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.

The Traveling Tournament Problem with Maximum Tour Length Two: A Practical Algorithm with An Improved Approximation Bound

The Traveling Tournament Problem is a well-known benchmark problem in tournament timetabling, which asks us to design a schedule of home/away games of n teams (n is even) under some feasibility requirements such that the total traveling distance of all the n teams is minimized. In this paper, we study TTP-2, the traveling tournament problem where at most two consecutive home games or away games are allowed, and give an effective algorithm for n/2 being odd. Experiments on the well-known benchmark sets show that we can beat previously known solutions for all instances with n/2 being odd by an average improvement of 2.66%. Furthermore, we improve the theoretical approximation ratio from 3/2+O(1/n) to 1+O(1/n) for n/2 being odd, answering a challenging open problem in this area.

Quit When You Can: Efficient Evaluation of Ensembles by Optimized Ordering

Given a classifier ensemble and a dataset, many examples may be confidently and accurately classified after only a subset of the base models in the ensemble is evaluated. Dynamically deciding to classify early can reduce both mean latency and CPU without harming the accuracy of the original ensemble. To achieve such gains, we propose jointly optimizing the evaluation order of the base models and early-stopping thresholds. Our proposed objective is a combinatorial optimization problem, but we provide a greedy algorithm that achieves a 4-approximation of the optimal solution under certain assumptions, which is also the best achievable polynomial-time approximation bound. Experiments on benchmark and real-world problems show that the proposed Quit When You Can (QWYC) algorithm can speed up average evaluation time by 1.8–2.7 times on even jointly trained ensembles, which are more difficult to speed up than independently or sequentially trained ensembles. QWYC’s joint optimization of ordering and thresholds also performed better in experiments than previous fixed orderings, including gradient boosted trees’ ordering.

A Tight Bound for Stochastic Submodular Cover

We show that the Adaptive Greedy algorithm of Golovin and Krause achieves an approximation bound of (ln(Q/η)+1) for Stochastic Submodular Cover: here Q is the “goal value” and η is the minimum gap between Q and any attainable utility value Q'<Q. Although this bound was claimed by Golovin and Krause in the original version of their paper, the proof was later shown to be incorrect by Nan & Saligrama. The subsequent corrected proof of Golovin and Krause gives a quadratic bound of (ln(Q/η)+1)2. A bound of 56(ln(Q/η)+1) is implied by work of Im et al. Other bounds for the problem depend on quantities other than Q and η. Our bound restores the original bound claimed by Golovin and Krause, generalizing the well-known (ln m+1) approximation bound on the greedy algorithm for the classical Set Cover problem, where m is the size of the ground set.

Tight Approximation for Unconstrained XOS Maximization

A set function is called XOS if it can be represented by the maximum of additive functions. When such a representation is fixed, the number of additive functions required to define the XOS function is called the width. In this paper, we study the problem of maximizing XOS functions in the value oracle model. The problem is trivial for the XOS functions of width 1 because they are just additive, but it is already nontrivial even when the width is restricted to 2. We show two types of tight bounds on the polynomial-time approximability for this problem. First, in general, the approximation bound is between O(n) and [Formula: see text], and exactly [Formula: see text] if randomization is allowed, where n is the ground set size. Second, when the width of the input XOS functions is bounded by a constant k ≥ 2, the approximation bound is between k − 1 and k − 1 − ɛ for any ɛ > 0. In particular, we give a linear-time algorithm to find an exact maximizer of a given XOS function of width 2, whereas we show that any exact algorithm requires an exponential number of value oracle calls even when the width is restricted to 3.

Near-Optimal Graph Signal Sampling by Pareto Optimization

In this paper, we focus on the bandlimited graph signal sampling problem. To sample graph signals, we need to find small-sized subset of nodes with the minimal optimal reconstruction error. We formulate this problem as a subset selection problem, and propose an efficient Pareto Optimization for Graph Signal Sampling (POGSS) algorithm. Since the evaluation of the objective function is very time-consuming, a novel acceleration algorithm is proposed in this paper as well, which accelerates the evaluation of any solution. Theoretical analysis shows that POGSS finds the desired solution in quadratic time while guaranteeing nearly the best known approximation bound. Empirical studies on both Erdos-Renyi graphs and Gaussian graphs demonstrate that our method outperforms the state-of-the-art greedy algorithms.

Limit theorems for random polytopes with vertices on convex surfaces

We consider the random polytope Kn, defined as the convex hull of n points chosen independently and uniformly at random on the boundary of a smooth convex body in ℝd. We present both lower and upper variance bounds, a strong law of large numbers, and a central limit theorem for the intrinsic volumes of Kn. A normal approximation bound from Stein's method and estimates for surface bodies are among the tools involved.

Revisiting the Approximation Bound for Stochastic Submodular Cover

Deshpande et al. presented a k(ln R + 1) approximation bound for Stochastic Submodular Cover, where k is the state set size, R is the maximum utility of a single item, and the utility function is integer-valued. This bound is similar to the ln Q/(eta+1) bound given by Golovin and Krause, whose analysis was recently found to have an error. Here Q >= R is the goal utility and eta is the minimum gap between Q and any attainable utility Q' < Q. We revisit the proof of the k(ln R + 1) bound of Deshpande et al., fill in the details of the proof of a key lemma, and prove two bounds for real-valued utility functions: k(ln R_1 + 1) and (ln R_E + 1). Here R_1 equals the maximum ratio between the largest increase in utility attainable from a single item, and the smallest non-zero increase attainable from that same item (in the same state). The quantity R_E equals the maximum ratio between the largest expected increase in utility from a single item, and the smallest non-zero expected increase in utility from that same item. Our bounds apply only to the stochastic setting with independent states.