Following recent advances in combining approximation algorithms with fixed-parameter tractability (FPT), we study FPT-time approximation algorithms for minimum-norm k-clustering problems, parameterized by the number k of open facilities. For the capacitated setting, we give a tight (3+ε)-approximation for the general-norm capacitated k-clustering problem in FPT-time parameterized by k and ε. Prior to our work, such a result was only known for the capacitated k-median problem [Cohen-Addad and Li, 2019]. As a special case, our result yields an FPT-time 3-approximation for capacitated k-center. The problem has not been studied in the FPT-time setting, with the previous best known polynomial-time approximation ratio being 9 [An et al., 2015]. In the uncapacitated setting, we consider the top-cn norm k-clustering problem, where the goal of the problem is to minimize the top-cn norm of the connection distance vector. Our main result is a tight (1 + 2/(ec) + ε)-approximation algorithm for the problem with c ∈ (1/e, 1]. (For the case c ≤ 1/e, there is a simple tight (3+ε)-approximation.) Our framework can be easily extended to give a tight (3, 1 + 2/e + ε)-bi-criteria approximation for the (k-center, k-median) problem in FPT time, improving the previous best polynomial-time (4, 8) guarantee [Soroush Alamdari and David B. Shmoys, 2017]. All results are based on a unified framework: computing a (1+ε)-approximate solution using O((k log n)/ε) facilities S via LP rounding, sampling a few client representatives R based on the solution S, guessing a few pivots from S ∪ R and some radius information on the pivots, and solving the problem using the guesses. We believe this framework can lead to further results on k-clustering problems.

In this paper, we study the parameterized complexity of local search, whose goal is to find a good nearby solution from the given current solution. Formally, given an optimization problem where the goal is to find the largest feasible subset S of a universe U, the new input consists of a current solution P (not necessarily feasible) as well as an ordinary input for the problem. Given the existence of a feasible solution S^*, the goal is to find a feasible solution as good as S^* in parameterized time f(k)⋅n^O(1), where k denotes the distance |PΔ S^*|. This model generalizes numerous classical parameterized optimization problems whose parameter k is the minimum number of elements removed from U to make it feasible, which corresponds to the case P = U. We apply this model to widely studied Constraint Satisfaction Problems (CSPs), where U is the set of constraints, and a subset U' of constraints is feasible if there is an assignment to the variables satisfying all constraints in U'. We give a complete characterization of the parameterized complexity of all boolean-alphabet symmetric CSPs, where the predicate’s acceptance depends on the number of true literals.

Relaxations for the constraint satisfaction problem (CSP) include bounded width, linear program (LP), semidefinite program (SDP), affine integer program (AIP), and the combined LP+AIP of Brakensiek, Guruswami, Wrochna, and Živný (SICOMP 2020). Tightening relaxations systematically leads to hierarchies and stronger algorithms. For the LP+AIP hierarchy, a constant level lower bound for approximate graph coloring was given by Ciardo and Živný (STOC 2023). We prove the first linear (and hence optimal) level lower bound for LP+AIP and its stronger variant, SDP+AIP. For each hierarchy, our bound holds for random instances of a broad class of CSPs that we call 𝜏-wise neutral. We extend to other hierarchies the LP lower bound techniques in Benabbas, Georgiou, Magen and Tulsiani (ToC 2012) and Kothari, Mori, O’Donnell, and Witmer (STOC 2017), and simplify the SDP solution construction in the latter.