Optimal Distributed Covering Algorithms

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+epsilon). Let Delta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log{Delta} / log log Delta) rounds, for constants epsilon in (0,1] and f in N^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and epsilon, our algorithm improves over the (f+epsilon)-approximation algorithm of [Fabian Kuhn et al., 2006] whose running time is O(log Delta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(f log n) rounds, improving over the classical result of [Samir Khuller et al., 1994] that achieves a running time of O(f log^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [Koufogiannakis and Young, 2011]. We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f+epsilon)-approximate integral solution in O((1+f/log n)* ((log Delta)/(log log Delta) + (f * log M)^{1.01}* log epsilon^{-1}* (log Delta)^{0.01})) rounds, where f bounds the number of variables in a constraint, Delta bounds the number of constraints a variable appears in, and M=max {1, ceil[1/a_{min}]}, where a_{min} is the smallest normalized constraint coefficient. This improves over the results of [Fabian Kuhn et al., 2006] for the integral case, which combined with rounding achieves the same guarantees in O(epsilon^{-4}* f^4 * log f * log(M * Delta)) rounds

Kernelization for Finding Lineal Topologies (Depth-First Spanning Trees) with Many or Few Leaves

For a given graph $G$, a depth-first search (DFS) tree $T$ of $G$ is an $r$-rooted spanning tree such that every edge of $G$ is either an edge of $T$ or is between a \textit{descendant} and an \textit{ancestor} in $T$. A graph $G$ together with a DFS tree is called a \textit{lineal topology} $\mathcal{T} = (G, r, T)$. Sam et al. (2023) initiated study of the parameterized complexity of the \textsc{Min-LLT} and \textsc{Max-LLT} problems which ask, given a graph $G$ and an integer $k\geq 0$, whether $G$ has a DFS tree with at most $k$ and at least $k$ leaves, respectively. Particularly, they showed that for the dual parameterization, where the tasks are to find DFS trees with at least $n-k$ and at most $n-k$ leaves, respectively, these problems are fixed-parameter tractable when parameterized by $k$. However, the proofs were based on Courcelle's theorem, thereby making the running times a tower of exponentials. We prove that both problems admit polynomial kernels with \Oh(k^3) vertices. In particular, this implies FPT algorithms running in k^{\Oh(k)}\cdot n^{O(1)} time. We achieve these results by making use of a \Oh(k)-sized vertex cover structure associated with each problem. This also allows us to demonstrate polynomial kernels for \textsc{Min-LLT} and \textsc{Max-LLT} for the structural parameterization by the vertex cover number.Comment: 16 pages, accepted for presentation at FCT 202

Settlement Fund Circulation Problem

In the economic activities, the central bank has an important role to cover payments of banks, when they are short of funds to clear their debts. For this purpose, the central bank timely puts funds so that the economic activities go smooth. Since payments in this mechanism are processed sequentially, the total amount of funds put by the central bank critically depends on the order of the payments. Then an interest goes to the amount to prepare if the order of the payments can be controlled by the central bank, or if it is determined under the worst case scenario. This motivates us to introduce a brand-new problem, which we call the settlement fund circulation problem. The problems are formulated as follows: Let G=(V,A) be a directed multigraph with a vertex set V and an arc set A. Each arc ain A is endowed debt d(a)ge 0, and the debts are settled sequentially under a sequence pi of arcs. Each vertex vin V is put fund in the amount of p_{pi}(v)ge 0 under the sequence. The minimum/maximum settlement fund circulation problem (Min-SFC/Max-SFC) in a given graph G with debts d: Arightarrow mathbb{R}_{+}cup {0} asks to find a bijection pi:Ato {1,2,dots,|A|} that minimizes/maximizes the total funds sum _{vin V}p_{pi }(v). In this paper, we show that both Min-SFC and Max-SFC are NP-hard; in particular, Min-SFC is (I) strongly NP-hard even if G is (i) a multigraph with |V|=2 or (ii) a simple graph with treewidth at most two,and is (II) (not necessarily strongly) NP-hard for simple trees of diameter four, while it is solvable in polynomial time for stars. Also, we identify several polynomial time solvable cases for both problems

Parameterized Streaming Algorithms for Min-Ones d-SAT

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)

Stackelberg Max Closure with Multiple Followers

In a Stackelberg max closure game, we are given a digraph whose vertices correspond to projects from which firms can choose and whose arcs represent precedence constraints. Some projects are under the control of a leader who sets prices in the first stage of the game, while in the second stage, the firms choose a feasible subset of projects of maximum value. For a single follower, the leader’s problem of finding revenue-maximizing prices can be solved in strongly polynomial time. In this paper, we focus on the setting with multiple followers and distinguish two situations. In the case in which only one copy of each project is available (limited supply), we show that the two-follower problem is solvable in strongly polynomial time, whereas the problem with three or more followers is NP-hard. In the case of unlimited supply, that is, when sufficient copies of each project are available, we show that the two-follower problem is already APX-hard. As a side result, we prove that Stackelberg min vertex cover on bipartite graphs with a single follower is APX-hard

A Constant Factor Approximation for Capacitated Min-Max Tree Cover

Given a graph G = (V,E) with non-negative real edge lengths and an integer parameter k, the (uncapacitated) Min-Max Tree Cover problem seeks to find a set of at most k trees which together span V and each tree is a subgraph of G. The objective is to minimize the maximum length among all the trees. In this paper, we consider a capacitated generalization of the above and give the first constant factor approximation algorithm. In the capacitated version, there is a hard uniform capacity (?) on the number of vertices a tree can cover. Our result extends to the rooted version of the problem, where we are given a set of k root vertices, R and each of the covering trees is required to include a distinct vertex in R as the root. Prior to our work, the only result known was a (2k-1)-approximation algorithm for the special case when the total number of vertices in the graph is k? [Guttmann-Beck and Hassin, J. of Algorithms, 1997]. Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al.,OR Letters 2004] [Khani et al.,Algorithmica 2010]. Instead, we use Steiner trees that cover ? vertices along with an iterative refinement procedure that ensures that the output trees have low cost and the vertices are well distributed among the trees