Constant-Factor FPT Approximation for Capacitated k-Median

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon

Fast and Deterministic Approximations for k-Cut

In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time? We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut

On Computational Power of Quantum Read-Once Branching Programs

In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting technique, we show that any Boolean function with linear polynomial presentation can be computed by a quantum read-once branching program using a relatively small (usually logarithmic in the size of input) number of qubits. Then we show that the described class of Boolean functions is closed under the polynomial projections.Comment: In Proceedings HPC 2010, arXiv:1103.226

Subexponential parameterized algorithms for graphs of polynomial growth

We show that for a number of parameterized problems for which only $2^{O(k)} n^{O(1)}$ time algorithms are known on general graphs, subexponential parameterized algorithms with running time $2^{O(k^{1-\frac{1}{1+\delta}} \log^2 k)} n^{O(1)}$ are possible for graphs of polynomial growth with growth rate (degree) $\delta$, that is, if we assume that every ball of radius $r$ contains only $O(r^\delta)$ vertices. The algorithms use the technique of low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for planar graphs; here we show how this strategy can be made to work for graphs with polynomial growth. Formally, we prove that, given a graph $G$ of polynomial growth with growth rate $\delta$ and an integer $k$, one can in randomized polynomial time find a subset $A \subseteq V(G)$ such that on one hand the treewidth of $G[A]$ is $O(k^{1-\frac{1}{1+\delta}} \log k)$, and on the other hand for every set $X \subseteq V(G)$ of size at most $k$, the probability that $X \subseteq A$ is $2^{-O(k^{1-\frac{1}{1+\delta}} \log^2 k)}$. Together with standard dynamic programming techniques on graphs of bounded treewidth, this statement gives subexponential parameterized algorithms for a number of subgraph search problems, such as Long Path or Steiner Tree, in graphs of polynomial growth. We complement the algorithm with an almost tight lower bound for Long Path: unless the Exponential Time Hypothesis fails, no parameterized algorithm with running time $2^{k^{1-\frac{1}{\delta}-\varepsilon}}n^{O(1)}$ is possible for any $\varepsilon > 0$ and an integer $\delta \geq 3$

Subexponential Parameterized Algorithms for Graphs of Polynomial Growth

We show that for a number of parameterized problems for which only 2^{O(k)} n^{O(1)} time algorithms are known on general graphs, subexponential parameterized algorithms with running time 2^{O(k^{1-1/(1+d)} log^2 k)} n^{O(1)} are possible for graphs of polynomial growth with growth rate (degree) d, that is, if we assume that every ball of radius r contains only O(r^d) vertices. The algorithms use the technique of low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for planar graphs; here we show how this strategy can be made to work for graphs of polynomial growth. Formally, we prove that, given a graph G of polynomial growth with growth rate d and an integer k, one can in randomized polynomial time find a subset A of V(G) such that on one hand the treewidth of G[A] is O(k^{1-1/(1+d)} log k), and on the other hand for every set X of vertices of size at most k, the probability that X is a subset of A is 2^{-O(k^{1-1/(1+d)} log^2 k)}. Together with standard dynamic programming techniques on graphs of bounded treewidth, this statement gives subexponential parameterized algorithms for a number of subgraph search problems, such as Long Path or Steiner Tree, in graphs of polynomial growth. We complement the algorithm with an almost tight lower bound for Long Path: unless the Exponential Time Hypothesis fails, no parameterized algorithm with running time 2^{k^{1-1/d-epsilon}}n^{O(1)} is possible for any positive epsilon and any integer d >= 3