Polynomial-time algorithms for permutation groups

Furst, Merrick L.; Hopcroft, John E.; Luks, Eugene M.

doi:10.1109/sfcs.1980.34

Cited by 196 publications

(131 citation statements)

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“…However, given a permutation group G with an arbitrary set of generators, we can always nd a set of generators such that every permutation 2 G is the product of at most n of the new generators, where n is the size of the set on which G acts. This result follows immediately using an algorithm described in [32]. Given a group G acting on a set of size n and generated by g 1 ; ; g k , w e can construct a table T with n rows (labeled 0 to n 1) and n columns (labeled 1 to n), of permutations with the following property: 2 G if and only if can be expressed as a 0 a 1 a r , w ere a i is a member of the i-th row.…”

Section: De Nition 344 the Exact Cover By 3-sets (X3c) Is De Ned Asmentioning

confidence: 98%

“…W e h a v e a c hain of subgroups I = G n 1 G 1 G 0 =G . This chain of subgroups and the right traversal of G i =G i+1 can be found in polynomial time [32]. We perform the following steps: Iteratively, descending down the chain gives us the answer after n 1 steps.…”

Section: Complexity Of the Orbit Problemmentioning

confidence: 99%

“…The theorem given above gives a polynomial time algorithm solving the orbit problem in a special case. The condition that G be given by a t most O(n 2 ) generators is not a restriction because it is always possible ( see [32] In [37] it is proved that all these problems are polynomially equivalent.…”

Section: Set Stabilizer In a Coset (Ssc)mentioning

confidence: 99%

“…Given a symmetry group G acting on [n], let H G be the subgroup generated by the set of transpositions S G given below. S = f(i; j) j (i; j) 2 Gg Since one can test (i; j) 2 G in polynomial time [32], the set S can be found in polynomial time (there are only n 2 transpositions). By lemma 4.6.4 COPfor H can be solved in polynomial time.…”

Section: Finding Easy Subgroupsmentioning

confidence: 99%

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Symmetry and induction in model checking

Clarke

Jha

1995

Computer Science Today

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With the increasing complexity of digital systems, testing of digital systems is becoming increasingly important. Perhaps, the most popular method for testing hardware is simulation. The incompleteness of simulation based testing methods has spurred the recent surge in the research on formal veri cation. In formal veri cation, one builds a precise model of the hardware under scrutiny and proves that the model satis es a speci cation of interest. For example, suppose one wants to verify that a router chip does not deadlock. In this case the user will build a precise model of the router and the speci cation will express the property of deadlock freedom. The two approaches to formal veri cation are model checking and theorem proving. In this thesis we will only discuss model checking. Most model checking procedures su er from the state explosion problem, i.e., the size of the state space of the system can be exponential in the number of state variables of the system. For certain systems, exploiting the inherent symmetry can alleviate the state-explosion problem. We discuss Model Checking procedures which exploit symmetry. Current model checkers can only verify a single state-transition system at a time. We also want to extend the model checking techniques to handle in nite families of nite-state systems. In practice, nite state concurrent systems often exhibit considerable symmetry. W e i n v estigate techniques for reducing the complexity o f temporal logic model checking in the presence of symmetry. In particular, we show that symmetry can frequently be used to reduce the size of the state space that must be explored during model checking. We also investigate complexity o f v arious problems related to exploiting symmetry in model checking. Partial order based reduction techniques exploit independence of actions. We demonstrate that partial order and symmetry based reduction techniques can be applied simultaneously. The ability to reason automatically about entire families of similar state-transition systems is an important research goal. Such families arise frequently in the design of reactive systems in both hardware and software. The in nite family of token rings is a simple example. More complicated examples are trees of processes consisting of one root, several internal and leaf nodes, and hierarchical buses with di erent n umbers of processors and caches. A technique for verifying entire families of state-transition systems based on network grammars and process invariants is presented here.

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Section: De Nition 344 the Exact Cover By 3-sets (X3c) Is De Ned Asmentioning

confidence: 98%

Section: Complexity Of the Orbit Problemmentioning

confidence: 99%

Section: Set Stabilizer In a Coset (Ssc)mentioning

confidence: 99%

Section: Finding Easy Subgroupsmentioning

confidence: 99%

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Symmetry and induction in model checking

Clarke

Jha

1995

Computer Science Today

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“…For the known bounded degree algorithm and the known bounded genus algorithms, the degree of the polynomial bounding the running time increases with increasing parameter (i.e., they have a running time of O(n f (k) )). Algorithms with uniformly polynomial running time (i.e., having a running time of O(f (k) · n d ) with d fixed) have only been devised for the parameters eigenvalue multiplicity [9], color multiplicity [12], feedback vertex set number [15], and rooted tree distance width [21]. In parametrized complexity theory such algorithms are called fixed-parameter tractable.…”

Section: Introductionmentioning

confidence: 99%

Isomorphism of (mis)Labeled Graphs

Schweitzer

2011

Algorithms – ESA 2011

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Abstract. For similarity measures of labeled and unlabeled graphs, we study the complexity of the graph isomorphism problem for pairs of input graphs which are close with respect to the measure. More precisely, we show that for every fixed integer k we can decide in quadratic time whether a labeled graph G can be obtained from another labeled graph H by relabeling at most k vertices. We extend the algorithm solving this problem to an algorithm determining the number ℓ of vertices that must be deleted and the number k of vertices that must be relabeled in order to make the graphs equivalent. The algorithm is fixed-parameter tractable in k + ℓ. Contrasting these tractability results, we also show that for those similarity measures that change only by finite amount d whenever one edge is relocated, the problem of deciding isomorphism of input pairs of bounded distance d is equivalent to solving graph isomorphism in general.

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