A Graph Theoretic Approach for Multi-Objective Budget Constrained Capsule Wardrobe Recommendation

Traditionally, capsule wardrobes are manually designed by expert fashionistas through their creativity and technical prowess. The goal is to curate minimal fashion items that can be assembled into several compatible and versatile outfits. It is usually a cost and time intensive process, and hence lacks scalability. Although there are a few approaches that attempt to automate the process, they tend to ignore the price of items or shopping budget. In this article, we formulate this task as a multi-objective budget constrained capsule wardrobe recommendation ( MOBCCWR ) problem. It is modeled as a bipartite graph having two disjoint vertex sets corresponding to top-wear and bottom-wear items, respectively. An edge represents compatibility between the corresponding item pairs. The objective is to find a 1-neighbor subset of fashion items as a capsule wardrobe that jointly maximize compatibility and versatility scores by considering corresponding user-specified preference weight coefficients and an overall shopping budget as a means of achieving personalization. We study the complexity class of MOBCCWR , show that it is NP-Complete, and propose a greedy algorithm for finding a near-optimal solution in real time. We also analyze the time complexity and approximation bound for our algorithm. Experimental results show the effectiveness of the proposed approach on both real and synthetic datasets.

Computing top-k temporal closeness in temporal networks

The closeness centrality of a vertex in a classical static graph is the reciprocal of the sum of the distances to all other vertices. However, networks are often dynamic and change over time. Temporal distances take these dynamics into account. In this work, we consider the harmonic temporal closeness with respect to the shortest duration distance. We introduce an efficient algorithm for computing the exact top-k temporal closeness values and the corresponding vertices. The algorithm can be generalized to the task of computing all closeness values. Furthermore, we derive heuristic modifications that perform well on real-world data sets and drastically reduce the running times. For the case that edge traversal takes an equal amount of time for all edges, we lift two approximation algorithms to the temporal domain. The algorithms approximate the transitive closure of a temporal graph (which is an essential ingredient for the top-k algorithm) and the temporal closeness for all vertices, respectively, with high probability. We experimentally evaluate all our new approaches on real-world data sets and show that they lead to drastically reduced running times while keeping high quality in many cases. Moreover, we demonstrate that the top-k temporal and static closeness vertex sets differ quite largely in the considered temporal networks.

Quotients of the Gordian and H(2)-Gordian graphs

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.

Channels, Billiards, and Perfect Matching 2-Divisibility

Let $m_G$ denote the number of perfect matchings of the graph $G$. We introduce a number of combinatorial tools for determining the parity of $m_G$ and giving a lower bound on the power of 2 dividing $m_G$. In particular, we introduce certain vertex sets called channels, which correspond to elements in the kernel of the adjacency matrix of $G$ modulo $2$. A result of Lovász states that the existence of a nontrivial channel is equivalent to $m_G$ being even. We give a new combinatorial proof of this result and strengthen it by showing that the number of channels gives a lower bound on the power of $2$ dividing $m_G$ when $G$ is planar. We describe a number of local graph operations which preserve the number of channels. We also establish a surprising connection between 2-divisibility of $m_G$ and dynamical systems by showing an equivalency between channels and billiard paths. We exploit this relationship to show that $2^{\frac{\gcd(m+1,n+1)-1}{2}}$ divides the number of domino tilings of the $m\times n$ rectangle. We also use billiard paths to give a fast algorithm for counting channels (and hence determining the parity of the number of domino tilings) in simply connected regions of the square grid.

The Minimum Merrifield–Simmons Index of Unicyclic Graphs with Diameter at Most Four

The Merrifield–Simmons index i G of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, i.e., the number of independent vertex sets of G . In this paper, we determine the minimum Merrifield–Simmons index of unicyclic graphs with n vertices and diameter at most four.

Efficient bi-triangle counting for large bipartite networks

A bipartite network is a network with two disjoint vertex sets and its edges only exist between vertices from different sets. It has received much interest since it can be used to model the relationship between two different sets of objects in many applications (e.g., the relationship between users and items in E-commerce). In this paper, we study the problem of efficient bi-triangle counting for a large bipartite network, where a bi-triangle is a cycle with three vertices from one vertex set and three vertices from another vertex set. Counting bi-triangles has found many real applications such as computing the transitivity coefficient and clustering coefficient for bipartite networks. To enable efficient bi-triangle counting, we first develop a baseline algorithm relying on the observation that each bi-triangle can be considered as the join of three wedges. Then, we propose a more sophisticated algorithm which regards a bi-triangle as the join of two super-wedges, where a wedge is a path with two edges while a super-wedge is a path with three edges. We further optimize the algorithm by ranking vertices according to their degrees. We have performed extensive experiments on both real and synthetic bipartite networks, where the largest one contains more than one billion edges, and the results show that the proposed solutions are up to five orders of magnitude faster than the baseline method.

More on linear and metric tree maps

We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.

THE ALGORITHM FOR MARKING VERTEXES. COMPARISON OF ALGORITHMS FOR SEARCHING FOR CONNECTIVITY COMPONENTS IN A GRAPH USING THE EXAMPLE OF COMBINING WALLETS IN A BITCOIN DATABASE

The article is devoted to comparing the efficiency of algorithms for processing Bitcoin blockchain transaction database. The article describes the algorithm of vertex marking developed by the group. Based on the comparison of this and other algorithms, it is expected to identify the most effective algorithm for clustering addresses based on belonging to a single user. The Bitcoin database contains information about millions of financial transactions. Even though information about transactions is anonymous, there are methods for combining user addresses into wallets. In this article, we study algorithms of searching connectivity components, which are based on one of the methods of combining wallets based on the heuristic feature of the «total waste» of one user. The emphasis is placed on the practical aspects of implementation – hardware limitations in processing big data sets, as well as the choice of a solution for many graph connectivity components – the maximum connected set of graph vertices, in other words, a set of nonempty vertex sets and a set of vertex pairs.