اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe Minimum Wiener Connector
33
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph $G=(V,E)$ and a set $Qsubseteq V$ of query vertices, find a subgraph of $G$ that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless $mathbf{P} = mathbf{NP}$. Our main contribution is a constant-factor approximation algorithm running in time $widetilde{O}(|Q||E|)$. A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set $Q$ a small number of important vertices (i.e., vertices with high centrality).
قيم البحث
اقرأ أيضاً
We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the $1$-dimensional homology classes with $mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studi
ed in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N^omega + N^2 g)$ time, where $N$ denotes the number of simplices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $omega$ denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $tilde{O}(m^omega)$ time, where $m$ denotes the number of edges in $K$, whereas the second algorithm runs in $O(m^omega + N m^{omega-1})$ time. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^omega)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $tilde{O}(m^omega)$ time.
A recent breakthrough by Ambainis, Balodis, Iraids, Kokainis, Pr=usis and Vihrovs (SODA19) showed how to construct faster quantum algorithms for the Traveling Salesman Problem and a few other NP-hard problems by combining in a novel way quantum searc
h with classical dynamic programming. In this paper, we show how to apply this approach to the minimum Steiner tree problem, a well-known NP-hard problem, and construct the first quantum algorithm that solves this problem faster than the best known classical algorithms. More precisely, the complexity of our quantum algorithm is $mathcal{O}(1.812^kpoly(n))$, where $n$ denotes the number of vertices in the graph and $k$ denotes the number of terminals. In comparison, the best known classical algorithm has complexity $mathcal{O}(2^kpoly(n))$.
We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, Delta_U, Delta_V)$, where $Del
ta_U$ and $Delta_V$ denote the maximum degree in $U$ and $V$, respectively, is lexicographically minimum over all such graphs. $G$ is $k$-critical bipartite if deleting at most $k=n-m$ vertices from $U$ creates $G$ that has a complete matching, i.e., a matching of size $m$. We show that, if $m(n-m+1)/n$ is an integer, then a solution of the M$k$CBG-$(n,m)$ problem can be found among $(a,b)$-regular bipartite graphs of order $(n,m)$, with $a=m(n-m+1)/n$, and $b=n-m+1$. If $a=m-1$, then all $(a,b)$-regular bipartite graphs of order $(n,m)$ are $k$-critical bipartite. For $a<m-1$, it is not the case. We characterize the values of $n$, $m$, $a$, and $b$ that admit an $(a,b)$-regular bipartite graph of order $(n,m)$, with $b=n-m+1$, and give a simple construction that creates such a $k$-critical bipartite graph whenever possible. Our techniques are based on Halls marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.
The minimum cut problem in an undirected and weighted graph $G$ is to find the minimum total weight of a set of edges whose removal disconnects $G$. We completely characterize the quantum query and time complexity of the minimum cut problem in the ad
jacency matrix model. If $G$ has $n$ vertices and edge weights at least $1$ and at most $tau$, we give a quantum algorithm to solve the minimum cut problem using $tilde O(n^{3/2}sqrt{tau})$ queries and time. Moreover, for every integer $1 le tau le n$ we give an example of a graph $G$ with edge weights $1$ and $tau$ such that solving the minimum cut problem on $G$ requires $Omega(n^{3/2}sqrt{tau})$ many queries to the adjacency matrix of $G$. These results contrast with the classical randomized case where $Omega(n^2)$ queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when $G$ has $m$ edges the classical randomized complexity of the minimum cut problem is $tilde Theta(m)$. We show that the quantum query and time complexity are $tilde O(sqrt{mntau})$ and $tilde O(sqrt{mntau} + n^{3/2})$, respectively, where again the edge weights are between $1$ and $tau$. For dense graphs we give lower bounds on the quantum query complexity of $Omega(n^{3/2})$ for $tau > 1$ and $Omega(tau n)$ for any $1 leq tau leq n$. Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Kargers tree packing technique (STOC 1996).
A stable cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. In this paper we study Min Sta
ble Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We show that the problem is weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this with a pseudo-polynomial DP algorithm running in time $(Deltacdot W)^{O(tw)}n^{O(1)}$, where $tw$ is the treewidth, $Delta$ the maximum degree, and $W$ the maximum weight. On the other hand, bounding $Delta$ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both $tw+Delta$ and obtain an FPT algorithm running in time $2^{O(Delta tw)}(n+log W)^{O(1)}$. Our main result is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in $(nW)^{o(pw)}$ or $2^{o(Delta pw)}(n+log W)^{O(1)}$, then the ETH is false. Complementing this, we show that we can obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions. Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time $Delta^{O(tw)}n^{O(1)}$. We show that this is also probably essentially optimal: an algorithm running in $n^{o(pw)}$ would contradict the ETH.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
