth
- 1998
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Citation Context ...e focus on the second method, which involves the relationship between bundle size and utility. The size/utility relationship is easy to visualize in the uni-dimensional case. Figure 2, after Pisinger =-=[36], -=-illustrates four possibilities; Martello et al. and Kellerer et al. sizesdescribe others [20,37]. Strongly-correlated instances are among the hardest for today’s best KP solvers and are the subject ... |

64 | Some tractable combinatorial auctions
- Tennenholtz
- 2000
(Show Context)
Citation Context ...hat the multi-good-type/multi-unit WDP is “tractable” when the number of types of goods is fixed, and describes a longest-paths dynamic programming algorithm in the context of a two-good-type exam=-=ple [21]-=-. It is not clear whether the intended meaning is that polynomial or pseudo-polynomial solutions exist (the former cannot be true, because this WDP includes NP-hard problems MCKP and 0-1 KP as special... |

62 | Generalized Vickrey auctions
- MacKie-Mason, Varian
- 1994
(Show Context)
Citation Context ...nsive taxonomy [7]. We consider only discrete allocation (integral quantities of goods). Our results apply to the allocator of proper economic mechanisms such as the Generalized Vickrey Auction (GVA) =-=[8]-=- or Vickrey-Clarke-Groves (VCG) mechanisms [9], but we do not consider incentive issues surrounding auctions. Finally, although a wide range of approximation schemes for KPs have been proposed, we res... |

60 | An algorithm for multi-unit combinatorial auctions
- Leyton-Brown, Shoham, et al.
- 2000
(Show Context)
Citation Context ...nd exact winner determination in more general problems. The few papers that have considered multi-unit CAs (MUCAs) report that the WDP is NP hard when problem size is measured by number of good types =-=[1, 4, 5].-=- ¡ Sixth Workshop on Agent-Mediated Electronic Commerce (AMEC-VI), New York, July 2004. http://ana.lcs.mit.edu/peyman/amecvi.htm $Id: auction_knapsack_connection.tex,v 1.54 2004/06/01 06:30:48 kteren... |

54 | Approximately-strategyproof and tractable multi-unit auctions
- Kothari, Parkes, et al.
- 2003
(Show Context)
Citation Context ...e that no change in goods owned implies no change in utility. Some prior work on single-good-type/multi-unit auctions has restricted the form of bids, e.g., demand must be monotonic in per-unit price =-=[22]-=- or atomic bids are forbidden [23]; monotonicity restrictions have also appeared in multi-good CA analyses [24]. In the single-good-type case, divisibility is required for existence of a uniform price... |

50 |
Fredrik Ygge. Integer programming for combinatorial auction winner determination
- Andersson, Tenhunen
- 2000
(Show Context)
Citation Context ...ally no restrictions on the expression of agent utility functions. The connection between CA WDPs and generalized KPs allows us to retain much of the flexibility and generality of integer programming =-=[2]-=- while exploiting the special structure of KPs to obtain simple and efficient exact solvers. In special cases such as single-good multi-unit auctions, textbook uni-dimensional KP solvers compare rathe... |

50 |
de Vries and Rakesh V. Vohra. Combinatorial Auctions: A survey
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Citation Context ...ate: 2004/02/16 07:55:51 $ Terence Kelly kterence@hpl.hp.com Hewlett-Packard Laboratories 1501 Page Mill Road m/s 1125 Palo Alto CA 94304 USA 15 February 2004 motivated the 1990s surge of CA research =-=[10, 51]-=-. Special properties of spectrum auctions—particularly the restriction that only a single unit of each type of good is available— received much attention in E-commerce research literature. An importan... |

42 | A minimal algorithm for the multiple-choice knapsack problem
- Pisinger
- 1995
(Show Context)
Citation Context ...ble formulation. In fact, simple MCKP solvers in modern texts scale rather well with problem size (see Section 5.2), and efficient specialized solvers are the subject of sophisticated recent research =-=[27]-=-. Holte observes that Operations Researchers have long investigated MDKPs that are substantively identical to multi-unit CA WDPs [28], contrary to claims in recent E-commerce literature that MUCA WDPs... |

