Maximum st-flow in directed planar graphs via shortest paths

Minimum cuts have been closely related to shortest paths in planar graphs via planar duality - so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason - so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no constraints on the source and sink. We believe this a promising avenue for developing algorithms that are more practical than the current asymptotically best algorithms for maximum st-flow.Comment: 20 pages, 4 figures. Short version to be published in proceedings of IWOCA'1

On Approximating Restricted Cycle Covers

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change

EFFECT OF GENETIC AND ENVIRONMENTAL FACTORS ON SEX RATIO IN CROSSBRED PIGS

The study was initiated with an idea to investigate few genetic and environmental factors that affect sex ratio of Khasi local and their different crossbreds with Hampshire pigs. Individual data were collected of pure Khasi local and its crossbred with 50, 75 and 87.5 % Hampshire inheritance in different seasons like rainy (July to October), summer (March-June) and winter (Nov- Feb). The sex ratio for Khasi local crossbred with 50, 75 and 87.5 % Hampshire inheritance was 1.21 ± 0.16, 1.32 ± 0.16, 1.48 ± 0.16 an 1.32 ± 0.16 respectively with an overall mean sex ratio 1.38 ± 0.16, whereas, the sex ratio for spring, rainy and winter season was 1.31 ± 0.17, 1.29 ± 0.16 and 1.32 ± 0.15, respectively. Similarly, the sex ratio for larger litters and smaller litters was 1.40 ± 0.13 and 1.45 ± 0.13 respectively. This study concludes that crossbreds at different levels of inheritance, season and litter size had no effect on sex ratio

The complexity of the Pk partition problem and related problems in bipartite graphs

International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $\alpha$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $\alpha = 1$ by Ding et al. For any constant $\alpha > 1$, we give an $O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})$-approximation algorithm. When $\alpha \geq 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $\alpha =2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $\epsilon > 0$, the problem admits no polynomial-time $2^{\log^{1-\epsilon}n}$-approximation algorithm, improving upon the $\Omega(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption)

A 2k2k-Vertex Kernel for Maximum Internal Spanning Tree

We consider the parameterized version of the maximum internal spanning tree problem, which, given an $n$-vertex graph and a parameter $k$, asks for a spanning tree with at least $k$ internal vertices. Fomin et al. [J. Comput. System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a $3k$-vertex kernel. Here we propose a novel way to use the same reduction rule, resulting in an improved $2k$-vertex kernel. Our algorithm applies first a greedy procedure consisting of a sequence of local exchange operations, which ends with a local-optimal spanning tree, and then uses this special tree to find a reducible structure. As a corollary of our kernel, we obtain a deterministic algorithm for the problem running in time $4^k \cdot n^{O(1)}$

Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

The facility location problem is an NP-hard optimization problem. Therefore, approximation algorithms are often used to solve large instances. Such algorithms often perform much better than worst-case analysis suggests. Therefore, probabilistic analysis is a widely used tool to analyze such algorithms. Most research on probabilistic analysis of NP-hard optimization problems involving metric spaces, such as the facility location problem, has been focused on Euclidean instances, and also instances with independent (random) edge lengths, which are non-metric, have been researched. We would like to extend this knowledge to other, more general, metrics. We investigate the facility location problem using random shortest path metrics. We analyze some probabilistic properties for a simple greedy heuristic which gives a solution to the facility location problem: opening the $\kappa$ cheapest facilities (with $\kappa$ only depending on the facility opening costs). If the facility opening costs are such that $\kappa$ is not too large, then we show that this heuristic is asymptotically optimal. On the other hand, for large values of $\kappa$, the analysis becomes more difficult, and we provide a closed-form expression as upper bound for the expected approximation ratio. In the special case where all facility opening costs are equal this closed-form expression reduces to $O(\sqrt[4]{\ln(n)})$ or $O(1)$ or even $1+o(1)$ if the opening costs are sufficiently small.Comment: A preliminary version accepted to CiE 201

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems