Computational complexity of reconstruction and isomorphism testing for designs and line graphs
aa r X i v : . [ c s . CC ] J un COMPUTATIONAL COMPLEXITY OFRECONSTRUCTION AND ISOMORPHISM TESTING FORDESIGNS AND LINE GRAPHS
MICHAEL HUBERWilhelm-Schickard-Institute for Computer ScienceUniversity of TuebingenSand 13, D-72076 Tuebingen, GermanyE-mail: [email protected]
Abstract.
Graphs with high symmetry or regularity are the mainsource for experimentally hard instances of the notoriously difficult graphisomorphism problem. In this paper, we study the computational com-plexity of isomorphism testing for line graphs of t -( v, k, λ ) designs. Forthis class of highly regular graphs, we obtain a worst-case running timeof O ( v log v + O (1) ) for bounded parameters t, k, λ . In a first step, our ap-proach makes use of the Babai–Luks algorithm to compute canonicalforms of t -designs. In a second step, we show that t -designs can bereconstructed from their line graphs in polynomial-time. The first isalgebraic in nature, the second purely combinatorial. For both, pro-found structural knowledge in design theory is required. Our resultsextend earlier complexity results about isomorphism testing of graphsgenerated from Steiner triple systems and block designs. Introduction
The Graph Isomorphism (GI) problem consists in deciding whether twogiven finite graphs are isomorphic – that is, whether there exists an edge-preserving bijection between the vertex sets of the graphs. Besides of itspractical importance, the inability to directly classify the GI problem intoeither of the conventional complexity classes P or NP -complete until nowhave made it one of the central topics in structural complexity theory. Con-sequently, it is of interest to identify the difficult instances of the problem.The best worst-case algorithm for arbitrary graphs with v vertices hasrunning time exp (cid:0) O ( √ v log v ) (cid:1) , see [5, 6]. This has mainly been achievedby a combination of Luks’ seminal polynomial-time algorithm for graphs of Received by the editors August 11, 2009; and in revised form June 21, 2010.2000
Mathematics Subject Classification.
Key words and phrases.
Computational complexity, reconstructibility, isomorphismtesting, combinatorial design, line graph, graph isomorphism problem, hypergraph iso-morphism problem.The author gratefully acknowledges support by the Deutsche Forschungsgemeinschaft(DFG) via a Heisenberg grant (Hu954/4) and a Heinz Maier-Leibnitz Prize grant(Hu954/5). bounded degree [29], together with a combinatorial degree reduction due toZemlyachenko et al. [42]. After a quarter-century, this moderately exponen-tial bound for graph isomorphism still remains the state of the art despiteextensive efforts.Apparently, many graphs that seem to capture much of the computa-tional difficulty are obtained from highly regular combinatorial structures,like combinatorial designs and related configurations, see [16, 31]. Hence, itis a primary goal to reduce for these types of graphs the leading √ v termin the exponent to v / − ǫ for some constant ǫ >
0. For important specialcases, that of strongly regular graphs and that of line graphs derived fromSteiner 2-designs, Spielman [40] reduced the exponent of the exponent to1 / /
4, respectively. For the former, Babai [2] had initially given anelementary combinatorial algorithm in v O ( √ v log v ) time. Far more efficientisomorphism tests (polynomial-time or even better) are known for severalparameterized classes with bounded values for their parameters. The mostprominent classes are planar graphs, graphs of bounded degree, boundedgenus, bounded color class, or bounded eigenvalue multiplicity. For a unify-ing treatment of these parameterized classes, see [17]. A strict generalizationof the results for bounded degree and bounded genus was obtained in [33, 34].On the other hand, GI-completeness (i.e. there exists a polynomial-timeTuring reduction from the GI problem) has been proved for a number ofrestricted graph classes, including regular graphs, bipartite graphs, chordalgraphs, self-complementary graphs, split graphs, and perfect graphs (cf. [42]for some further classes).In this paper, we consider the computational problem of testing isomor-phism of line graphs derived from t -( v, k, λ ) designs. For bounded parame-ters t, k, λ , we obtain a sub-exponential algorithm for this important specialclass of the GI problem. This extends earlier complexity results about iso-morphism testing of graphs generated from Steiner triple systems and blockdesigns. Moreover, as t -( v, k, λ ) designs can be viewed as k -uniform hyper-graphs on v vertices, this problem is also interesting in view of the recentmoderately exponential bound for hypergraph isomorphism: Babai and Co-denotti [4] have shown that isomorphism of hypergraphs of bounded rankwith v vertices can be tested in time exp (cid:0) e O ( √ v ) (cid:1) (where, as usual, the e O -notation suppresses polylogarithmic factors).We state our main result: Main Theorem.
