Generalized Assignment via Submodular Optimization with Reserved Capacity

We study a variant of the generalized assignment problem (GAP) with group constraints. An instance of (Group GAP) is a set I of items, partitioned into L groups, and a set of m uniform (unit-sized) bins. Each item i in I has a size s_i >0, and a profit p_{i,j} >= 0 if packed in bin j. A group of items is satisfied if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total profit from satisfied groups is maximized. We point to central applications of Group GAP in Video-on-Demand services, mobile Device-to-Device network caching and base station cooperation in 5G networks. Our main result is a 1/6-approximation algorithm for Group GAP instances where the total size of each group is at most m/2. At the heart of our algorithm lies an interesting derivation of a submodular function from the classic LP formulation of GAP, which facilitates the construction of a high profit solution utilizing at most half the total bin capacity, while the other half is reserved for later use. In particular, we give an algorithm for submodular maximization subject to a knapsack constraint, which finds a solution of profit at least 1/3 of the optimum, using at most half the knapsack capacity, under mild restrictions on element sizes. Our novel approach of submodular optimization subject to a knapsack with reserved capacity constraint may find applications in solving other group assignment problems

Lower Bounds on the Depth of Monotone Arithmetic Computations

AbstractConsider an arithmetic expression of lengthninvolving only the operations {+,×} and non-negative constants. We prove lower bounds on the depth of any binary computation tree over the same sets of operations and constants that computes such an expression. We exhibit a family of arithmetic expressions that requires computation trees of depth at least 1.5log2n−O(1), thus proving a conjecture of S. R. Kosaraju (1986,in“Proc. of the 18th ACM Symp. on Theory Computing,” pp. 231–239). In contrast, Kosaraju showed how to compute such expressions with computation trees of depth 2log2n+O(1)

Approximating Connected Maximum Cuts via Local Search

The Connected Max Cut (CMC) problem takes in an undirected graph G(V,E) and finds a subset S ? V such that the induced subgraph G[S] is connected and the number of edges connecting vertices in S to vertices in V?S is maximized. This problem is closely related to the Max Leaf Degree (MLD) problem. The input to the MLD problem is an undirected graph G(V,E) and the goal is to find a subtree of G that maximizes the degree (in G) of its leaves. [Gandhi et al. 2018] observed that an ?-approximation for the MLD problem induces an ?(?)-approximation for the CMC problem. We present an ?(log log |V|)-approximation algorithm for the MLD problem via local search. This implies an ?(log log |V|)-approximation algorithm for the CMC problem. Thus, improving (exponentially) the best known ?(log |V|) approximation of the Connected Max Cut problem [Hajiaghayi et al. 2015]

The Preemptive Resource Allocation Problem

We revisit a classical scheduling model to incorporate modern trends in data center networks and cloud services. Addressing some key challenges in the allocation of shared resources to user requests (jobs) in such settings, we consider the following variants of the classic resource allocation problem (RAP). The input to our problems is a set J of jobs and a set M of homogeneous hosts, each has an available amount of some resource. A job is associated with a release time, a due date, a weight and a given length, as well as its resource requirement. A feasible schedule is an allocation of the resource to a subset of the jobs, satisfying the job release times/due dates as well as the resource constraints. A crucial distinction between classic RAP and our problems is that we allow preemption and migration of jobs, motivated by virtualization techniques. We consider two natural objectives: throughput maximization (MaxT), which seeks a maximum weight subset of the jobs that can be feasibly scheduled on the hosts in M, and resource minimization (MinR), that is finding the minimum number of (homogeneous) hosts needed to feasibly schedule all jobs. Both problems are known to be NP-hard. We first present an Omega(1)-approximation algorithm for MaxT instances where time-windows form a laminar family of intervals. We then extend the algorithm to handle instances with arbitrary time-windows, assuming there is sufficient slack for each job to be completed. For MinR we study a more general setting with d resources and derive an O(log d)-approximation for any fixed d >= 1, under the assumption that time-windows are not too small. This assumption can be removed leading to a slightly worse ratio of O(log d log^* T), where T is the maximum due date of any job

Generalized Assignment of Time-Sensitive Item Groups

We study the generalized assignment problem with time-sensitive item groups (chi-AGAP). It has central applications in advertisement placement on the Internet, and in virtual network embedding in Cloud data centers. We are given a set of items, partitioned into n groups, and a set of T identical bins (or, time-slots). Each group 1 0 and a non-negative utility u_{it} when packed into bin t in chi_j. A bin can accommodate at most one item from each group and the total size of the items in a bin cannot exceed its capacity. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from groups that are completely packed is maximized. Our main result is an Omega(1)-approximation algorithm for chi-AGAP. Our approximation technique relies on a non-trivial rounding of a configuration LP, which can be adapted to other common scenarios of resource allocation in Cloud data centers