가상 머신 배치에서 디스크 반대 공존 제약 조건을 위한 혼합 정수 계획법 개선 방안 탐색

📝 원문 정보

• Title: Exploring Mixed Integer Programming Reformulations for Virtual Machine Placement with Disk Anti-Colocation Constraints
• ArXiv ID: 1903.02139
• 발행일: 2019-03-07
• 저자: Xiaoying Zheng and Ye Xia

📝 초록 (Abstract)

💡 논문 핵심 해설 (Deep Analysis)

Core Summary: The paper introduces three formulations—F1, F2, and COMB—to solve the complex problem of placing virtual machines on physical machines.

Problem Statement: In data centers, efficiently placing multiple VMs on PMs is crucial. One major challenge is disk anticolocation, where VM disks must be distributed across different PMs to avoid resource contention issues. This task becomes increasingly difficult due to the large number of VMs and PMs involved, along with varying resource requirements.

Solution Approach: The paper proposes three formulations:

• F1: A straightforward approach that assigns each VM individually to a PM. However, this method can become computationally expensive as the number of variables and constraints increases.
• F2: This formulation uses precomputed configurations for each PM type. It is effective when there are not too many possible configurations but becomes impractical with an excessive number of possibilities.
• COMB: A hybrid approach that combines F1 and F2, useful in scenarios where the number of feasible configurations is large.

Key Results: The paper compares the performance of these formulations and demonstrates their effectiveness under different conditions. COMB proves particularly advantageous when dealing with PM types having a high number of possible configurations.

Significance & Application: These approaches help optimize resource allocation in data centers, potentially reducing costs and improving performance. They also provide flexibility by offering suitable formulations for various scenarios.

📄 논문 본문 발췌 (Translation)

\begin{align}
\textbf{F1:} \min_{x,y,z} & \sum_{j \in \mathcal{P}} \hat{c}_j z_j \\
\text{s.t.} & y_{ikjl} \leq x_{ij}, \ i \in \mathcal{V}, j \in \mathcal{P}, k \in R_i, l \in D_j \\
& \sum_{j \in \mathcal{P}} \sum_{l \in D_j} y_{ikjl} = 1, \ i \in \mathcal{V}, k \in R_i \\
& \sum_{j \in \mathcal{P}} x_{ij} \leq 1, \ i \in \mathcal{V} \\
& \sum_{k \in R_i} y_{ikjl} \leq 1, \ i \in \mathcal{V}, j \in \mathcal{P}, l \in D_j
\end{align}

\begin{align}
\textbf{F2:} \min_{\gamma,z} & \sum_{v \in \mathcal{T}^p, j \in \mathcal{Y}_v} \hat{c}_j z_j \\
\text{s.t.} & \sum_{t \in \mathcal{C}_v} \gamma_{jt} \leq 1, \ v \in \mathcal{T}^p, j \in \mathcal{Y}_v \\
& \sum_{v \in \mathcal{T}^p} \sum_{j \in \mathcal{Y}_v} \sum_{t \in \mathcal{C}_v} \gamma_{jt} w_u^t \geq m_u, \ u \in \mathcal{T}^v \\
& z_j \leq \sum_{t \in \mathcal{C}_v} \gamma_{jt}, \ v \in \mathcal{T}^p, j \in \mathcal{Y}_v
\end{align}

\begin{align}
\textbf{COMB:} \min_{x,y,z,\gamma} & \sum_{j \in \mathcal{P}} \hat{c}_j z_j \\
\text{s.t.} & y_{ikjl} \leq x_{ij}, \ i \in \mathcal{V}, j \in \mathcal{P}_1, k \in R_i, l \in D_j \\
& \sum_{j \in \mathcal{P}_1} \sum_{l \in D_j} y_{ikjl} = \sum_{j \in \mathcal{P}_1} x_{ij}, \ i \in \mathcal{V}, k \in R_i \\
& \sum_{j \in \mathcal{P}_1} x_{ij} \leq 1, \ i \in \mathcal{V}
\end{align}

\begin{align}
& \sum_{i \in \mathcal{V}} \alpha_i x_{ij} \leq C_j, \ j \in \mathcal{P}_1 \\
& \sum_{i \in \mathcal{V}} \beta_i x_{ij} \leq M_j, \ j \in \mathcal{P}_1
\end{align}

\begin{align}
& z_j \leq \sum_{t \in \mathcal{C}_v} \gamma_{jt}, \ v \in \mathcal{T}^p_2, j \in \mathcal{Y}_v \\
& Bz_j \geq \sum_{t \in \mathcal{C}_v} \gamma_{jt}, \ v \in \mathcal{T}^p_2, j \in \mathcal{Y}_v
\end{align}

Reference