Unavoidable doubly connected large graphs

A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs: (1) Pn, a path on n vertices; (2) K1,ns, the graph obtained from K 1,n by subdividing an edge once; (3) K2,n\e, the graph obtained from K2,n by deleting an edge;(4) K2,n+, the graph obtained from K2,n by adding an edge between the two degree-n vertices x1 and x2, and a pendent edge at each xi. Two applications of this result are also discussed in the paper. © 2003 Elsevier B.V. All rights reserved

Formulas for approximating pseudo-Boolean random variables

We consider { 0, 1 }n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. We then derive explicit formulas for approximating a pseudo-Boolean random variable by a linear function if the measure is permutation-invariant, and by a function of degree at most k if the measure is a product measure. These formulas generalize results due to Hammer-Holzman and Grabisch-Marichal-Roubens. We also derive a formula for the best faithful linear approximation that extends a result due to Charnes-Golany-Keane-Rousseau concerning generalized Shapley values. We show that a theorem of Hammer-Holzman that states that a pseudo-Boolean function and its best approximation of degree at most k have the same derivatives up to order k does not generalize to this setting for arbitrary probability measures, but does generalize if the probability measure is a product measure. © 2007 Elsevier B.V. All rights reserved

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments

Ranking tournaments with no errors II: Minimax relation

A tournament T=(V,A) is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In the first paper we have given a structural description of all Möbius-free tournaments, and have proved that every CM tournament is Möbius-free. In this second paper we establish the converse by using our structural theorems and linear programming approach

Transforms of pseudo-Boolean random variables

As in earlier works, we consider {0, 1}n as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. Under the assumption that the coordinate random variables are independent, we show it is very easy to give an orthonormal basis for the space of pseudo-Boolean random variables of degree at most k. We use this orthonormal basis to find the transform of a given pseudo-Boolean random variable and to answer various least squares minimization questions. © 2009 Elsevier B.V. All rights reserved

Out-of-Distributed Semantic Pruning for Robust Semi-Supervised Learning

Recent advances in robust semi-supervised learning (SSL) typically filter out-of-distribution (OOD) information at the sample level. We argue that an overlooked problem of robust SSL is its corrupted information on semantic level, practically limiting the development of the field. In this paper, we take an initial step to explore and propose a unified framework termed OOD Semantic Pruning (OSP), which aims at pruning OOD semantics out from in-distribution (ID) features. Specifically, (i) we propose an aliasing OOD matching module to pair each ID sample with an OOD sample with semantic overlap. (ii) We design a soft orthogonality regularization, which first transforms each ID feature by suppressing its semantic component that is collinear with paired OOD sample. It then forces the predictions before and after soft orthogonality decomposition to be consistent. Being practically simple, our method shows a strong performance in OOD detection and ID classification on challenging benchmarks. In particular, OSP surpasses the previous state-of-the-art by 13.7% on accuracy for ID classification and 5.9% on AUROC for OOD detection on TinyImageNet dataset. The source codes are publicly available at https://github.com/rain305f/OSP.Comment: Accpected by CVPR 202