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Register a new userParameterized Algorithms for Maximum Cut with Connectivity Constraints
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Tesshu Hanaka
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We study two variants of textsc{Maximum Cut}, which we call textsc{Connected Maximum Cut} and textsc{Maximum Minimal Cut}, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas textsc{Maximum Cut} on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size.
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Given a graph $G=(V,E)$, capacities $w(e)$ on edges, and a subset of terminals $mathcal{T} subseteq V: |mathcal{T}| = k$, a mimicking network for $(G,mathcal{T})$ is a graph $(H,w)$ that contains copies of $mathcal{T}$ and preserves the value of minimum cuts separating any subset $A, B subseteq mathcal{T}$ of terminals. Mimicking networks of size $2^{2^k}$ are known to exist and can be constructed algorithmically, while the best known lower bound is $2^{Omega(k)}$; therefore, an exponential size is required if one aims at preserving cuts exactly. In this paper, we study mimicking networks that preserve connectivity of the graph exactly up to the value of $c$, where $c$ is a parameter. This notion of mimicking network is sufficient for some applications, as we will elaborate. We first show that a mimicking of size $3^c cdot k$ exists, that is, we can preserve cuts with small capacity using a network of size linear in $k$. Next, we show an algorithm that finds such a mimicking network in time $2^{O(c^2)} operatorname{poly}(m)$.
Let $G$ be an $n$-vertex graph with $m$ edges. When asked a subset $S$ of vertices, a cut query on $G$ returns the number of edges of $G$ that have exactly one endpoint in $S$. We show that there is a bounded-error quantum algorithm that determines all connected components of $G$ after making $O(log(n)^6)$ many cut queries. In contrast, it follows from results in communication complexity that any randomized algorithm even just to decide whether the graph is connected or not must make at least $Omega(n/log(n))$ many cut queries. We further show that with $O(log(n)^8)$ many cut queries a quantum algorithm can with high probability output a spanning forest for $G$. En route to proving these results, we design quantum algorithms for learning a graph using cut queries. We show that a quantum algorithm can learn a graph with maximum degree $d$ after $O(d log(n)^2)$ many cut queries, and can learn a general graph with $O(sqrt{m} log(n)^{3/2})$ many cut queries. These two upper bounds are tight up to the poly-logarithmic factors, and compare to $Omega(dn)$ and $Omega(m/log(n))$ lower bounds on the number of cut queries needed by a randomized algorithm for the same problems, respectively. The key ingredients in our results are the Bernstein-Vazirani algorithm, approximate counting with OR queries, and learning sparse vectors from inner products as in compressed sensing.
Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well known problems like Set Cover, Vertex Cover, Dominating Set and Facility Location to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (non-partial) version of all these problems have been intensively studied in planar graphs and in graphs excluding a fixed graph $H$ as a minor. However, the techniques developed for parameterized version of non-partial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems.
We consider two matrix completion problems, in which we are given a matrix with missing entries and the task is to complete the matrix in a way that (1) minimizes the rank, or (2) minimizes the number of distinct rows. We study the parameterized complexity of the two aforementioned problems with respect to several parameters of interest, including the minimum number of matrix rows, columns, and rows plus columns needed to cover all missing entries. We obtain new algorithmic results showing that, for the bounded domain case, both problems are fixed-parameter tractable with respect to all aforementioned parameters. We complement these results with a lower-bound result for the unbounded domain case that rules out fixed-parameter tractability w.r.t. some of the parameters under consideration.
Given a graph $G=(V,E)$ with two distinguished vertices $s,tin V$ and an integer parameter $L>0$, an {em $L$-bounded cut} is a subset $F$ of edges (vertices) such that the every path between $s$ and $t$ in $Gsetminus F$ has length more than $L$. The task is to find an $L$-bounded cut of minimum cardinality. Though the problem is very simple to state and has been studied since the beginning of the 70s, it is not much understood yet. The problem is known to be $cal{NP}$-hard to approximate within a small constant factor even for $Lgeq 4$ (for $Lgeq 5$ for the vertex cuts). On the other hand, the best known approximation algorithm for general graphs has approximation ratio only $mathcal{O}({n^{2/3}})$ in the edge case, and $mathcal{O}({sqrt{n}})$ in the vertex case, where $n$ denotes the number of vertices. We show that for planar graphs, it is possible to solve both the edge- and the vertex-version of the problem optimally in time $mathcal{O}(L^{3L}n)$. That is, the problem is fixed parameter tractable (FPT) with respect to $L$ on planar graphs. Furthermore, we show that the problem remains FPT even for bounded genus graphs, a super class of planar graphs. Our second contribution deals with approximations of the vertex version of the problem. We describe an algorithm that for a given a graph $G$, its tree decomposition of treewidth $tau$ and vertices $s$ and $t$ computes a $tau$-approximation of the minimum $L$-bounded $s-t$ vertex cut; if the decomposition is not given, then the approximation ratio is $mathcal{O}(tau sqrt{log tau})$. For graphs with treewidth bounded by $mathcal{O}(n^{1/2-epsilon})$ for any $epsilon>0$, but not by a constant, this is the best approximation in terms of~$n$ that we are aware of.
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