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.
In this paper, we study the online version of generalized last open-end bin packing problem. In this problem items arrive sequentially and need to be packed into a bin with a load strictly less than 1. If no such bin exists a new one needs to be opened. We introduce a unit overload cost c. Each bin has a unit capacity and unit cost. If the total size of the items packed into the bin is greater than 1, an overload cost is needed for the overloaded portion, and the objective changes from packing all arriving items as required using the minimized number of bins to minimizing the total cost of all bins used. In this paper, we use an algorithm based on the First Fit strategy for this problem, and proved that the asymptotic competitive ratio of this algorithm is at most 2/1 + c when 0 ≤ c ≤ 1 and at most 2c/1 + c when c > 1 .
With the integration of next-generation mobile communication systems and artificial intelligence, multi-robot systems utilize integrated sensing and communication (ISAC) technologies to meet the perception and communication needs for high performance, stability, and security. We introduce distributed communication strategies to enhance multi-robot cooperation, especially in sequential decision-making and complex adversarial environments. In this paper, we model these problems as a robust sequential submodular optimization framework, which restricts the multiple robots to satisfy the partition matroid constraints. We adopt the divide-and-conquer strategy to design the Distributed Robust Sequential Submodular Maximization (DRSSM) algorithm to enhance the information sharing and cooperative perception of robots. The DRSSM algorithm is proven to obtain curvature-dependent approximation guarantees. The numerical experiments demonstrate that the algorithm outperforms the centralized greedy algorithm and the centralized robust algorithm in solution quality, runtime, and robustness under attack.
This paper analyzes strategic transaction submission behavior in Ethereum’s EIP-1559, focusing on non-myopic users who optimize costs over finite horizons. While prior work assumes myopic users who schedule their transactions in the immediate block, we formalize the dynamic decision-making of non-myopic bidders facing stochastic base fee adjustments, variable block capacities, and transaction deferral risks. We model the evolving base fee and user interactions as a semi-Markov process (sMP) and frame the optimal bidding strategy as a finite-horizon optimal stopping problem. By establishing equivalence to a semi-Markov decision process (sMDP), we prove the existence of deterministic stationary optimal policies and characterize the value function as the minimal solution to a recursive optimality equation. A value iteration algorithm is proposed to compute ϵ -optimal stopping policies, with explicit convergence bounds dependent on the semi-Markov kernel’s properties.
In this article, an alternated inertial projection algorithm is introduced to solve the splitting feasibility problems, and the convergence of the iterative sequence generated by the proposed method is obtained under appropriate conditions. When the splitting feasibility problem satisfies some bounded linear regularity, it is proved that the generated sequences converge linearly. Finally, some numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.
Split feasibility problem where the sets C and Q are nonconvex is considered in this paper. This class of problems are widely applied in compressed sensing, matrix factorization, outlier detection and many other fields. In this setting, split feasibility problem can be formulated as a non-convex and non-smooth problem with variable separation, so that alternating minimization algorithm can be used. We focus on the alternating minimization algorithm with Bregman distance studied in Chao et al. [7]. To further enhance efficiency, Armijo-type search method and linearization technique are employed. Under suitable assumptions on the sets involved, we show that, the sequence generated by our algorithm is bounded and globally convergent when applied to the nonconvex split feasibility problem. Some mild conditions that ensure the sequence converges to the solution of problem are proposed in this paper.
For a set S of n points in three dimensions, the 2-center problem asks for two congruent balls of the minimum radius, whose union covers all points of S. We first present an O(n^2 log n) time and O(n) space algorithm for the set S of points in convex position. Our algorithm is characterized by finding a (small) set of planes such that it contains an optimal plane that partitions S into two disjoint subsets; each can be covered by a ball of the minimum radius. The convexity of S allows to efficiently enumerate all possible situations needed in computing the 2-center of S. It is further extended to solve the 2-center problem for a set of arbitrarily given points in O(n^2 log n) time and O(n) space. This significantly improves upon two previously known results; one is a deterministic algorithm with O(n^3+ϵ) running time for any ϵ > 0 [Agarwal et al., SIAM J. Computing 29 (2000) 912-953], and the other is a randomized algorithm with roughly O(n^2 log ^4 n loglog n) expected time and O(n^2) space [Agarwal et al., Comput. Geometry 46 (2013) 734-746]. Moreover, our algorithm can be easily generalized to the 2-center problem in fixed d-dimension, d ≥ 4 . For small values of d, our result also gives an improvement upon the previously known n^O(1) time bound [Agarwal and Procopiuc, Algorithmica 33 (2002) 201-226].
