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20
Hardtosolve bimatrix games
 ECONOMETRICA
, 2006
"... The Lemke–Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, th ..."
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Cited by 24 (1 self)
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The Lemke–Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. This paper presents a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in dspace. The construction is extended to nonsquare games where, in addition to exponentially long Lemke–Howson computations, finding an equilibrium by support enumeration takes on average exponential time.
Bound Propagation
 Journal of Artificial Intelligence Research
, 2003
"... In this article we present an algorithm to compute bounds on the marginals of a graphical model. For several small clusters of nodes upper and lower bounds on the marginal values are computed independently of the rest of the network. The range of allowed probability distributions over the surroundin ..."
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Cited by 20 (0 self)
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In this article we present an algorithm to compute bounds on the marginals of a graphical model. For several small clusters of nodes upper and lower bounds on the marginal values are computed independently of the rest of the network. The range of allowed probability distributions over the surrounding nodes is restricted using earlier computed bounds. As we will show, this can be...
Beating simplex for fractional packing and covering linear programs. FOCS
, 2007
"... We give an approximation algorithm for packing and covering linear programs (linear programs with nonnegative coefficients). Given a constraint matrix with n nonzeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a f ..."
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Cited by 11 (2 self)
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We give an approximation algorithm for packing and covering linear programs (linear programs with nonnegative coefficients). Given a constraint matrix with n nonzeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1 + ε of OPT (the optimal cost) in time O(n + (r + c)log(n)/ε 2). For dense problems (with r, c = O ( √ n)) the time is O(n + √ n log(n)/ε 2) — linear even as ε → 0. In comparison, previous Lagrangianrelaxation algorithms generally take at least Ω(n log(n)/ε 2) time, while (for small ε) the Simplex algorithm typically takes at least Ω(n min(r, c)) time. 1.
Approximation schemes for packing with item fragmentation. Theory Comput
 Syst
"... We consider two variants of the classical bin packing problem in which items may be fragmented. This can potentially reduce the total number of bins needed for packing the instance. However, since fragmentation incurs overhead, we attempt to avoid it as much as possible. In bin packing with size inc ..."
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Cited by 10 (2 self)
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We consider two variants of the classical bin packing problem in which items may be fragmented. This can potentially reduce the total number of bins needed for packing the instance. However, since fragmentation incurs overhead, we attempt to avoid it as much as possible. In bin packing with size increasing fragmentation (BPSIF), fragmenting an item increases the input size (due to a header/footer of fixed size that is added to each fragment). In bin packing with size preserving fragmentation (BPSPF), there is a bound on the total number of fragmented items. These two variants of bin packing capture many practical scenarios, including message transmission in community TV networks, VLSI circuit design and preemptive scheduling on parallel machines with setup times/setup costs. While both BPSPF and BPSIF do not belong to the class of problems that admit a polynomial time approximation scheme (PTAS), we show in this paper that both problems admit a dual PTAS and an asymptotic PTAS. We also develop for each of the problems a dual asymptotic fully polynomial time approximation scheme (AFPTAS). Our AFPTASs are based on a nonstandard transformation of the mixed packing and covering linear program formulations of our problems into pure covering programs, which enables to efficiently solve these programs.
Linear programming and unique sink orientations
 in: Proc. 17th Annual Symposium on Discrete Algorithms (SODA), 2006
, 2006
"... We show that any linear program (LP) in n nonnegative variables and m equality constraints defines in a natural way a unique sink orientation of the ndimensional cube. From the sink of the cube, we can either read off an optimal solution to the LP, or we obtain certificates for infeasibility or unb ..."
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Cited by 8 (2 self)
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We show that any linear program (LP) in n nonnegative variables and m equality constraints defines in a natural way a unique sink orientation of the ndimensional cube. From the sink of the cube, we can either read off an optimal solution to the LP, or we obtain certificates for infeasibility or unboundedness. This reduction complements the implicit local neighborhoods induced by the vertexedge structure of the feasible region with an explicit neighborhood structure that allows random access to all 2 n candidate solutions. Using the currently best sinkfinding algorithm for general unique sink orientations, we obtain the fastest deterministic LP algorithm in the RAM model, for the central case n = 2m. 1
Model predictive control applied to constraint handling in active noise and vibration control
 IEEE Trans. Control Systems Technology
, 2008
"... Abstract—The difficulties imposed by actuator limitations in a range of active vibration and noise control problems are well recognized. This paper proposes and examines a new approach of employing model predictive control (MPC). MPC permits limitations on allowable control action to be explicitly i ..."
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Cited by 7 (3 self)
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Abstract—The difficulties imposed by actuator limitations in a range of active vibration and noise control problems are well recognized. This paper proposes and examines a new approach of employing model predictive control (MPC). MPC permits limitations on allowable control action to be explicitly included in the computation of an optimal control action. Such techniques have been widely and successfully applied in many other areas. However, due to the relatively high computational requirements of MPC, existing applications have been limited to systems with slow dynamics. This paper illustrates that MPC can be implemented on inexpensive hardware at high sampling rates using traditional online quadratic programming methods for nontrivial models and with significant control performance dividends. Index Terms—Active noise and vibration control, active structures, model predictive control (MPC), piezoelectric actuators. I.
Detecting Infeasibility in InfeasibleInteriorPoint Methods for Optimization
 Foundations of Computational Mathematics, Minneapolis 2002, London Mathematical Society Lecture Note Series 312
, 2003
"... We study interiorpoint methods for optimization problems in the case of infeasibility or unboundedness. While many such methods are designed to search for optimal solutions even when they do not exist, we show that they can be viewed as implicitly searching for welldefined optimal solutions to rel ..."
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Cited by 4 (1 self)
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We study interiorpoint methods for optimization problems in the case of infeasibility or unboundedness. While many such methods are designed to search for optimal solutions even when they do not exist, we show that they can be viewed as implicitly searching for welldefined optimal solutions to related problems whose optimal solutions give certificates of infeasibility for the original problem or its dual. Our main development is in the context of linear programming, but we also discuss extensions to more general convex programming problems.
Constructing Convex 3Polytopes from Two Triangulations of a Polygon
"... Guibas conjectured that given a convex polygon P in the xyplane along with two triangulations of it, T 1 and T 2 that share no diagonals, it is always possible to assign height values to the vertices of P such that P [T 1 [T 2 becomes a convex 3polytope. Dekster found a counter example but left ..."
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Cited by 1 (0 self)
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Guibas conjectured that given a convex polygon P in the xyplane along with two triangulations of it, T 1 and T 2 that share no diagonals, it is always possible to assign height values to the vertices of P such that P [T 1 [T 2 becomes a convex 3polytope. Dekster found a counter example but left open the questions of deciding if a given con guration corresponds to a convex 3polytope, and constructing such realizations when they exist. This paper gives a proof that a relaxed version of Guibas' conjecture always holds true. The question of deciding the realizability of Guibas' conjecture is characterized in terms of a linear programming problem. This leads to an algorithm for deciding and constructing such realizations that incorporates a linear programming step with O(n ) inequality constraints and n variables. 1