The claw, diamond-free edge deletion problem admits an O(k^3) -vertex kernel [Theoretical Computer Science 2022]. In this paper, we improve it to O(k^2) vertices. We use the modulator technique and then bound the number of vertices outside the modulator. Let M be an arbitrary modulator. We find that, the propagation between the maximal cliques in G-M is always due to the claw graph. Based on the observation, the number of vertices in the maximal cliques untouched to M can be bounded by O(k^2) . Since |M|=O(k) and the maximal cliques in G-M touched to M is bounded by some normal kernel frameworks, we thus have the claimed kernelization size.
We propose and study a combination model of l-centrum and the diversity-aware k-median problem, called the diversity-aware l-centrum problem. In this problem, we are given a set of clients, a set of facilities in a metric space, integers k and l, and r[i] (i∈ [t]) for each facility group. The diversity constraint refers to the requirement that each group must contain at least r[i] open facilities. The goal is to find a subset of facilities of size at most k so as to minimize the sum of the l largest distances from a client to its nearest open facility while satisfying the diversity constraint. As our main contribution, this work provides the first constant approximation algorithm for the diversity-aware l-centrum problem in FPT time, based on a combination of leader selection and coreset construction techniques.
When solving complex Markov Decision Processes (MDPs), the state value function can be represented by a parameterized function or a lookup table. Parameterized functions are space-efficient but computationally complex, while lookup tables are efficient but struggle with large state spaces, leading to memory waste or high query complexity. This study proposes a lookup table design with O(1) query complexity and no memory waste for complex MDPs commonly found in dynamic vehicle routing problems. We define a state key as a numerical triple and design two 3D arrays. Though initially memory-wasteful, we apply operations like mirroring, rotation, and translation to remove unused memory while maintaining O(1) query complexity. Our contributions are: (1) making state aggregation optional in certain MDPs, and (2) efficiently managing memory for querying complex data. An illustrative example is used throughout to enhance clarity.
In this paper, we consider the parallel-machine customer order scheduling with weighted and submodular rejection penalties, which is formally defined as follows. We are given a set 𝒪 of n customer orders and a set of m dedicated machines. Each order consists of m different product types and has an associated weight, where each product type must be processed on a corresponding dedicated machine. Each order can either be rejected, in which case a rejection penalty must be paid, or accepted and processed on the m dedicated machines. If an order is accepted, the weighted completion time is computed as the product of its completion time and its weight. The completion time of an order is defined as the maximum completion time among all its product types. The objective is to select a rejected order set ℛ⊆𝒪 and schedule the remaining orders 𝒜=𝒪∖ℛ such that the sum of the maximum weighted completion time W_max(𝒜) of the accepted orders in 𝒜 and the penalty π (ℛ) is minimized, where the penalty is determined by a nondecreasing submodular function π (· ) . We present a deterministic polynomial-time e/e-1 -approximation algorithm based on the randomized (α ,β ) -rounding algorithm.
This paper addresses the fair division of indivisible chores in the additive setting. We propose a polynomial-time algorithm that guarantees either a 3.69-EFX or 1.24-proportional allocation for three agents. Additionally, we introduce a new fairness concept called discrete perfect (DP), which applies to chore allocations. We prove the existence of DP for two agents by providing an efficient algorithm and extend this by proposing an algorithm that approximates DP for the general case with n agents. We believe the DP concept holds independent value and could have broader applications.
