Approximating minimum power covers of intersecting families and directed edge-connectivity problems

AbstractGiven a (directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Let G=(V,E) be a graph with edge costs {c(e):e∈E} and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Powerk-Edge-Outconnected Subgraph problem we are given a root r∈V, and require that G contains k pairwise edge-disjoint rv-paths for all v∈V−r. In the Min-Powerk-Edge-Connected Subgraph problem G is required to be k-edge-connected. For k=1, these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln|V|) approximation threshold. For k=Ω(nε), they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O(kln|V|). Our algorithms are based on a more general O(ln|V|)-approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F is intersecting if X∩Y,X∪Y∈F for any intersecting X,Y∈F, and an edge set I covers F if for every X∈F there is an edge in I entering X

Geographic max-flow and min-cut under a circular disk failure model

Failures in fiber-optic networks may be caused by natural disasters, such as floods or earthquakes, as well as other events, such as an Electromagnetic Pulse (EMP) attack. These events occur in specific geographical locations, therefore the geography of the network determines the effect of failure events on the network's connectivity and capacity. In this paper we consider a generalization of the min-cut and max-flow problems under a geographic failure model. Specifically, we consider the problem of finding the minimum number of failures, modeled as circular disks, to disconnect a pair of nodes and the maximum number of failure disjoint paths between pairs of nodes. This model applies to the scenario where an adversary is attacking the network multiple times with intention to reduce its connectivity. We present a polynomial time algorithm to solve the geographic min-cut problem and develop an ILP formulation, an exact algorithm, and a heuristic algorithm for the geographic max-flow problem.National Science Foundation (U.S.) (Grant CNS-0830961)National Science Foundation (U.S.) (Grant CNS-1017714)National Science Foundation (U.S.) (Grant CNS-1017800)National Science Foundation (U.S.) (CAREER Grant 0348000)United States. Defense Threat Reduction Agency (Grant HDTRA1-07-1-0004)United States. Defense Threat Reduction Agency (Grant HDTRA-09-1-005

Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints

For a given graph $G$ with positive integral cost and delay on edges, distinct vertices $s$ and $t$, cost bound $C\in Z^{+}$ and delay bound $D\in Z^{+}$, the $k$ bi-constraint path ($k$BCP) problem is to compute $k$ disjoint $st$-paths subject to $C$ and $D$. This problem is known NP-hard, even when $k=1$ \cite{garey1979computers}. This paper first gives a simple approximation algorithm with factor-$(2,2)$, i.e. the algorithm computes a solution with delay and cost bounded by $2*D$ and $2*C$ respectively. Later, a novel improved approximation algorithm with ratio $(1+\beta,\,\max\{2,\,1+\ln\frac{1}{\beta}\})$ is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method. As a consequence, we can obtain a factor-$(1.369,\,2)$ approximation algorithm by setting $1+\ln\frac{1}{\beta}=2$ and a factor-$(1.567,\,1.567)$ algorithm by setting $1+\beta=1+\ln\frac{1}{\beta}$. Besides, by setting $\beta=0$, an approximation algorithm with ratio $(1,\, O(\ln n))$, i.e. an algorithm with only a single factor ratio $O(\ln n)$ on cost, can be immediately obtained. To the best of our knowledge, this is the first non-trivial approximation algorithm for the $k$BCP problem that strictly obeys the delay constraint.Comment: 12 page

Edge- and Node-Disjoint Paths in P Systems

In this paper, we continue our development of algorithms used for topological network discovery. We present native P system versions of two fundamental problems in graph theory: finding the maximum number of edge- and node-disjoint paths between a source node and target node. We start from the standard depth-first-search maximum flow algorithms, but our approach is totally distributed, when initially no structural information is available and each P system cell has to even learn its immediate neighbors. For the node-disjoint version, our P system rules are designed to enforce node weight capacities (of one), in addition to edge capacities (of one), which are not readily available in the standard network flow algorithms.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005

Finding k partially disjoint paths in a directed planar graph

The {\it partially disjoint paths problem} is: {\it given:} a directed graph, vertices $r_1,s_1,\ldots,r_k,s_k$, and a set $F$ of pairs $\{i,j\}$ from $\{1,\ldots,k\}$, {\it find:} for each $i=1,\ldots,k$ a directed $r_i-s_i$ path $P_i$ such that if $\{i,j\}\in F$ then $P_i$ and $P_j$ are disjoint. We show that for fixed $k$, this problem is solvable in polynomial time if the directed graph is planar. More generally, the problem is solvable in polynomial time for directed graphs embedded on a fixed compact surface. Moreover, one may specify for each edge a subset of $\{1,\ldots,k\}$ prescribing which of the $r_i-s_i$ paths are allowed to traverse this edge