Approximating submodular kk-partition via principal partition sequence

In submodular $k$-partition, the input is a non-negative submodular function $f$ defined over a finite ground set $V$ (given by an evaluation oracle) along with a positive integer $k$ and the goal is to find a partition of the ground set $V$ into $k$ non-empty parts $V_1, V_2, ..., V_k$ in order to minimize $\sum_{i=1}^k f(V_i)$. Narayanan, Roy, and Patkar (Journal of Algorithms, 1996) designed an algorithm for submodular $k$-partition based on the principal partition sequence and showed that the approximation factor of their algorithm is $2$ for the special case of graph cut functions (subsequently rediscovered by Ravi and Sinha (Journal of Operational Research, 2008)). In this work, we study the approximation factor of their algorithm for three subfamilies of submodular functions -- monotone, symmetric, and posimodular, and show the following results: 1. The approximation factor of their algorithm for monotone submodular $k$-partition is $4/3$. This result improves on the $2$-factor achievable via other algorithms. Moreover, our upper bound of $4/3$ matches the recently shown lower bound under polynomial number of function evaluation queries (Santiago, IWOCA 2021). Our upper bound of $4/3$ is also the first improvement beyond $2$ for a certain graph partitioning problem that is a special case of monotone submodular $k$-partition. 2. The approximation factor of their algorithm for symmetric submodular $k$-partition is $2$. This result generalizes their approximation factor analysis beyond graph cut functions. 3. The approximation factor of their algorithm for posimodular submodular $k$-partition is $2$. We also construct an example to show that the approximation factor of their algorithm for arbitrary submodular functions is $\Omega(n/k)$.Comment: Accepted to APPROX'2

Submodularity and Its Applications in Wireless Communications

This monograph studies the submodularity in wireless communications and how to use it to enhance or improve the design of the optimization algorithms. The work is done in three different systems. In a cross-layer adaptive modulation problem, we prove the submodularity of the dynamic programming (DP), which contributes to the monotonicity of the optimal transmission policy. The monotonicity is utilized in a policy iteration algorithm to relieve the curse of dimensionality of DP. In addition, we show that the monotonic optimal policy can be determined by a multivariate minimization problem, which can be solved by a discrete simultaneous perturbation stochastic approximation (DSPSA) algorithm. We show that the DSPSA is able to converge to the optimal policy in real time. For the adaptive modulation problem in a network-coded two-way relay channel, a two-player game model is proposed. We prove the supermodularity of this game, which ensures the existence of pure strategy Nash equilibria (PSNEs). We apply the Cournot tatonnement and show that it converges to the extremal, the largest and smallest, PSNEs within a finite number of iterations. We derive the sufficient conditions for the extremal PSNEs to be symmetric and monotonic in the channel signal-to-noise (SNR) ratio. Based on the submodularity of the entropy function, we study the communication for omniscience (CO) problem: how to let all users obtain all the information in a multiple random source via communications. In particular, we consider the minimum sum-rate problem: how to attain omniscience by the minimum total number of communications. The results cover both asymptotic and non-asymptotic models where the transmission rates are real and integral, respectively. We reveal the submodularity of the minimum sum-rate problem and propose polynomial time algorithms for solving it. We discuss the significance and applications of the fundamental partition, the one that gives rise to the minimum sum-rate in the asymptotic model. We also show how to achieve the omniscience in a successive manner