Eugene M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences, 25(1):42--65, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....complexity class NPI, particularly because both the counting and decision versions of this problem are equally hard, a property not shared by NP complete problems. However, polynomial time graph isomorphism algorithms exist for several classes of graphs. Examples are graphs with bounded degree [42], trees 0, and partial k trees [11] Most of these polynomial isomorphism algorithms are, however, not applicable in practice because of large constant multipliers. The exceptions are planar graphs and circular arc graphs [18] for which efficient isomorphism algorithms exist. It is possible to ....

Luks, E. M. (1982). Isomorphism of graphs of bounded valence can be tested in polynomial time. JCSSI, 25, 42-65.

....a similar incidence list for each facet. Moreover, using this data structure, none of the four steps in the for loop needs more than n) time (the critical part being Step 2) Since the body of the forloop is not executed more than n d times, this yields an ) time algorithm. Luks [22] gave a polynomial time algorithm for the graph isomorphism problem on graphs of bounded maximal degree. Since the graph of a simple d polytope is d regular, his algorithm runs in polynomial time on graphs of simple polytopes of bounded dimension. Proposition 3 The isomorphism problem for graphs ....

Luks, E.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. System Sci. 25, 42--65 (1982)

.... complete even if V and H descript ions of P and Q are part of the input data [34] In constant dimension the problem can be solved in polynomial time by a reduction [34] to the graph isomorphism problem for graphs of bounded degree, for which a polynomial time algorithm is known (Luks [41]) Problem 21 can polynomially be reduced to this problem. For polytopes of bounded dimension both problems are polynomial time equivalent. 23. Selfduality of Polytopes Input: Face Lattice of a polytope P Output: Yes if P is isomorphic to its dual, No otherwise This is a special case of ....

E. M. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comput. Syst. Sci., 25 (1982), pp. 42-65.

....a similar incidence list for each facet. Moreover, using this data structure, none of the four steps in the for loop needs more than n) time (the critical part being Step 2) Since the body of the forloop is not executed more than n d times, this yields an ) time algorithm. Luks [22] gave a polynomial time algorithm for the graph isomorphism problem on graphs of bounded maximal degree. Since the graph of a simple d polytope is d regular, his algorithm runs in polynomial time on graphs of simple polytopes of bounded dimension. Proposition 3 The isomorphism problem for graphs ....

Luks, E.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. System Sci. 25, 42--65 (1982)

....and the parameter is the pair (s; k) where s is the size of alphabet and k is the number of sequences, then both problems are hard for W [t] for all t [BDFHW95, Hal96] 6. The Graph Isomorphism problem where the parameter k is the maximum degree of the graphs. This is in XP by the results of Luks [Luks82]. This problem would seem to be a reasonable candidate for representing a parameterized degree intermediate between FPT and W [1] for the same reasons that Graph Isomorphism classically seems to represent a polynomial time degree intermediate between P and NP. On the other hand, parameterized ....

E. Luks, "Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time," J. Comput. and Systems Sci. 25 (1982), 42--65.

....(1) is known, and it is perfectly possible that none exists, there are significant classes of groups G for which step (1) does have an efficient implementation. Luks has shown that p groups groups in which every element has order a power of p for some prime p is an example of such a class [18]. Returning to the Markov chain itself, we note immediately that M is ergodic, since every state can be reached from every other in a single transition, by selecting the identity permutation in step (1) The easiest way to get at the stationary distribution is perhaps by considering a random walk ....

Eugene M. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, Journal of Computer and System Sciences 25 (1982), pp. 42--65.

....the stabilizers of vectors and subspaces and finding centralizers and intersections of subgroups. For an integer constant d 0, let d denote the class of finite groups all of whose nonabelian composition factors lie in S d (evidently, d includes all solvable groups) An earlier result of Luks [12] asserts that, for permutation groups in d , one can find set stabilizers in polynomial time (see also the work of Babai, Cameron, and Palfy [3] for important related results) In a future paper [16] we will generalize the methods of [13] that solve these graphisomorphism inspired matrix group ....