Isomorphism of line graphs of t - ( v, k, λ ) designs can bedetermined in O ( v log v + O (1) ) time for bounded parameters t, k, λ . In a first step, our approach makes use of the Babai–Luks algorithmto compute canonical forms of t -designs. In a second step, we show that t -designs can be reconstructed from their line graphs in polynomial-time.The first is algebraic in nature, the second purely combinatorial. For both,profound structural knowledge in design theory is required. Specifically, we ECONSTRUCTION AND ISOMORPHISM TESTING 3 make use of the Ray-Chauduri–Wilson theorem on the minimal number ofblocks, an extension of the Erd˝os–Ko–Rado theorem to t -designs due toRand, as well as a recent result of Kreher and Rees concerning the maximalsize of a subdesign in a t -design. Related Work.
There are only a few known complexity results aboutisomorphism problems related to combinatorial t -designs: Prior to Spiel-man’s result for Steiner 2-designs, Miller [32] had shown that the specificcase of isomorphism of line graphs derived from Steiner triple systems (i.e.Steiner 2-designs with block size 3) can be determined in sub-exponential, O ( v log v + O (1) ), time. His proof uses the fact that a Steiner triple system canbe represented as a quasigroup, and hence has a set of at most 1 + log v gen-erators. He also obtained the same bound for testing isomorphism of graphsfrom Latin squares. Moreover, he gave an O ( v log log v + O (1) ) isomorphism al-gorithm for affine and projective planes. Miller’s algorithm has been appliedby M. Colbourn [13] to perform isomorphism of Steiner t -designs with blocksize t + 1 in O ( v log v + O (1) ) time. Concerning isomorphism testing of blockdesigns (i.e. 2-designs with arbitrary λ ), Babai and Luks [6] derived as aconsequence of Luks’ techniques [29] an algorithm for bounded block size k and bounded λ in time O ( v log v + f ( k,λ ) ). On the other hand, C. Colbourn andM. Colbourn [10] verified that the isomorphism problem for block designsis GI-complete, even for triple systems. For a few other results regardingspecific designs, we refer to the survey [14, Sect. 3]. We note that the com-plexity of the Steiner t -design isomorphism problem in relation to the GIproblem is still unresolved (even for fixed t ). This is also the case for theisomorphism problem of Steiner triple and quadruple systems, respectively. Overview.
Relevant definitions and concepts from combinatorial designtheory including line graphs will be summarized in Section 2. The readermay want to skim this section and return to it when necessary. In Section 3,we apply the Babai–Luks algorithm to compute canonical forms of t -designs.In Section 4, we show that t -designs can be reconstructed from their linegraphs in polynomial-time. We finally combine the results of these sectionsto prove our main theorem.For further detailed discussion in particular on the GI problem, we refer tothe excellent literature: the books by Hoffmann [19], K¨obler, Sch¨oning andTor´an [26] as well as the surveys by Arvind and Tor´an [1], Babai [3], Boothand Colbourn [8], Goldberg [18], K¨obler [25], Read and Corneil [38], andZemlyachenko et al. [42]. The current standard reference on the complexityof group-theoretic computation is Seress [39]. MICHAEL HUBER Designs and Line Graphs
Combinatorial Designs.