41 | Side constraints and non-price attributes in markets
- Sandholm, Suri
- 2001
(Show Context)
Citation Context ... well known, without saying by whom [3,24,75]. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits [4,15,29,33,44,50,55,65,66,77]; twelve CiteSeer hits =-=[3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]-=-; and eleven Science Direct hits [13, 23, 25, 39,60, 61, 64,67, 70,72, 75]. Somewhat ironically, the only detailed discussion of the connection between combinatorial auction WDPs and generalized KPs o... |

38 |
The theory and computation of knapsack functions
- Gilmore, Gomory
- 1966
(Show Context)
Citation Context ...re note that in MDMCK we do not seek to fill a container in the sense of tiling in two dimensions; the term “multidimensional knapsack” has been applied to this very different kind of packing problem =-=[14]-=-. Some authors prefer “multi-constraint” for problems like MDMCK to avoid confusion but “multi-dimensional” appears to be more standard [26, 43]. 3.1. Application to Auctions In an auction setting, we... |

35 | User Profile Replication for Faster Location Lookup in Mobile Environments
- Shivakumar, Widom
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Citation Context ..., and a handful casually state that the connection is well known, without saying by whom [3,24,75]. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits =-=[4,15,29,33,44,50,55,65,66,77]-=-; twelve CiteSeer hits [3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]; and eleven Science Direct hits [13, 23, 25, 39,60, 61, 64,67, 70,72, 75]. Somewhat ironically, the only detailed discussion of the c... |

32 |
A remark on a problem of
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(Show Context)
Citation Context ...rd instances of classic uni-dimensional knapsack problems. The first is to make the coefficients enormous; Chvátal describes how large they must be in order to foil a range of common solution methods=-= [35]-=-. We shall continue to assume that coefficients are bounded and therefore focus on the second method, which involves the relationship between bundle size and utility. The size/utility relationship is ... |

31 | New trends in exact algorithms for the 0-1 knapsack problem
- Martello, Pisinger, et al.
- 2000
(Show Context)
Citation Context ...The size/utility relationship is easy to visualize in the uni-dimensional case. Figure 2, after Pisinger [36], illustrates four possibilities; Martello et al. and Kellerer et al. sizesdescribe others =-=[20,37]. -=-Strongly-correlated instances are among the hardest for today’s best KP solvers and are the subject of ongoing research [37,38]. An extended version of this paper describes how to construct generali... |

28 |
Combinatorial auction design
- Pekec, Rothkopf
(Show Context)
Citation Context ...of goods. Although CAs are applicable to a wide range of allocation problems, the U.S. Federal Communications Commission’s spectrum allocation problem largely motivated the 1990s surge of CA researc=-=h [1,2]. Sp-=-ecial properties of spectrum auctions—particularly the restriction that only a single unit of each type of good is available—received much attention in E-commerce research literature. An important... |

25 | Peruser profile replication in mobile environments: Algorithms, analysis, and simulation results,” ACM/Baltzer
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Citation Context |

21 | Virtual worlds: fast and strategyproof auctions for dynamic resource allocation
- Ng, Parkes, et al.
- 2003
(Show Context)
Citation Context ... well known, without saying by whom [3,24,75]. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits [4,15,29,33,44,50,55,65,66,77]; twelve CiteSeer hits =-=[3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]-=-; and eleven Science Direct hits [13, 23, 25, 39,60, 61, 64,67, 70,72, 75]. Somewhat ironically, the only detailed discussion of the connection between combinatorial auction WDPs and generalized KPs o... |