E. M. LUKS, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comput. System Sci. 25 (1982), 42--65.

....[5, p. 1461] No general worst case polynomial time algorithms are known for this problem, but it is commonly believed not to be NP complete (unless, of course, P=NP) 30] Polynomial time algorithms are available in many special cases [5, p. 1511] in particular for graphs of bounded degrees [33, 3]. Observe that many practical applications entail graphs of bounded degree because the objects involved (logic gates in VLSI chips, facts stored in knowledge bases, etc. are interconnected sparsely. In contrast, Boolean Satisfiability instances of bounded degree, e.g. 3 SAT, are known to be ....

....problem is polynomial time solvable if the length of the longest clause and the number of occurrences of the most common literal are bounded by a constant. That is because the degree of graph vertices is bounded by that constant, in which case the graph automorphism problem is poly time solvable [33, 5]. If applied literally, the proposed construction only addresses symmetries that map p to q for particular p and q, rather than arbitrary symmetries. In order to find even a single non trivial symmetry, one may need to traverse all pairs of variables. Thus, no isomorphism of symmetry groups is ....

E. Luks, "Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time", Proc. IEEE Symp. on Foundations of Comp. Sci. (1980), pp. 42-49.

....developed an ecient implementation. A computer program is presented in [2] In that paper, we also showed how to modify the algorithm such that isomorphism of coherent algebras is checked. In spite of the introduction of successful group theoretic methods in graph isomorphism testing (see e.g. [13]) and a recent negative result concerning a generalization of the WLstabilization (it is shown in [4] that even a high dimensional version does not necessarily provide the automorphism partition of a graph) the obtained speedup lets the WL method become a signi cant heuristic tool in graph ....

E.M. Luks, "Isomorphism of graphs of bounded valence can be tested in polynomial time", J. Comput. Sys. Sci. 25 (1982) 42-65.

....stabilizers of vectors and subspaces and finding centralizers and intersections of subgroups. For an integer constant d 0, let # d denote the class of finite groups all of whose nonabelian composition factors lie in S d (evidently, # d includes all solvable groups) An earlier result of Luks [12] asserts that, for permutation groups in # d , one can find set stabilizers in polynomial time (see also the work of Babai, Cameron, and Palfy [3] for important related results) In a future paper [16] we will generalize the methods of [13] that solve these graphisomorphism inspired matrix group ....

E. M. LUKS, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comput. System Sci. 25 (1982), 42--65.

....theoretical computer science in several ways. One of the motivations for the invention of interactive proofs was Babai s investigation of complexity in finite matrix groups [Ba85] Much of our understanding of the complexity of graph isomorphism comes from research in permutation group algorithms [Ba79, FHL, Lu82]. Numerous applications have been found in cryptography [BCY, BY, IY, M, MM89, MM92] Asymptotic complexity theory has also benefited computational group theory. While polynomial time and practical are two different notions, often the structure used to design a polynomial time algorithm can be ....

E. M. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comput. System Sci. 25 (1982), 42#65.

....the graph isomorphism problem restricted to graphs of simple polytopes of bounded dimension can be solved in polynomial time (Proposition 2. 2) which is a direct consequence of a polynomial time algorithm for the graph isomorphism problem restricted to graphs of bounded maximal degree (due to Luks [13]) For concepts and notations concerning polytope theory, we refer to Ziegler s book [20] 1 Hardness Results for Arbitrary Dimension The results in this section are based on the following construction that produces from any given graph G = V; E) on jV j = n nodes a certain polytope P (G) see ....

....of a planar three connected graph) Since the graph isomorphism problem for planar graphs can be solved in linear time by an algorithm due to Hopcroft and Wong [9] both the graph and the polytope isomorphism problem for three dimensional polytopes can thus be solved in linear time. Luks [13] gave a polynomial time algorithm for the graph isomorphism problem on graphs of bounded maximal degree. Since the graph of a simple d polytope is d regular, his algorithm runs in polynomial time on graphs of simple polytopes of bounded dimension. Proposition 2.1. The isomorphism problem for ....

E. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comput. Syst. Sci. , 25 (1982), pp. 42-65.

....for the automorphism group may be harder. The exact status of these two problems is unresolved. They belong to a select group of problems which belong to NP but are not known either to belong to P nor to be NP complete. For some particular classes of graphs, notably graphs of bounded valency [42] and graphs with bounded eigenvalue multiplicity [7] the isomorphism problem is known to be polynomial. See [23] for the fundamentals of computational complexity. In practice, these questions can be resolved for graphs with thousands of vertices. Chapter gives an account of the algorithms ....

E. M. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comp. Sys. Sci. 25 (1982), 42-65.

....so on. b) verify whether f is an isomorphism from A to B. 3. if none of the above tests succeeds, A and B are not isomorphic. This procedure clearly runs in polynomial time. The converse to Proposition 5.2 does not hold. Indeed, isomorphism of graphs of bounded degree is in polynomial time [13]. However, the class of graphs of bounded degree of the following form: is not polynomially rigid (a graph of length n has 2 n automorphisms) and thus not polynomially orderable, by Proposition 5.1. 6 Open problems We have seen that if a class is polynomially orderable, then fixpoint logic ....

E.M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences, 25:42--65, 1982.

....from two sided games to one sided games for the k embedding; the proof of NP completeness of the maximal embedding, for the cooperative game, is a trivial reduction from the problem of nding a k clique. To add to our joy, isomorphism of graphs of bounded degree is decidable in polynomial time [Luk80] Unfortunately, this limitation does not make sense for our application domain: after we translate from generalized structures to structures or graphs, limiting the degree of the structure (that is, for any member of the structure, the number of tuples containing it that are members in ....

Eugene Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. In 21st Annual Symposium on Foundations of Computer Science, pages 42-49, Syracuse, New York, 1980. IEEE.

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E.M. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comp. Syst. Sci. 25 (1982), 42-65. -13-

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E.M. Luks, Isomorphism of graphs of bounded valence can betestedinpolynomial time, J. Comp. Syst. Sci. 25 (1982), 42--65.

....above problems, solutions geared to special group classes have facilitated polynomial time algorithms for significant instances of ISO. For example, the solution to A just for 2 groups yielded the first (and still the only known) polynomial time approach to testing isomorphism of trivalent graphs [16], and subsequently a polynomialtime set stabilizer algorithm for groups with bounded nonabelian composition factors ( d groups, see below) yielded ISO in polynomial time for graphs of bounded valence or bounded genus [16] 23] The polynomial time solution of A in d groups led immediately to ....

.... known) polynomial time approach to testing isomorphism of trivalent graphs [16] and subsequently a polynomialtime set stabilizer algorithm for groups with bounded nonabelian composition factors ( d groups, see below) yielded ISO in polynomial time for graphs of bounded valence or bounded genus [16], 23] The polynomial time solution of A in d groups led immediately to similar success with B and C. However, the normalizer question for d has remained open (see [19, Question 16] The main result of this paper is its resolution. Specifically, Problem D may be stated Normalizer (NORM) ....

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E. M. LUKS, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comput. System Sci. 25 (1982), 42--65.

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Eugene M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences, 25(1):42--65, 1982.

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E. M. Luks, Isomorphism of graphs of bounded valence can be tested in polynomial time, JCSS, 25 (1982), pp. 42--65.

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E. Luks, "Isomorphism of graphs of bounded valence can be tested in polynomial time," JCSS 25 (1982) 42-65.

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E. Luks, "Isomorphism of graphs of bounded valence can be tested in polynomial time," Proc. IEEE Symp. Foundations Comput. Sci., pp. 42--49, 1980.

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E.M. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer System Science, pages 4265, 1982.

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E. Luks, "Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time," J. Comput. and Systems Sci. 25 (1982), 42--65.

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E. M. LUKS, Isomorphism of graphs of bounded valence can be tested in polynomial time, J. Comput. System Sci. 25 (1982), 42--65.

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