Canonisation and Definability for Graphs of Bounded Rank Width

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width $k$ is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time $n^{f(k)}$ for a non-elementary function $f$. Our result yields an isomorphism test for graphs of rank width $k$ running in time $n^{O(k)}$. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.Comment: 32 page

A subexponential-time quantum algorithm for the dihedral hidden subgroup problem

We present a quantum algorithm for the dihedral hidden subgroup problem with time and query complexity $O(\exp(C\sqrt{\log N}))$. In this problem an oracle computes a function $f$ on the dihedral group $D_N$ which is invariant under a hidden reflection in $D_N$. By contrast the classical query complexity of DHSP is $O(\sqrt{N})$. The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins with the quantum character transform on the group, just as for other hidden subgroup problems. Then it tensors irreducible representations of $D_N$ and extracts summands to obtain target representations. Finally, state tomography on the target representations reveals the hidden subgroup.Comment: 11 pages. Revised in response to referee reports. Early sections are more accessible; expanded section on other hidden subgroup problem

Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields

Process model comparison based on cophenetic distance

The automated comparison of process models has received increasing attention in the last decade, due to the growing existence of process models and repositories, and the consequent need to assess similarities between the underlying processes. Current techniques for process model comparison are either structural (based on graph edit distances), or behavioural (through activity profiles or the analysis of the execution semantics). Accordingly, there is a gap between the quality of the information provided by these two families, i.e., structural techniques may be fast but inaccurate, whilst behavioural are accurate but complex. In this paper we present a novel technique, that is based on a well-known technique to compare labeled trees through the notion of Cophenetic distance. The technique lays between the two families of methods for comparing a process model: it has an structural nature, but can provide accurate information on the differences/similarities of two process models. The experimental evaluation on various benchmarks sets are reported, that position the proposed technique as a valuable tool for process model comparison.Peer ReviewedPostprint (author's final draft

Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe

Äriprotsessimudelite ühildamine

Väitekirja elektrooniline versioon ei sisalda publikatsioone.Ettevõtted, kellel on aastatepikkune kogemus äriprotsesside haldamises, omavad sageli protsesside repositooriumeid, mis võivad endas sisaldada sadu või isegi tuhandeid äriprotsessimudeleid. Need mudelid pärinevad erinevatest allikatest ja need on loonud ning neid on muutnud erinevad osapooled, kellel on erinevad modelleerimise oskused ning praktikad. üheks sagedaseks praktikaks on uute mudelite loomine, kasutades olemasolevaid mudeleid, kopeerides neist fragmente ning neid seejärel muutes. See omakorda loob olukorra, kus protsessimudelite repositoorium sisaldab mudeleid, milles on identseid mudeli fragmente, mis viitavad samale alamprotsessile. Kui sellised fragmendid jätta konsolideerimata, siis võib see põhjustada repositooriumis ebakõlasid -- üks ja sama alamprotsess võib olla erinevates protsessides erinevalt kirjeldatud. Sageli on ettevõtetel mudelid, millel on sarnased eesmärgid, kuid mis on mõeldud erinevate klientide, toodete, äriüksuste või geograafiliste regioonide jaoks. Näiteks on äriprotsessid kodukindlustuse ja autokindlustuse jaoks sama ärilise eesmärgiga. Loomulikult sisaldavad nende protsesside mudelid mitmeid identseid alamfragmente (nagu näiteks poliisi andmete kontrollimine), samas on need protsessid mitmes punktis erinevad. Nende protsesside eraldi haldamine on ebaefektiivne ning tekitab liiasusi. Doktoritöös otsisime vastust küsimusele: kuidas identifitseerida protsessimudelite repositooriumis korduvaid mudelite fragmente, ning üldisemalt -- kuidas leida ning konsolideerida sarnasusi suurtes äriprotsessimudelite repositooriumites? Doktoritöös on sisse toodud kaks üksteist täiendavat meetodit äriprotsessimudelite konsolideerimiseks, täpsemalt protsessimudelite ühildamine üheks mudeliks ning mudelifragmentide ekstraktimine. Esimene neist võtab sisendiks kaks või enam protsessimudelit ning konstrueerib neist ühe konsolideeritud protsessimudeli, mis sisaldab kõikide sisendmudelite käitumist. Selline lähenemine võimaldab analüütikutel hallata korraga tervet perekonda sarnaseid mudeleid ning neid muuta sünkroniseeritud viisil. Teine lähenemine, alamprotsesside ekstraktimine, sisaldab endas sagedasti esinevate fragmentide identifitseerimist (protsessimudelites kloonide leidmist) ning nende kapseldamist alamprotsessideks

A general computational tool for structure synthesis

Synthesis of structures is a very difficult task even with only a small number of components that form a system; yet it is the catalyst of innovation. Molecular structures and nanostructures typically have a large number of similar components but different connections, which manifests a more challenging task for their synthesis. This thesis presents a novel method and its related algorithms and computer programs for the synthesis of structures. This novel method is based on several concepts: (1) the structure is represented by a graph and further by the adjacency matrix; and (2) instead of only exploiting the eigenvalue of the adjacency matrix, both the eigenvalue and the eigenvector are exploited; specifically the components of the eigenvector have been found very useful in algorithm development. This novel method is called the Eigensystem method. The complexity of the Eigensystem method is equal to that of the famous program called Nauty in the combinatorial world. However, the Eigensystem method can work for the weighted and both directed and undirected graph, while the Nauty program can only work for the non-weighted and both directed and undirected graph. The cause for this is the different philosophies underlying these two methods. The Nauty program is based on the recursive component decomposition strategy, which could involve some unmanageable complexities when dealing with the weighted graph, albeit no such an attempt has been reported in the literature. It is noted that in practical applications of structure synthesis, weighted graphs are more useful than non-weighted graphs for representing physical systems. Pivoted at the Eigensystem method, this thesis presents the algorithms and computer programs for the three fundamental problems in structure synthesis, namely the isomorphism/automorphism, the unique labeling, and the enumeration of the structures or graphs