Computer Science

This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian Completion Problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.

By providing a new framework, we extend previous results on locally checkable problems in bounded treewidth graphs. As a consequence, we show how to solve, in polynomial time for bounded treewidth graphs, double Roman domination and Grundy domination, among other problems for which no such algorithm was previously known. Moreover, by proving that fixed powers of bounded degree and bounded treewidth graphs are also bounded degree and bounded treewidth graphs, we can enlarge the family of problems that can be solved in polynomial time for these graph classes, including distance coloring problems and distance domination problems (for bounded distances).

Using three supercomputers, we broke a record set in 2011, in the enumeration of non-isomorphic regular graphs by expanding the sequence of A006820 in Online Encyclopedia of Integer Sequences (OEIS), to achieve the number for 4-regular graphs of order 23 as 429,668,180,677,439, while discovering serval optimal regular graphs with minimum average shortest path lengths (ASPL) that can be used as interconnection networks for parallel computers. The number of 4-regular graphs and the optimal graphs, extremely time-consuming to calculate, result from a method we adapt from GENREG, a classical regular graph generator, to fit for supercomputers' strengths of using thousands of processor cores.

One method to obtain a proper vertex coloring of graphs using a reasonable number of colors is to start from any arbitrary proper coloring and then repeat some local re-coloring techniques to reduce the number of color classes. The Grundy (First-Fit) coloring and color-dominating colorings of graphs are two well-known such techniques. The color-dominating colorings are also known and commonly referred as {\rm b}-colorings. But these two topics have been studied separately in graph theory. We introduce a new coloring procedure which combines the strategies of these two techniques and satisfies an additional property. We first prove that the vertices of every graph G can be effectively colored using color classes say C 1 ,…, C k such that (i) for any two colors i and j with 1≤i<j≤k , any vertex of color j is adjacent to a vertex of color i , (ii) there exists a set { u 1 ,…, u k } of vertices of G such that u j ∈ C j for any j∈{1,…,k} and u k is adjacent to u j for each 1≤j≤k with j≠k , and (iii) for each i and j with i≠j , the vertex u j has a neighbor in C i . This provides a new vertex coloring heuristic which improves both Grundy and color-dominating colorings. Denote by z(G) the maximum number of colors used in any proper vertex coloring satisfying the above properties. The z(G) quantifies the worst-case behavior of the heuristic. We prove the existence of { G n } n≥1 such that min{Γ( G n ),b( G n )}→∞ but z( G n )≤3 for each n . For each positive integer t we construct a family of finitely many colored graphs D t satisfying the property that if z(G)≥t for a graph G then G contains an element from D t as a colored subgraph. This provides an algorithmic method for proving numeric upper bounds for z(G) .

In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring γ of a digraph D is proper if no two arcs with the same head or with the same tail are assigned the same colour. For each vertex u of D , we denote by S − γ (u) and S + γ (u) the sets of colours that appear on the incoming arcs and on the outgoing arcs of u , respectively. An arc colouring γ of D is \emph{neighbour-distinguishing} if, for every two adjacent vertices u and v of D , the ordered pairs ( S − γ (u), S + γ (u)) and ( S − γ (v), S + γ (v)) are distinct. The neighbour-distinguishing index of D is then the smallest number of colours needed for a neighbour-distinguishing arc-colouring of D .We prove upper bounds on the neighbour-distinguishing index of various classes of digraphs.

The standard model and the bandit model are two generalizations of the mistake-bound model to online multiclass classification. In both models the learner guesses a classification in each round, but in the standard model the learner recieves the correct classification after each guess, while in the bandit model the learner is only told whether or not their guess is correct in each round. For any set F of multiclass classifiers, define op t std (F) and op t bandit (F) to be the optimal worst-case number of prediction mistakes in the standard and bandit models respectively. Long (Theoretical Computer Science, 2020) claimed that for all M>2 and infinitely many k , there exists a set F of functions from a set X to a set Y of size k such that op t std (F)=M and op t bandit (F)??1?�o(1))(|Y|ln|Y|)op t std (F) . The proof of this result depended on the following lemma, which is false e.g. for all prime p?? , s=1 (the all 1 vector), t=2 (the all 2 vector), and all z . Lemma: Fix n?? and prime p , and let u be chosen uniformly at random from {0,??p??} n . For any s,t??{1,??p??} n with s?�t and for any z?�{0,??p??} , we have Pr(t?�u=zmodp | s?�u=zmodp)= 1 p . We show that this lemma is false precisely when s and t are multiples of each other mod p . Then using a new lemma, we fix Long's proof.

Let F be a quadratic APN function of n variables. The associated Boolean function γ F in 2n variables ( γ F (a,b)=1 if a≠0 and equation F(x)+F(x+a)=b has solutions) has the form γ F (a,b)= Φ F (a)⋅b+ φ F (a)+1 for appropriate functions Φ F : F n 2 → F n 2 and φ F : F n 2 → F 2 . We summarize the known results and prove new ones regarding properties of Φ F and φ F . For instance, we prove that degree of Φ F is either n or less or equal to n−2 . Based on computation experiments, we formulate a conjecture that degree of any component function of Φ F is n−2 . We show that this conjecture is based on two other conjectures of independent interest.

Let k and d be such that k≥d+2 . Consider two k -colourings of a d -degenerate graph G . Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The k -reconfiguration graph of G is the graph whose vertices are the proper k -colourings of G , with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d+2) -reconfiguration graph of any d -degenerate graph on n vertices is O( n 2 ) . So far, the existence of a polynomial diameter is open even for d=2 . In this paper, we prove that the diameter of the k -reconfiguration graph of a d -degenerate graph is O( n d+1 ) for k≥d+2 . Moreover, we prove that if k≥ 3 2 (d+1) then the diameter of the k -reconfiguration graph is quadratic, improving the previous bound of k≥2d+1 . We also show that the 5 -reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.

We consider the routing flow shop problem with two machines on an asymmetric network. For this problem we discuss properties of an optimal schedule and present a polynomial time algorithm assuming the number of nodes of the network to be bounded by a constant. To the best of our knowledge, this is the first positive result on the complexity of the routing flow shop problem with an arbitrary structure of the transportation network, even in the case of a symmetric network. This result stands in contrast with the complexity of the two-machine routing open shop problem, which was shown to be NP-hard even on the two-node network.

We propose a quantum algorithm to estimate the Gowers U 2 norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are ϵ -far from the set of linear Boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers U 3 norms of Boolean functions.