Combinatorial design theory is a rich subject onthe interface of several disciplines, including coding and information theory,cryptography, combinatorics, group theory, and geometry. In particular, thestudy of designs with high symmetry properties has a very long history andestablishes deep connections between these areas (see, e.g., [12, 15, 20, 21,22, 23, 30]).For positive integers t ≤ k ≤ v and λ , we define a t - ( v, k, λ ) design to bea finite incidence structure D = ( X, B , I ), where X denotes a set of points , | X | = v , and B a set of blocks , |B| = b , satisfying the following regularityproperties: each block B ∈ B is incident with k points, and each t -subsetof X is incident with λ blocks. A flag of D is an incident point-block pair( x, B ) ∈ I with x ∈ X and B ∈ B . If t < k < v holds, then we speak of a non-trivial t -design. In this paper, ‘repeated blocks’ are not allowed, that is,the same k -element subset of points may not occur twice as a block. Thus,alternatively a t -( v, k, λ ) design can be viewed as a k -uniform hypergraphon v vertices with the property that every set of t vertices is contained in λ common edges.Incidence preserving maps which take points to points and blocks toblocks are of fundamental importance. We recall the formal definition ofan isomorphism between incidence structures: Let S = ( X , B , I ) and S = ( X , B , I ) be two incidence structures. A bijective map α : X ∪ B −→ X ∪ B is an isomorphism of S onto S , if the following holds:(i) for x ∈ X and B ∈ B , we have x α ∈ X and B α ∈ B ,(ii) for all x ∈ X and all B ∈ B , we have( x, B ) ∈ I ⇐⇒ ( x α , B α ) ∈ I . In this case, the incidence structures S and S are isomorphic . An iso-morphism of an incidence structure S onto itself is called an automorphism of S . The full group of automorphisms of an incidence structure S will bedenoted by Aut( S ).For historical reasons, a t -( v, k, λ ) design with λ = 1 is called a Steiner t -design (sometimes also a Steiner system ). The special case of a Steinerdesign with parameters t = 2 and k = 3 is called a Steiner triple system
STS( v ) of order v . A Steiner design with parameters t = 3 and k = 4 iscalled a Steiner quadruple system
SQS( v ) of order v . For example if weconsider Steiner quadruple systems, the vector space Z d with the set B ofblocks taken to be the set of all subsets of four distinct elements of Z d whose vector sum is zero, is a boolean SQS(2 d ). More geometrically, theseSQS(2 d ) consist of the points and planes of the d -dimensional binary affinespace AG ( d, v ) is that v ≡ v = 8 ECONSTRUCTION AND ISOMORPHISM TESTING 5
Figure 1.
Illustration of the unique SQS(8), with three types ofblocks: faces, opposite edges, and inscribed regular tetrahedra. and v = 10 there exists a SQS( v ) in each case, unique up to isomorphism.These are the affine space AG (3 ,
2) (cf. Figure 1) and the M¨obius plane oforder 3. For v = 14 there are exactly 4, and for v = 16 exactly 1 , , v ,the number N ( v ) of non-isomorphic SQS( v ) grows exponentially, i.e.lim inf v →∞ log N ( v ) v > . For a detailed treatment of combinatorial designs, we refer the reader tothe encyclopedic accounts [7, 11].We provide some combinatorial tools which will be helpful for the re-mainder of the paper. For the existence of t -designs, the following basicnecessary conditions can be obtained via elementary counting arguments(see, for instance, [7]): Lemma 1.
Let D = ( X, B , I ) be a t - ( v, k, λ ) design, and for a positiveinteger s ≤ t , let S ⊆ X with | S | = s . Then the number of blocks incidentwith each element of S is given by λ s = λ (cid:0) v − st − s (cid:1)(cid:0) k − st − s (cid:1) . In particular, for t ≥ , a t - ( v, k, λ ) design is also an s - ( v, k, λ s ) design. It is customary to set r := λ denoting the number of blocks incident witha given point. Lemma 2.
