Papers by Peter Bro Miltersen
Artificial Intelligence, Sep 1, 2023
Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, whi... more Multi-unit auctions are a paradigmatic model, where a seller brings multiple units of a good, while several buyers bring monetary endowments. It is well known that Walrasian equilibria do not always exist in this model, however compelling relaxations such as Walrasian envy-free pricing do. In this paper we design an optimal envy-free mechanism for multi-unit auctions with budgets. When the market is even mildly competitive, the approximation ratios of this mechanism are small constants for both the revenue and welfare objectives, and in fact for welfare the approximation converges to 1 as the market becomes fully competitive. We also give an impossibility theorem, showing that truthfulness requires discarding resources, and in particular, is incompatible with (Pareto) efficiency.
BRICS Report Series, Dec 11, 2003
We study how big the blow-up in size can be when one switches between the CNF and DNF representat... more We study how big the blow-up in size can be when one switches between the CNF and DNF representations of boolean functions. For a function f : {0, 1} n → {0, 1}, cnfsize(f ) denotes the minimum number of clauses in a CNF for f ; similarly, dnfsize(f ) denotes the minimum number of terms in a DNF for f . For 0 ≤ m ≤ 2 n-1 , let dnfsize(m, n) be the maximum dnfsize(f ) for a function f : {0, 1} n → {0, 1} with cnfsize(f ) ≤ m. We show that there are constants c1, c2 ≥ 1 and > 0, such that for all large n and all m ∈ [ 1 n, 2 n ], we have In particular, when m is the polynomial n c , we get dnfsize(n c , n) = 2 n-θ(c -1 n log n ) .
BRICS Report Series, Dec 5, 2002
We consider the computational power of constant width polynomial size cylindrical circuits and no... more We consider the computational power of constant width polynomial size cylindrical circuits and nondeterministic branching programs. We show that every function computed by a Π 2 • MOD • AC 0 circuit can also be computed by a constant width polynomial size cylindrical nondeterministic branching program (or cylindrical circuit) and that every function computed by a constant width polynomial size cylindrical circuit belongs to ACC 0 .
arXiv (Cornell University), Jun 26, 2008
We consider approximating the minmax value of a multiplayer game in strategic form. Tightening re... more We consider approximating the minmax value of a multiplayer game in strategic form. Tightening recent bounds by Borgs et al., we observe that approximating the value with a precision of ǫ log n digits (for any constant ǫ > 0) is NP-hard, where n is the size of the game. On the other hand, approximating the value with a precision of c log log n digits (for any constant c ≥ 1) can be done in quasi-polynomial time. We consider the parameterized complexity of the problem, with the parameter being the number of pure strategies k of the player for which the minmax value is computed. We show that if there are three players, k = 2 and there are only two possible rational payoffs, the minmax value is a rational number and can be computed exactly in linear time. In the general case, we show that the value can be approximated with any polynomial number of digits of accuracy in time n O(k) . On the other hand, we show that minmax value approximation is W [1]-hard and hence not likely to be fixed parameter tractable. Concretely, we show that if k-CLIQUE requires time n Ω(k) then so does minmax value computation.
arXiv (Cornell University), Feb 7, 2012
Emek et al. presented a model of probabilistic single-item second price auctions where an auction... more Emek et al. presented a model of probabilistic single-item second price auctions where an auctioneer who is informed about the type of an item for sale, broadcasts a signal about this type to uninformed bidders. They proved that finding the optimal (for the purpose of generating revenue) pure signaling scheme is strongly NP-hard. In contrast, we prove that finding the optimal mixed signaling scheme can be done in polynomial time using linear programming. For the proof, we show that the problem is strongly related to a problem of optimally bundling divisible goods for auctioning. We also prove that a mixed signaling scheme can in some cases generate twice as much revenue as the best pure signaling scheme and we prove a generally applicable lower bound on the revenue generated by the best mixed signaling scheme.