Given an edge-weighted undirected connected graph G = (V, E, ρ , 𝒳, 𝒴) , where ρ : E →R^+∪{ 0 } is an edge-weight function, 𝒳⊂ V is a subset of clients, and 𝒴⊂ V is a subset of candidates, and a positive integer k < |𝒴| , the k-Supplier Problem (k SP) asks for an optimal subset of 𝒴 of cardinality at most k to minimize the radius from 𝒳 to the subset. In this paper, we focus on the case of 𝒳∩𝒴∅ , 𝒳, 𝒴 , and consider the scenario where the shortest path distances d(· , · ) in G satisfy a parameterized triangle inequality between 𝒳 and 𝒴 , i.e., d(x, y) + d(y, z) ≥α· d(x, z), ∀ x, y, z ∈{ u, v, w }, x y, y z, z x , where 1 ≤α≤ 2 is a parameter, for any three distinct vertices, v, u ∈𝒳 and w ∈𝒴 . We present a two-stage dual approximation algorithm ALG for the kSP with parameter triangle inequality between 𝒳 and 𝒴 . If it stops at the end of Stage 1 then it achieves a 2/α -approximation, and if it stops at the end of Stage 2 then it achieves a (2/α ^2 + 1/α) -approximation. ALG runs in a polynomial time and the above two parameterized performance factors of it are both strictly monotonic decreasing with respect to the value of parameter α . For the kSP instances with parameterized triangle inequality having 1 < α≤ 2 , it is implied by 1 ≤2/α < 2 and 1 ≤2/α ^2 + 1/α < 3 that ALG has better approximation ratios than the previously best polynomial-time 3-approximation algorithm of Hochbaum and Shmoys (J. ACM. 33: 533–550, 1986). Furthermore, regardless of the parameterized aspect, ALG achieves a 2-approximation if it stops at the end of Stage 1 and a 3-approximation if it stops at the end of Stage 2, for the general kSP.
Given a set P of n points on which facilities can be placed and an integer k,we want to place k facilities on k points in P so that the minimum distance between facilities is maximized. The problem is called the k-dispersion problem. In this paper, we consider the 4-dispersion problem when P is a set of points on a plane (2-dimensional space). Note that the solution of the 2-dispersion problem corresponds to the diameter of P. We give an O(n^3) time algorithm to solve the 4-dispersion problem in the L_∞ metric, and an O(n^3) time algorithm to solve the 4-dispersion problem in the L_1 metric. Also, we give an O(n^3log ^2 n) time algorithm to solve the 4-dispersion problem in the L_2 metric. Also, we give faster (but complicated) O(n^2log n) -time algorithms to solve the max-min 4-dispersion problem for L_∞ metric and the L_1 metric.
Mastermind is a classic code-guessing game in which a game master secretly selects a multi-digit code with no duplicates and a codebreaker iteratively submits candidate codes. For each submission, the game master compares it to the secret code and tells the number of “hits” and “blows” as feedback. In this paper, we design a novel method to play Mastermind without a game master using card-based cryptographic techniques. In particular, we design feedback protocols and a code generation protocol, where the former outputs the number of hits and blows without revealing the code itself, and the latter outputs a sequence of cards representing a code chosen uniformly at random. By using these protocols, we can enjoy Mastermind without the need for a game master. All of the proposed protocols are extremely simple and can be executed in real gameplay. This enables people to learn fundamental principles of cryptography through playing Mastermind.