Let D = ( X, B , I ) be a t - ( v, k, λ ) design. Then the followingholds: (a) bk = vr. (b) (cid:18) vt (cid:19) λ = b (cid:18) kt (cid:19) . (c) r ( k −
1) = λ ( v − for t ≥ . Lemma 3.
Let D = ( X, B , I ) be a t - ( v, k, λ ) design. Then λ (cid:18) v − st − s (cid:19) ≡ (cid:18) mod (cid:18) k − st − s (cid:19)(cid:19) for each positive integer s ≤ t . MICHAEL HUBER
A generalized version of
Fisher’s Inequality for t -designs by Ray-Chaudhuriand Wilson [37, Thm. 1] gives lower bounds on the number of blocks: Theorem 4. (Ray-Chaudhuri and Wilson, 1975).
Let D = ( X, B , I ) be a t - ( v, k, λ ) design. If t is even, say t = 2 s , and v ≥ k + s , then b ≥ (cid:0) vs (cid:1) . If t is odd, say t = 2 s + 1 , and v − ≥ k + s , then b ≥ (cid:0) v − s (cid:1) . Line Graphs.
For an incidence structure S = ( X, B , I ), the line graph G ( S ) of S has as set of vertices the set B of blocks, whereas any twovertices are adjacent if and only if their corresponding blocks are incidentwith at least one common point. Line graphs of incidence structures aresometimes alternatively called block graphs or block intersection graphs (or Steiner graphs in the case of Steiner t -designs). As an example, we considera Steiner 2-(7 , ,
1) design, the well-known
Fano plane , which is the smallestdesign arising from a finite projective geometry. Since any two of its sevenblocks have a point in common, its line graph is isomorphic to the completegraph K (see Figure 2). We note that a line graph of a Steiner 2-designis a strongly regular graph , i.e. each pair of adjacent vertices has the samenumber of common neighbors, and each pair of non-adjacent vertices hasthe same number of common neighbors. Figure 2.
The Fano plane
P G (2 , K . Some Further Notation.
An incidence structure S = ( X , B , I ) iscalled a substructure of an incidence structure S = ( X, B , I ), if the followingholds:(i) X ⊆ X and B ⊆ B ,(ii) for all x ∈ X and all B ∈ B , we have( x, B ) ∈ I ⇐⇒ ( x, B ) ∈ I. A subdesign of a t -( v, k, λ ) design is a substructure of the incidence structurewhich itself is a t -( w, k, λ ) design. The subdesign is proper if w < v .A composition series for a finite group G is a chain of normal subgroupsof the form 1 = G m ✁ · · · ✁ G ✁ G ✁ G = G, in which the quotients G i /G i +1 are simple groups. The factor groups are the composition factors of G . They are independent of the choice of composition ECONSTRUCTION AND ISOMORPHISM TESTING 7 series by the Jordan–H¨older theorem. The composition width of G , denotedby cw( G ), is defined to be the smallest positive integer n such that everynon-Abelian composition factor of G embeds in the symmetric group S n .Throughout this paper, logarithms are taken base 2. All other notationis standard. 3. Isomorphism Testing of Designs
A standard algorithmic approach for testing isomorphism of graphs is totry to assign to each graph a canonical label ( canonical form ), so that twographs are isomorphic if and only if the have the same label. For instance,one could start out by labeling the vertices by their degrees, and then refine this labeling by further distinguishing equal labels through other local prop-erties of the vertices. If, after refinement, it is possible to endow a uniquelabel to every vertex, then a canonical label for the graph has been found.This procedure with its numerous variations has provided good algorithmsfor a variety of special classes of graphs. On the other hand, obstacles mayoccur if the graphs exhibit a high degree of regularity or symmetry, e.g.for regular graphs or graphs associated with highly regular combinatorialstructures. In some cases it is possible to break up the symmetry by in-dividualizing particular vertices before endowing them with unique labels.For further details on the different methods used for canonical labeling, werefer to [6, 38, 41] and [9, Sect. 2].Particularly important for our purposes, Miller [32] showed that a canon-ical labeling can be found in O ( v log v + O (1) ) time for Steiner triple systems.His proof relies on the fact that a Steiner triple system can be representedas a quasigroup, and hence has a set of at most 1 + log v generators. Byindividualizing these, it is then possible to order in polynomial-time theremaining vertices in a canonical way. Babai and Luks [6] extended thisapproach by an algebraization of the problem which involves informationabout the groups of automorphisms. Applied to 2-designs, they obtainedthe subsequent result. Theorem 5. (Babai and Luks, 1983).