Let M(m; n) be the minimum number of comparators in a comparator network that merges two ordered ... more Let M(m; n) be the minimum number of comparators in a comparator network that merges two ordered chains x 1 x 2 : : : x m and y 1 y 2 : : : y n , where n m. Batcher's odd-even merge yields the following upper bound: and then Yao and Yao (for M(m; n)) have shown the following lower bounds: M(m; n) 1 2 n log 2 (m + 1); M(n; n) 1 2 n log 2 n + O(n): We prove a new lower bound that matches the upper bound asymptotically: M(m; n) 1 2 (m + n) log 2 (m + 1) O(m); e.g., M(n; n) n log 2 n O(n): Our proof technique extends to give similarly tight lower bounds for the size of monotone Boolean circuits for merging, and for the size of switching networks capable of realizing the set of permutations that arise from merging.
Information Processing Letters, Jul 1, 2005
In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boo... more In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boolean function on n input bits. The best known bounds appear to be However, the bounds do not seem to be explicitly stated in the literature. We give a simple direct elementary proof of the lower bound valid for the full binary basis, and we give an explicit proof of the upper bound valid for the basis {¬, ∧, ∨}.
BRICS Report Series, 1997
Automata, Languages and Programming, 1996
We consider solving the static dictionary problem with n keys from the universe f0; : : : ; m?1g ... more We consider solving the static dictionary problem with n keys from the universe f0; : : : ; m?1g on a RAM with direct and indirect addressing, conditional jump, addition, bitwise Boolean operations, and arbitrary shifts (a Practical RAM). For any > 0, tries yield constant query time using space m , provided that n = m o(1) . We show that this is essentially optimal: Any scheme with constant query time requires space m for some > 0, even if n (log m) 2 .
Lecture Notes in Computer Science, 2009
We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., poly... more We study the following question, communicated to us by Miklós Ajtai: Can all explicit (e.g., polynomial time computable) functions f : ({0, 1} w ) 3 → {0, 1} w be computed by word circuits of constant size? Here, a word circuit is an acyclic circuit where each wire holds a word (i.e., an element of {0, 1} w ) and each gate G computes some binary operation gG : ({0, 1} w ) 2 → {0, 1} w , defined for all word lengths w. As our main result, we present an explicit function so that its w'th slice for any w ≥ 8 cannot be computed by word circuits with at most 4 gates. Also, we formally relate Ajtai's question to open problems concerning ACC 0 circuits.
Some Meet-in-the-Middle Circuit Lower Bounds
Lecture Notes in Computer Science, 2004
We observe that a combination of known top-down and bottom-up lower bound techniques of circuit c... more We observe that a combination of known top-down and bottom-up lower bound techniques of circuit complexity may yield new circuit lower bounds. An important example is this: Razborov and Wigderson showed that a certain function f in ACC 0 cannot be computed by polynomial size circuits consisting of two layers of MAJORITY gates at the top and a layer of AND gates at the bottom. We observe that a simple combination of their result with the Håstad switching lemma yields the following seemingly much stronger result: The same ...
Linear Hashing
Journal of The ACM - JACM
Journal of the ACM, 1996
Let M(m,n) be the minimum number of comparators needed in a comparator network that merges m elem... more Let M(m,n) be the minimum number of comparators needed in a comparator network that merges m elements x 1 ≤ x 2 ≤ … ≤ x m and n elements y 1 ≤ y 2 ≤ … ≤ y m , where n ≥ m. Batcher's odd-even merge yields the following upper bound: M(m,n) ≤ ½(m + n)log 2 m + O(n); in particular, M(n,n) ≤ n log 2 n + o(n) We prove the following lower bound that matches the upper bound above asymptotically as n ≥ m →∞; M(m,n) ≥ ½(m+n)log 2 m - O(m) in particular, M(n,n) ≥ n log 2 - O(n). Our proof technique extends to give similarily tight lower bounds for the size of monotone Boolean circuits for merging, and for the size of switching networks capable of realizing the set of permutations that arise from merging.
