Algorithmic construction of sets for
            <i>k</i>
            -restrictions

Alon, Noga; Moshkovitz, Dana; Safra, Shmuel

doi:10.1145/1150334.1150336

Cited by 245 publications

(226 citation statements)

References 21 publications

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“…However, the game used in the proof of Proposition 6.1 has only one level, one state, and no transitions, so none of these limitations apply to it, and the ILP will find the optimal abstraction for games in that class. The NP-hardness reduction in the proof is from SET COVER, which cannot be approximated in worst-case polynomial time to an approximation ratio better than a constant times ln q unless P = NP [Alon et al 2006]. Therefore, the gap in solution quality-in terms of the size of the (action) abstraction for the level-found by the greedy algorithm (which runs in polynomial time) versus the ILP (which finds an optimal solution in this game class) cannot be bounded by any constant.…”

Section: Bottom-up Single-pass Level-by-level Greedy Abstraction Algomentioning

confidence: 99%

Lossy stochastic game abstraction with bounds

Sandholm¹,

Singh

2012

Proceedings of the 13th ACM Conference on Electronic Commerce

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Abstraction followed by equilibrium finding has emerged as the leading approach to solving games. Lossless abstraction typically yields games that are still too large to solve, so lossy abstraction is needed. Unfortunately, prior lossy game abstraction algorithms have no guarantees on solution quality. We developed a framework that enables the design of lossy game abstraction algorithms with guarantees on solution quality. It simultaneously handles state and action abstraction. We define a measure of reward approximation error and transition probability error achieved by state and action abstraction in stochastic games such that the regret of the equilibrium found in the abstract game when implemented in the original, unabstracted game is upper-bounded by a function of those measures. We then develop the first lossy game abstraction algorithms with bounds on solution quality. Both of them work level-by-level up from the end of the game. One of the algorithms is greedy and the other is an integer linear program. We also prove that the abstraction problem is NP-complete (even with just action abstraction, 2 agents, and a 1-step game), but point out that this does not mean that the game abstraction problems that occur in practice cannot be solved quickly.

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Section: Bottom-up Single-pass Level-by-level Greedy Abstraction Algomentioning

confidence: 99%

Lossy stochastic game abstraction with bounds

Sandholm¹,

Singh

2012

Proceedings of the 13th ACM Conference on Electronic Commerce

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“…This matches the lower bound given by Berlinkov, albeit for alphabets of arbitrary, instead of constant size. However, assuming P = NP, SET COVER has no c · log napproximation for some constant c > 0 [12]. This offers a significant improvement over the previous best lower bound.…”

Section: Resultsmentioning

confidence: 94%

“…Notice that w 1 · · · w l ∈ Σ * is a reset sequence for A if and only if 1≤i≤l S wi is a cover for X. SET COVER has no c log n-approximation for some constant c > 0 unless P = NP [12]. Thus, this lower bound also extends to STACK COVER.…”

Section: The Stack Cover Problemmentioning

confidence: 91%

Approximating Minimum Reset Sequences

Gerbush

Heeringa

2011

Implementation and Application of Automata

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Abstract. We consider the problem of finding minimum reset sequences in synchronizing automata. The well-knownČerný conjecture states that every n-state synchronizing automaton has a reset sequence with length at most (n − 1) 2 . While this conjecture gives an upper bound on the length of every reset sequence, it does not directly address the problem of finding the shortest reset sequence. We call this the MINIMUM RESET SEQUENCE (MRS) problem. We give an O(kmn k + n 4 /k)-time n−1 k−1 -approximation for the MRS problem for any k ≥ 2. We also show that our analysis is tight. When k = 2 our algorithm reduces to Eppstein's algorithm and yields an (n − 1)-approximation. When k = n our algorithm is the familiar exponential-time, exact algorithm. We define a nontrivial class of MRS which we call STACK COVER. We show that STACK COVER naturally generalizes two classic optimization problems: MIN SET COVER and SHORTEST COMMON SUPERSEQUENCE. Both these problems are known to be hard to approximate, although at present, SET COVER has a slightly stronger lower bound. In particular, it is NP-hard to approximate SET COVER to within a factor of c · log n for some c > 0. Thus, the MINIMUM RESET SEQUENCE problem is as least as hard to approximate as SET COVER. This improves the previous best lower bound which showed that it was NP-hard to approximate the MRS on binary alphabets to within any constant factor. Our result requires an alphabet of arbitrary size.

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“…shown that SCP does not admit an o(lg n) approximation under the weaker assumption that P = NP [14,2].…”

Section: Inapproximability Of Minimum Verification Set Problemmentioning

confidence: 99%

Hardness and inapproximability results for minimum verification set and minimum path decision tree problems

Türker¹,

Yenigün²

2012

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Minimization of decision trees is a well studied problem. In this work, we introduce two new problems related to minimization of decision trees.The problems are called minimum verification set (MinVS) and minimum path decision tree (MinPathDT) problems. Decision tree problems ask the question "What is the unknown given object?". MinVS problem on the other hand asks the question "Is the unknown object z?", for a given object z. Hence it is not an identification, but rather a verification problem. MinPathDT problem aims to construct a decision tree where only the cost of the root-to-leaf path corresponding to a given object is minimized, whereas decision tree problems in general try to minimize the overall cost of decision trees considering all the objects. Therefore, MinVS and MinPathDT are seemingly easier problems. However, in this work we prove that MinVS and MinPathDT problems are both NP-complete and cannot be approximated within a factor in o(lg n) unless P = NP.

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Algorithmic construction of sets for k -restrictions

Cited by 245 publications

References 21 publications

Lossy stochastic game abstraction with bounds

Lossy stochastic game abstraction with bounds

Approximating Minimum Reset Sequences

Hardness and inapproximability results for minimum verification set and minimum path decision tree problems

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