Canonical forms (and hence iso-morphism testing) for non-trivial - ( v, k, λ ) designs can be computed in O ( v log v + f ( k,λ ) ) time. In particular, the time bound is O ( v log v + O (1) ) forbounded parameters k, λ . A crucial observation in the Babai–Luks approach is the following well-known fact (see, e.g., [11, Ch. II.1]): If there is a 2-( v, k, λ ) design containinga proper 2-( w, k, λ ) subdesign, then v ≥ ( k − w + 1. As the set of allsubdesigns is closed under intersection, any subset ‘generates’ a subdesign.In order to extend Theorem 5 to t -designs, we need a recent result by Kreherand Rees [27]. MICHAEL HUBER
Theorem 6. (Kreher and Rees, 2001).
Suppose D is a non-trivial t - ( v, k, λ ) design with t ≥ containing a proper t - ( w, k, λ ) subdesign. Then v ≥ w when t is odd, while v ≥ w + 1 when t is even. We can now prove the following result.
Theorem 7.
Canonical forms (and hence isomorphism testing) for non-trivial t - ( v, k, λ ) designs with t ≥ can be computed in O ( v log v + f ( t,k,λ ) ) time. In particular, the time bound is O ( v log v + O (1) ) for bounded parameters t, k, λ .Proof. Let D = ( X, B , I ) be a non-trivial t -( v, k, λ ) design with t ≥
2. Inview of Theorem 6, we establish the key observation(1) D has a generating set S of size at most 1 + log v .By individualizing S , we may proceed for the remainder of the proof bystraightforwardly adapting the method of proof used for Theorem 5 (cf. [6,Thm. 4.6]). We note that this method relies on results of Luks [29]. In whatfollows, we describe the basic steps. We first obtain(2) For fixed t , the composition factors of the setwise stabilizer Aut S ( D )are subgroups of S n , where n = max( λ, k − t ). In particular, thecomposition width cw(Aut S ( D )) is at most n .This is then employed in an inductive procedure for finding canonical formsthrough a nested sequence of graphs. We indicate the underlying con-struction for the nested graphs. For a sequence S = ( u , . . . , u s ), a chain { Y i } i of subsets of X is constructed as follows: Y = { u } and while Y i = X , if Y i induces a subdesign then Y i +1 = Y i ∪ { first u j not in Y i } else Y i +1 = Y i ∪ { B ∈ B : | B ∩ Y i | ≥ t } . The nested graphs { H j } j are de-fined as bipartite graphs, H i − and H i , both having the set Y i on one sideand on the other side the vertices representing those blocks entirely in Y i (for H i − ) or those in Y i +1 (for H i ), and edges correspond to flags. The pro-cedure invokes as a subroutine an algorithm of Babai and Luks (describedin detail in [6, Sect. 4.2]) for finding canonical forms for a bipartite graphwith respect to a group action on one of its sides, the complexity of whichis sensitive to the maximum degree on that side and to the compositionwidth of the group. With respect to the given construction of the nestedsequence, it can be shown (again via applying techniques of Luks [29]) thatthe maximum degree on the side of group action is bounded by k − t . Wetherefore obtain(3) For fixed t , the total running time is O ( v log v + ω (max( λ,k − t ))+ O (1) ).This establishes the claim. (cid:3) Reconstruction of Designs from Line Graphs
If we now give an efficient method of reconstructing a t -design from its linegraph, then isomorphism of line graphs of t -( v, k, λ ) designs can be tested in O ( v log v + O (1) ) time for bounded t, k, λ . To accomplish this task, we utilize ECONSTRUCTION AND ISOMORPHISM TESTING 9 an extension of the well-known Erd˝os–Ko–Rado theorem to t -designs, whichhas been obtained by Rands [36]. Theorem 8.
