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The fair k -center problem has been paid lots of attention recently. In the fair k -center problem, we are given a set X of points in a metric space and a parameter k ∈ Z + , where the points in X are divided into several groups, and each point is assigned a color to denote which group it is in. The goal is to partition X into k clusters such that the number of cluster centers with each color is equal to a given value, and the k -center problem objective is minimized. In this paper, we consider the fair k -center problem with outliers and capacity constraints, denoted as the fair k -center with outliers (F k CO) problem and the capacitated fair k -center (CF k C) problem, respectively. The outliers constraints allow up to z outliers to be discarded when computing the objective function, while the capacity constraints require that each cluster has size no more than L. In this paper, we design an Fixed-Parameter Tractability (FPT) approximation algorithm and a polynomial approximation algorithm for the above two problems. In particular, our algorithms give (1 + ϵ) -approximations with FPT time for the F k CO and CF k C problems in doubling metric space. Moreover, we also propose a 3-approximation algorithm in polynomial time for the F k CO problem with some reasonable assumptions. [ABSTRACT FROM AUTHOR]

The Restricted Subset Feedback Vertex Set problem (R-SFVS) takes a graph G = (V , E) , a terminal set T ⊆ V , and an integer k as the input. The task is to determine whether there exists a subset S ⊆ V ∖ T of at most k vertices, after deleting which no terminal in T is contained in a cycle in the remaining graph. R-SFVS is NP -complete even when the input graph is restricted to split graphs. In this paper, we mainly show that R-SFVS in chordal and split graphs can be solved in O (1.1550 | V |) time and exponential space or in O (1.1605 | V |) time and polynomial space, significantly improving all previous results. As a by-product, we show that the Maximum Independent Set problem parameterized by the edge clique cover number is fixed-parameter tractable. Furthermore, by using a simple reduction from R-SFVS to Vertex Cover , we obtain an O ⁎ (1.2738 k) -time parameterized algorithm and a tight O (k 2) -kernel for R-SFVS in chordal and split graphs. [ABSTRACT FROM AUTHOR]

We revisit the maximum satisfiability problem (Max-SAT) in the data stream model. In this problem, the stream consists of m clauses that are disjunctions of literals drawn from n Boolean variables. The objective is to find an assignment to the variables that maximizes the number of satisfied clauses. Chou et al. (FOCS 2020) showed that Ω (n) space is necessary to yield a 2 / 2 + ε approximation of the optimum value; they also presented an algorithm that yields a 2 / 2 − ε approximation of the optimum value using O (ε − 2 log ⁡ n) space. In this paper, we not only focus on approximating the optimum value, but also on obtaining the corresponding Boolean assignment using sublinear o (m n) space. We present randomized single-pass algorithms that w.h.p.1 yield: • A 1 − ε approximation using O ˜ (n / ε 3) space and exponential post-processing time • A 3 / 4 − ε approximation using O ˜ (n / ε) space and polynomial post-processing time. Our ideas also extend to dynamic streams. However, we show that the streaming k -SAT problem, which asks whether one can satisfy all size- k input clauses, must use Ω (n k) space. We also consider the related minimum satisfiability problem (Min-SAT), introduced by Kohli et al. (SIAM J. Discrete Math. 1994), that asks to find an assignment that minimizes the number of satisfied clauses. For this problem, we give a O ˜ (n 2 / ε 2) space algorithm, which is sublinear when m = ω (n) , that yields an α + ε approximation where α is the approximation guarantee of the offline algorithm. If each variable appears in at most f clauses, we show that a 2 f n approximation using O ˜ (n) space is possible. Finally, for the Max-AND-SAT problem where clauses are conjunctions of literals, we show that any single-pass algorithm that approximates the optimal value up to a factor better than 1/2 with success probability at least 2/3 must use Ω (m n) space. [ABSTRACT FROM AUTHOR]

To solve reachability is to determine whether there is a path from one vertex to the other in a graph. Standard graph traversal algorithms such as DFS and BFS take linear time to solve reachability; however, their space complexity is also linear. On the other hand, Savitch's algorithm takes quasipolynomial time, although the space-bound is O (log 2 ⁡ n). In this paper, we study space-efficient algorithms for deciding reachability that runs in polynomial time. We show a polynomial-time algorithm that solves reachability in directed graphs using O (w log ⁡ n) space. Our algorithm requires access to a tree decomposition of width w for the underlying undirected graph of the input. This requirement can be waived for graphs for which recursive balanced vertex separators can be computed space-efficiently. [ABSTRACT FROM AUTHOR]