Scheduling Lower Bounds via AND Subset Sum

Given $N$ instances $(X_1,t_1),\ldots,(X_N,t_N)$ of Subset Sum, the AND Subset Sum problem asks to determine whether all of these instances are yes-instances; that is, whether each set of integers $X_i$ has a subset that sums up to the target integer $t_i$. We prove that this problem cannot be solved in time $\tilde{O}((N \cdot t_{max})^{1-\epsilon})$, for $t_{max}=\max_i t_i$ and any $\epsilon > 0$, assuming the $\forall \exists$ Strong Exponential Time Hypothesis ($\forall \exists$-SETH). We then use this result to exclude $\tilde{O}(n+P_{max} \cdot n^{1-\epsilon})$-time algorithms for several scheduling problems on $n$ jobs with maximum processing time $P_{max}$, based on $\forall \exists$-SETH. These include classical problems such as $1||\sum w_jU_j$, the problem of minimizing the total weight of tardy jobs on a single machine, and $P_2||\sum U_j$, the problem of minimizing the number of tardy jobs on two identical parallel machines.Comment: 14 pages, ICALP'2

Optimal Cell Clustering and Activation for Energy Saving in Load-Coupled Wireless Networks

Optimizing activation and deactivation of base station transmissions provides an instrument for improving energy efficiency in cellular networks. In this paper, we study optimal cell clustering and scheduling of activation duration for each cluster, with the objective of minimizing the sum energy, subject to a time constraint of delivering the users' traffic demand. The cells within a cluster are simultaneously in transmission and napping modes, with cluster activation and deactivation, respectively. Our optimization framework accounts for the coupling relation among cells due to the mutual interference. Thus, the users' achievable rates in a cell depend on the cluster composition. On the theoretical side, we provide mathematical formulation and structural characterization for the energy-efficient cell clustering and scheduling optimization problem, and prove its NP hardness. On the algorithmic side, we first show how column generation facilitates problem solving, and then present our notion of local enumeration as a flexible and effective means for dealing with the trade-off between optimality and the combinatorial nature of cluster formation, as well as for the purpose of gauging the deviation from optimality. Numerical results demonstrate that our solutions achieve more than 60% energy saving over existing schemes, and that the solutions we obtain are within a few percent of deviation from global optimum.Comment: Revision, IEEE Transactions on Wireless Communication

Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines

We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times. Our main result is a $(3/2-c)$-approximation algorithm for some fixed $c>0$, improving upon the long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM, 2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce a new lift-and-project based SDP relaxation for the problem. This is necessary as the previous convex programming relaxations have an integrality gap of $3/2$. Second, we give a new general bipartite-rounding procedure that produces an assignment with certain strong negative correlation properties.Comment: 21 pages, 4 figure