The inertia of weighted unicyclic graphs

Let $G_w$ be a weighted graph. The \textit{inertia} of $G_w$ is the triple $In(G_w)=\big(i_+(G_w),i_-(G_w),$ $i_0(G_w)\big)$, where $i_+(G_w),i_-(G_w),i_0(G_w)$ are the number of the positive, negative and zero eigenvalues of the adjacency matrix $A(G_w)$ of $G_w$ including their multiplicities, respectively. $i_+(G_w)$, $i_-(G_w)$ is called the \textit{positive, negative index of inertia} of $G_w$, respectively. In this paper we present a lower bound for the positive, negative index of weighted unicyclic graphs of order $n$ with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order $n$ with two positive, two negative and at least $n-6$ zero eigenvalues, respectively.Comment: 23 pages, 8figure

Snevily's Conjecture about L\mathcal{L}-intersecting Families on Set Systems and its Analogue on Vector Spaces

The classical Erd\H{o}s-Ko-Rado theorem on the size of an intersecting family of $k$-subsets of the set $[n] = \{1, 2, \dots, n\}$ is one of the fundamental intersection theorems for set systems. After the establishment of the EKR theorem, many intersection theorems on set systems have appeared in the literature, such as the well-known Frankl-Wilson theorem, Alon-Babai-Suzuki theorem, and Grolmusz-Sudakov theorem. In 1995, Snevily proposed the conjecture that the upper bound for the size of an $\mathcal{L}$-intersecting family of subsets of $[n]$ is ${{n} \choose {s}}$ under the condition $\max \{l_{i}\} < \min \{k_{j}\}$, where $\mathcal{L} = \{l_{1}, \dots, l_{s}\}$ with $0 \leq l_{1} < \cdots < l_{s}$ and $k_{j}$ are subset sizes in the family. In this paper, we prove that Snevily's conjecture holds for $n \geq {{k^{2}} \choose {l_{1}+1}}s + l_{1}$, where $k$ is the maximum subset size in the family. We then derive an analogous result for $\mathcal{L}$-intersecting families of subspaces of an $n$-dimensional vector space over a finite field $\mathbb{F}_{q}$.Comment: arXiv admin note: text overlap with arXiv:1701.00585 by other author

Truthful Auctions for Automated Bidding in Online Advertising

Automated bidding, an emerging intelligent decision making paradigm powered by machine learning, has become popular in online advertising. Advertisers in automated bidding evaluate the cumulative utilities and have private financial constraints over multiple ad auctions in a long-term period. Based on these distinct features, we consider a new ad auction model for automated bidding: the values of advertisers are public while the financial constraints, such as budget and return on investment (ROI) rate, are private types. We derive the truthfulness conditions with respect to private constraints for this multi-dimensional setting, and demonstrate any feasible allocation rule could be equivalently reduced to a series of non-decreasing functions on budget. However, the resulted allocation mapped from these non-decreasing functions generally follows an irregular shape, making it difficult to obtain a closed-form expression for the auction objective. To overcome this design difficulty, we propose a family of truthful automated bidding auction with personalized rank scores, similar to the Generalized Second-Price (GSP) auction. The intuition behind our design is to leverage personalized rank scores as the criteria to allocate items, and compute a critical ROI to transform the constraints on budget to the same dimension as ROI. The experimental results demonstrate that the proposed auction mechanism outperforms the widely used ad auctions, such as first-price auction and second-price auction, in various automated bidding environments

Mobile Live Video Streaming Optimization via Crowdsourcing Brokerage

Nowadays, people can enjoy a rich real-time sensing cognition of what they are interested in anytime and anywhere by leveraging powerful mobile devices such as smartphones. As a key support for the propagation of these richer live media contents, cellular-based access technologies play a vital role to provide reliable and ubiquitous Internet access to mobile devices. However, these limited wireless network channel conditions vary and fluctuate depending on weather, building shields, congestion, etc., which degrade the quality of live video streaming dramatically. To address this challenge, we propose to use crowdsourcing brokerage in future networks which can improve each mobile user's bandwidth condition and reduce the fluctuation of network condition. Further, to serve mobile users better in this crowdsourcing style, we study the brokerage scheduling problem which aims at maximizing the user's QoE (quality of experience) satisfaction degree cost-effectively. Both offline and online algorithms are proposed to solve this problem. The results of extensive evaluations demonstrate that by leveraging crowdsourcing technique, our solution can cost-effectively guarantee a higher quality view experience