A tournament is an orientation of a complete graph. It is well-known that any tournament has a vertex from which every other vertex can be reached by a path of length at most 2. Such a vertex is called a king or a 2-king. It is also known that to find such a vertex, Ωn4/3 queries (to the adjacency matrix) are necessary and On3/2 probes are sufficient. It is a long standing open problem to narrow this gap between the upper and lower bound. We first show that – the adversary Ajtai et al. (2016) and Shen et al. (2003) used to prove the known Ωn4/3 lower bound cannot be used to prove a better lower bound, by giving an algorithm that achieves the bound against the same adversary; Clearly in any tournament there is a vertex from which every other vertex is reachable by a path of length at most d for any d≥2 and such a vertex is called a d-king. The bounds for finding a 2-king have been generalized (Ajtai et al., 2016) to obtain generalized upper and lower bounds to find a d-king. We show that – our algorithm against the weak adversary works against such an adversary for finding d-kings too. More generally, if we can find a 2-king in On4/3 time, then we can find a d-king in asymptotically optimal time for any d≥2. This was conjectured in an earlier paper. Then we address the complexity of finding a set of d-kings, i.e. a small subset of vertices such that every vertex is reachable from one of them by a path of length at most d. Such a set is called a d-cover. – We generalize the lower bound for finding a d-king to give a lower bound for finding k sized d-covers. We complement it with an algorithm matching this bound for k∈Ω(lgn). For d=1 for example, our results imply that we can find a (lgn−lglgn+k)-sized dominating set in On2/k time and that this bound is optimal. Finally we develop a dynamic data structure so that whenever a new vertex is added to the tournament, we can find a king of the new tournament in O(n) time.

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for $$d\text {-}\textsc {Hitting Set}{}$$ that runs in time $$n^{{{\,\mathrm{O}\,}}{(d^2 + (d / \epsilon ))}}$$ , uses $${{\,\mathrm{O}\,}}{((d^2 + (d / \epsilon ))\log {n})}$$ bits of space, and achieves an approximation ratio of $${{\,\mathrm{O}\,}}{((d / \epsilon ) n^{\epsilon })}$$ for any positive $$\epsilon \le 1$$ and any $$d \in {\mathbb {N}}$$ . In particular, this yields a factor- $${{\,\mathrm{O}\,}}{(\log {n})}$$ approximation algorithm which runs in time $$n^{{{\,\mathrm{O}\,}}{(\log {n})}}$$ and uses $${{\,\mathrm{O}\,}}{(\log ^2{n})}$$ bits of space (for constant d). As a corollary, we obtain similar bounds for $$\textsc {Vertex Cover}{}$$ and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for $$\textsc {Vertex Cover}{}$$ on graphs with maximum degree $$\varDelta$$ , and an algorithm for computing maximal independent sets, both of which run in time $$n^{{{\,\mathrm{O}\,}}{(\varDelta )}}$$ and use $${{\,\mathrm{O}\,}}{(\varDelta \log {n})}$$ bits of space. For the more general $$d\text {-}\textsc {Hitting Set}{}$$ problem, we devise a factor-d approximation algorithm which runs in time $$n^{{{\,\mathrm{O}\,}}{(d{\delta }^2)}}$$ and uses $${{\,\mathrm{O}\,}}{(d {\delta }^2 \log {n})}$$ bits of space on set families where each element appears in at most $$\delta$$ sets. For $$\textsc {Independent Set}{}$$ restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses $${{\,\mathrm{O}\,}}{(\log {n})}$$ bits of space. We also devise a factor- $${{\,\mathrm{O}\,}}{(d^2)}$$ approximation algorithm for $$\textsc {Dominating Set}{}$$ on d-degenerate graphs which runs in time $$n^{{{\,\mathrm{O}\,}}{(\log {n})}}$$ and uses $${{\,\mathrm{O}\,}}{(\log ^2{n})}$$ bits of space. For d-regular graphs, we show how a known randomized factor- $${{\,\mathrm{O}\,}}{(\log {d})}$$ approximation algorithm can be derandomized to run in time $$n^{{{\,\mathrm{O}\,}}{(1)}}$$ and use $${{\,\mathrm{O}\,}}{(\log n)}$$ bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.

In this work, we initiate the study of the Min-Ones d-SAT problem in the parameterized streaming model. An instance of the problem consists of a d-CNF formula F and an integer k, and the objective is to determine if F has a satisfying assignment which sets at most k variables to 1. In the parameterized streaming model, input is provided as a stream, just as in the usual streaming model. A key difference is that the bound on the read-write memory available to the algorithm is O(f(k) log n) (f: N -> N, a computable function) as opposed to the O(log n) bound of the usual streaming model. The other important difference is that the number of passes the algorithm makes over its input must be a (preferably small) function of k. We design a (k + 1)-pass parameterized streaming algorithm that solves Min-Ones d-SAT (d >= 2) using space O((kd^(ck) + k^d)log n) (c > 0, a constant) and a (d + 1)^k-pass algorithm that uses space O(k log n). We also design a streaming kernelization for Min-Ones 2-SAT that makes (k + 2) passes and uses space O(k^6 log n) to produce a kernel with O(k^6) clauses. To complement these positive results, we show that any k-pass algorithm for or Min-Ones d-SAT (d >= 2) requires space Omega(max{n^(1/k) / 2^k, log(n / k)}) on instances (F, k). This is achieved via a reduction from the streaming problem POT Pointer Chasing (Guha and McGregor [ICALP 2008]), which might be of independent interest. Given this, our (k + 1)-pass parameterized streaming algorithm is the best possible, inasmuch as the number of passes is concerned. In contrast to the results of Fafianie and Kratsch [MFCS 2014] and Chitnis et al. [SODA 2015], who independently showed that there are 1-pass parameterized streaming algorithms for Vertex Cover (a restriction of Min-Ones 2-SAT), we show using lower bounds from Communication Complexity that for any d >= 1, a 1-pass streaming algorithm for Min-Ones d-SAT requires space Omega(n). This excludes the possibility of a 1-pass parameterized streaming algorithm for the problem. Additionally, we show that any p-pass algorithm for the problem requires space Omega(n/p).