In this paper, we address three variations of the windy k-traveling salesman problem (the k-WTSP), which are related to the asymmetric traveling salesman problem. Concretely, given a weighted connected graph G=(V,E;w;r) of order n and an integer k≥ 1 , where r∈ V and a windy cost function w:A →ℝ^+ , where A={(v_i,v_j),(v_j,v_i) | v_iv_j∈ E} , i.e., for each edge v_iv_j∈ E , we denote w(v_i,v_j) to be the cost to traverse this edge v_iv_j from v_i to v_j and w(v_j,v_i) to be the cost to traverse the same edge v_iv_j from v_j to v_i , respectively, the k-WTSP is asked to find k circuits, starting and ending at r, such that each vertex of G is contained in at least one of these k circuits, the objective is to minimize the total costs of these k circuits. In addition, the minmax windy k-traveling salesman problem (the minmax k-WTSP) is asked to find k circuits as mentioned-above, the objective is to minimize the maximum cost of these k circuits. Finally, the minmax windy k-circuit cover problem (the minmax k-WCCP) is asked to find k circuits such that each vertex of G is contained in at least one of these k circuits, the objective is to minimize the maximum cost of these k circuits. We have three key contributions as follows. (1) We design a 3/2(α +1) -approximation algorithm to solve the k-WTSP in time O(n^3) , where α =max{w(v_i,v_j)/w(v_j,v_i) | v_iv_j∈ E} ; (2) We present a (3/2(α +1)+2(1-1/k)) -approximation algorithm to solve the minmax k-WTSP in time O(n^3) ; (3) We provide a (3/2(α +1)+k) -approximation algorithm to solve the minmax k-WCCP in time O(n^3) .
Let H=(V,E) be a hypergraph with n vertices and m edges. A⊆ E is an eliminating edge feedback set of H if H//A has no cycle and D_c'(H) denote the minimum cardinality of an eliminating edge feedback set of H. In this paper, we prove (i) for any hypergraph H, D_c'(H)≤ n/2 . (ii) for any hypergraph H with maximum degree ≤ 2 and girth k≥ 4 , D_c'(H)≤n/4 . (iii) for any hypergraph H with maximum degree ≤ 3 and girth k≥ 3 , D_c'(H)≤n/3 . Based on the proofs, some combinatorial algorithms on the eliminating edge feedback number are designed.
Unmanned Aerial Vehicles (UAVs) have been extensively utilized in environmental modeling tasks, including forest monitoring, agricultural mapping, and disaster assessment. These applications benefit from UAVs’ ability to collect high-resolution data for purposes such as deforestation detection, crop management, and damage evaluation. However, the common assumption that UAVs are homogeneous leads to significant inaccuracies due to differences in UAV platforms and sensor characteristics. This paper investigates the deployment of heterogeneous UAVs equipped with various sensors in fire-affected regions, modeled via a point of interest framework. The goal is to minimize measurement errors by explicitly leveraging UAV heterogeneity. We introduce a weighted frame potential metric based on matrix orthogonality, and develop a randomized greedy algorithm to optimize UAV positioning. The proposed algorithm guarantees at least a 50
In this paper, we design two algorithms for the problem of maximizing the difference of two DR-submodular function g - h (DDRS) on the integer lattice, where g (x) is monotone and non-negative DR-submodular, and h (x) is monotone and non-negative submodular with curvature c_h . Combining the lattice binary search with the threshold method, we present a one pass streaming algorithm for the unconstrained problems, and analyze the performance of the algorithm and obtain a provable approximation guarantees.
We study the problem of fairly allocating a set of indivisible chores among a set of agents, who possess heterogeneous costs over the chores. We assume that agents’ costs are bounded and lie in the interval [1, k]. We aim to find both fair and efficient allocations under this setting. We first show that a k-EFX and fPO allocation can be computed in polynomial time. Next, we generalize the result to the weighted setting and show the existence of a k-WEFX and fPO allocation. Finally, for k = 2 , we can compute an EFX allocation based on the well-known round-robin algorithm.
The paper outlines a possible computability-theoretic approach to the theory of Boolean-valued models. We introduce the notion of a decidable Boolean-valued model A . This notion is a natural extension of the classical definition of a decidable structure. Intuitively speaking, in order to formally define A , one needs to (algorithmically) work only with countably many truth values taken from a computable Boolean algebra. This convention allows us to view such A through the lens of computable structure theory. We show that the introduced approach provides a rich class of computable structures. We prove that any possible computable dimension can be realized by an appropriately constructed, decidable Boolean-valued model A . Consequently, there exists a model A having computable dimension 2 (that is, A possesses precisely two computable presentations, up to computable isomorphisms). In contrast, it is known that dimension 2 can be realized neither by a classical decidable structure nor by a computable Boolean algebra.