Let D = ( X, B , I ) be a t - ( v, k, λ ) design. Given < s < t ≤ k ,then there exists a function f ( k, t, s ) with the following property: supposethere is a subset A ⊆ B of blocks such that | A ∩ B | ≥ s for all A, B ∈ A ,then if v ≥ f ( k, t, s ) , it follows that |A| ≤ λ s (with λ s as in Lemma 1) , and the only families of blocks reaching this bound are those consisting of allblocks incident with an s -subset of X .Furthermore, the function f can be estimated as follows: f ( k, t, s ) ≤ s + (cid:18) ks (cid:19) ( k − s + 1)( k − s ) if s < t − s + ( k − s ) (cid:18) ks (cid:19) if s = t − . This result will enable us to efficiently find the maximum cliques in a linegraph and hence to reconstruct the points of the corresponding t -design.The idea of distinguishing cliques (i.e. sets of mutually adjacent vertices)by simple degree considerations, and using the maximum cliques in recon-struction goes back to Miller [32], while retrieving Latin squares, k -nets,and STS( v ). It has further been applied by Spielman [40] in case of Steiner2-designs, and by ¨Osterg˚ard et al. [24, 35] for STS( v ), SQS( v ), and Steiner t -designs via Rands’ theorem.We obtain the following result: Theorem 9.
Let G be a line graph on b vertices derived from a t - ( v, k, λ ) design D , where t ≥ . If b > k ( k − , then D can be reconstructed (up toisomorphism) in time polynomial in b .Proof. Let D = ( X, B , I ) be a t -( v, k, λ ) design with t ≥
2. Any point x ∈ X is incident with r distinct blocks. When we consider the line graph G ( D ) of D , these blocks correspond to vertices in G ( D ), and x induces edges betweenall mutual pairs of them. Hence, the blocks intersecting in x define a cliqueof size r in G ( D ). Choosing the case s = 1 in Theorem 8, only this typeof clique is of maximum size, if we presume that v ≥ f ( k, t, t ≥
2, always f ( k, t, ≤ k ( k − b ≥ v by Theorem 4. Thus,under the assumption that b > k ( k − D in polynomialtime in b . The claim follows. (cid:3) We note that b = Θ( v O (1) ) for bounded parameters t, k, λ in view ofLemma 2 (b). Remark 10.
Spielman [40, Prop. 10] elementary derived the stronger nec-essary condition √ b − > ( k − in the special case of Steiner 2-designs. We also remark that, in general, reconstructibility from line graphs fails forarbitrary incidence structures. The most natural and oldest graph repre-sentation of an incidence structure arguably is by its point-block incidencegraph (or
Levi graph ). However, this graph representation is normally lesscompact.
Proof of the Main Theorem : The result is obtained by putting togetherTheorem 7 and Theorem 9.
Acknowledgment.
I thank Peter Hauck, Michael Kaufmann and JacoboTor´an for interesting discussions about graph isomorphism, and for readingan early draft of this paper. I am also grateful for insightful suggestionsfrom one of the anonymous referees that helped improving the presentationof the paper.
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