Information Processing Letters, 2005
In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boo... more In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boolean function on n input bits. The best known bounds appear to be However, the bounds do not seem to be explicitly stated in the literature. We give a simple direct elementary proof of the lower bound valid for the full binary basis, and we give an explicit proof of the upper bound valid for the basis {¬, ∧, ∨}.
arXiv (Cornell University), Jan 17, 2012
One-clock priced timed games is a class of two-player, zero-sum, continuous-time games that was d... more One-clock priced timed games is a class of two-player, zero-sum, continuous-time games that was defined and thoroughly studied in previous works. We show that one-clock priced timed games can be solved in time m12 n n O(1) , where n is the number of states and m is the number of actions. The best previously known time bound for solving one-clock priced timed games was 2 O(n 2 +m) , due to Rutkowski. For our improvement, we introduce and study a new algorithm for solving one-clock priced timed games, based on the sweep-line technique from computational geometry and the strategy iteration paradigm from the algorithmic theory of Markov decision processes. As a corollary, we also improve the analysis of previous algorithms due to Bouyer, Cassez, Fleury, and Larsen; and Alur, Bernadsky, and Madhusudan.
arXiv (Cornell University), Feb 17, 2012
Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stoc... more Shapley's discounted stochastic games, Everett's recursive games and Gillette's undiscounted stochastic games are classical models of game theory describing two-player zero-sum games of potentially infinite duration. We describe algorithms for exactly solving these games. When the number of positions of the game is constant, our algorithms run in polynomial time.
In this paper we consider two party communication complexity when the input sizes of the two play... more In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the "asymmetric" case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds, and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our "round elimination" lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the "round reduction" techniques used in many previous lower bound proofs.
arXiv (Cornell University), Dec 22, 2011
Gimbert and Horn gave an algorithm for solving simple stochastic games with running time O(r!n) w... more Gimbert and Horn gave an algorithm for solving simple stochastic games with running time O(r!n) where n is the number of positions of the simple stochastic game and r is the number of its coin toss positions. Chatterjee et al. pointed out that a variant of strategy iteration can be implemented to solve this problem in time 4 r r O(1) n O(1). In this paper, we show that an algorithm combining value iteration with retrograde analysis achieves a time bound of O(r2 r (r log r + n)), thus improving both time bounds. While the algorithm is simple, the analysis leading to this time bound is involved, using techniques of extremal combinatorics to identify worst case instances for the algorithm.
BRICS Report Series, Dec 11, 2003
See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained... more See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting:
arXiv (Cornell University), Jul 6, 2013
We consider the fundamental mechanism design problem of approximate social welfare maximization u... more We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare maximization in this setting. With m being the number of alternatives, we exhibit a randomized truthful-in-expectation ordinal mechanism implementing an outcome whose expected social welfare is at least an Ω(m −3/4) fraction of the social welfare of the socially optimal alternative. On the other hand, we show that for sufficiently many agents and any truthful-in-expectation ordinal mechanism, there is a valuation profile where the mechanism achieves at most an O(m −2/3) fraction of the optimal social welfare in expectation. Furthermore, we prove that no truthful-in-expectation (not necessarily ordinal) mechanism can achieve 0.94-fraction of the optimal social welfare. We get tighter bounds for the natural special case of m = 3, and in that case furthermore obtain separation results concerning the approximation ratios achievable by natural restricted classes of truthful-in-expectation mechanisms. In particular, we show that for m = 3 and a sufficiently large number of agents, the best mechanism that is ordinal as well as mixed-unilateral has an approximation ratio between 0.610 and 0.611, the best ordinal mechanism has an approximation ratio between 0.616 and 0.641, while the best mixed-unilateral mechanism has an approximation ratio bigger than 0.660. In particular, the best mixed-unilateral non-ordinal (i.e., cardinal) mechanism strictly outperforms all ordinal ones, even the non-mixed-unilateral ordinal ones.
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Papers by Peter Bro Miltersen