17 | Combinatorial auctions, knapsack problems, and hill-climbing search
- Holte
- 2001
(Show Context)
Citation Context ...ecialized solvers are the subject of sophisticated recent research [27]. Holte observes that Operations Researchers have long investigated MDKPs that are substantively identical to multi-unit CA WDPs =-=[28]-=-, contrary to claims in recent E-commerce literature that MUCA WDPs were never before studied [4]. Years later, however, MUCA WDP research that cites Holte does not mention the connection he made [29]... |

14 |
knapsack problems
- Hard
- 1980
(Show Context)
Citation Context ...rd instances of classic uni-dimensional knapsack problems. The first is to make the coefficients enormous; Chvátal describes how large they must be in order to foil a range of common solution methods =-=[9]-=-. We shall continue to assume that coefficients are bounded and therefore focus on the second method, which involves the relationship between bundle size and utility. The size/utility relationship is ... |

10 |
Resource Allocation in Competitive Multiagent Systems
- Leyton-Brown
- 2003
(Show Context)
Citation Context ...[28], contrary to claims in recent E-commerce literature that MUCA WDPs were never before studied [4]. Years later, however, MUCA WDP research that cites Holte does not mention the connection he made =-=[29]. -=-A very recent text on KPs discusses Holte’s insight in considerable detail but does not make the connection between MDMCK and multi-unit CAs with XOR bids; instead it suggests the use of dummy goods... |

9 | Seller-focused algorithms for online auctioning
- Bagchi, Chaudhary, et al.
- 2001
(Show Context)
Citation Context ...ment of the WDP-KP connection. A few papers mention in passing a deep relationship between WDPs and KPs [8, 22], and a handful casually state that the connection is well known, without saying by whom =-=[3,24,75]-=-. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits [4,15,29,33,44,50,55,65,66,77]; twelve CiteSeer hits [3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]; ... |

9 |
Software agents for knowledge management: coordination in multi-agent supply chains and auctions. Expert Systems with Applications
- Wu
- 2001
(Show Context)
Citation Context ...ll of the “auction AND knapsack” results: Ten ACM Digital Library hits [4,15,29,33,44,50,55,65,66,77]; twelve CiteSeer hits [3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]; and eleven Science Direct hits =-=[13, 23, 25, 39,60, 61, 64,67, 70,72, 75]-=-. Somewhat ironically, the only detailed discussion of the connection between combinatorial auction WDPs and generalized KPs of which we are aware occurs in a very recent text written primarily by Ope... |

8 | Supporting coallocation in an auctioning-based resource allocator for grid systems
- Chen, Maheswaran, et al.
(Show Context)
Citation Context ... the documents at the intersection of the two keywords contained no detailed or systematic treatment of the WDP-KP connection. A few papers mention in passing a deep relationship between WDPs and KPs =-=[8, 22]-=-, and a handful casually state that the connection is well known, without saying by whom [3,24,75]. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits ... |

8 | Self-enforcing strategic demand reduction
- Reitsma, Stone, et al.
- 2001
(Show Context)
Citation Context ... well known, without saying by whom [3,24,75]. For completeness we include all of the “auction AND knapsack” results: Ten ACM Digital Library hits [4,15,29,33,44,50,55,65,66,77]; twelve CiteSeer hits =-=[3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]-=-; and eleven Science Direct hits [13, 23, 25, 39,60, 61, 64,67, 70,72, 75]. Somewhat ironically, the only detailed discussion of the connection between combinatorial auction WDPs and generalized KPs o... |

8 |
Emerging multiple issue e-auctions
- Teich, Wallenius, et al.
- 2004
(Show Context)
Citation Context ...ll of the “auction AND knapsack” results: Ten ACM Digital Library hits [4,15,29,33,44,50,55,65,66,77]; twelve CiteSeer hits [3, 4, 8, 16, 22, 24, 29,38, 45, 55,56, 58]; and eleven Science Direct hits =-=[13, 23, 25, 39,60, 61, 64,67, 70,72, 75]-=-. Somewhat ironically, the only detailed discussion of the connection between combinatorial auction WDPs and generalized KPs of which we are aware occurs in a very recent text written primarily by Ope... |