Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees

Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a $\tilde{O}(mn)$ time algorithm that computes a spanning tree of degree at most $\Delta^* +1$ is previously known [F\"urer \& Raghavachari 1994]; here $\Delta^*$ denotes the minimum tree degree of all the spanning trees. In this paper we give the first near-linear time approximation algorithm for this problem. Specifically speaking, we propose an $\tilde{O}(\frac{1}{\epsilon^7}m)$ time algorithm that computes a spanning tree with tree degree $(1+\epsilon)\Delta^* + O(\frac{1}{\epsilon^2}\log n)$ for any constant $\epsilon \in (0,\frac{1}{6})$. Thus, when $\Delta^*=\omega(\log n)$, we can achieve approximate solutions with constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page

Space Saving by Dynamic Algebraization

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithm based on tree decompositions in polynomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof such that the dynamic programming algorithm runs in time $O^*(2^h)$, where $h$ is the maximum number of vertices in the union of bags on the root to leaf paths on a given tree decomposition, which is a parameter closely related to the tree-depth of a graph. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning problems.Comment: 14 pages, 1 figur

An Efficient Local Search for Partial Latin Square Extension Problem

A partial Latin square (PLS) is a partial assignment of n symbols to an nxn grid such that, in each row and in each column, each symbol appears at most once. The partial Latin square extension problem is an NP-hard problem that asks for a largest extension of a given PLS. In this paper we propose an efficient local search for this problem. We focus on the local search such that the neighborhood is defined by (p,q)-swap, i.e., removing exactly p symbols and then assigning symbols to at most q empty cells. For p in {1,2,3}, our neighborhood search algorithm finds an improved solution or concludes that no such solution exists in O(n^{p+1}) time. We also propose a novel swap operation, Trellis-swap, which is a generalization of (1,q)-swap and (2,q)-swap. Our Trellis-neighborhood search algorithm takes O(n^{3.5}) time to do the same thing. Using these neighborhood search algorithms, we design a prototype iterated local search algorithm and show its effectiveness in comparison with state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver.Comment: 17 pages, 2 figure

The number of matchings in random graphs

We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical Mechanic

From Loom to Machine: Tibetan Aprons and the Configuration of Place

In this paper I examine how objects become connected to place in complex and contradictory ways. Over the past ten to fi fteen years, rapid transformations in Chinese manufacturing and transportation networks have signifi cantly altered the production, marketing, and consumption of commodities made in the Tibet Autonomous Region and traded in Kalimpong, India, and Kathmandu, Nepal. In an attempt to connect the ethnographic study of material culture with more macrolevel processes of geoeconomic change, I begin the piece with an examination of the changing production, materials, and styles of a very specifi c commodity, the Tibetan women’s apron. I then explore traders’ narratives about the values of handmade, machine-made, wool, and synthetic commodities, arguing that we ought to look beyond dichotomies of ‘old’ versus ‘new’ or ‘authentic’ versus ‘inauthentic’ objects to show in detail how the attachment of commodities to representations of place fi gures importantly in the contemporary study both of globalization and uneven development. Finally, I suggest that Karl Marx’s notion of dead labor is useful in analyzing the recent move towards the revitalization of Tibetan wool for both the domestic Chinese industry and the global tourist industry

Making subaltern shikaris: histories of the hunted in colonial central India

Academic histories of hunting or shikar in India have almost entirely focused on the sports hunting of British colonists and Indian royalty. This article attempts to balance this elite bias by focusing on the meaning of shikar in the construction of the Gond ‘tribal’ identity in late nineteenth and early twentieth-century colonial central India. Coining the term ‘subaltern shikaris’ to refer to the class of poor, rural hunters, typically ignored in this historiography, the article explores how the British managed to use hunting as a means of state penetration into central India’s forest interior, where they came to regard their Gond forest-dwelling subjects as essentially and eternally primitive hunting tribes. Subaltern shikaris were employed by elite sportsmen and were also paid to hunt in the colonial regime’s vermin eradication programme, which targeted tigers, wolves, bears and other species identified by the state as ‘dangerous beasts’. When offered economic incentives, forest dwellers usually willingly participated in new modes of hunting, even as impact on wildlife rapidly accelerated and became unsustainable. Yet as non-indigenous approaches to nature became normative, there was sometimes also resistance from Gond communities. As overkill accelerated, this led to exclusion of local peoples from natural resources, to their increasing incorporation into dominant political and economic systems, and to the eventual collapse of hunting as a livelihood. All of this raises the question: To what extent were subaltern subjects, like wildlife, ‘the hunted’ in colonial India

Introduction: Human ecology in the Himalaya

Knowledge of human adaptation in the Himalayas has developed more slowly than that for other world mountain systems. At the same time, the opening of the region to research has focused attention toward description in a “natural history” mode until quite recently. Where these studies have addressed issues of adaptation they have tended to do so more as a heuristic tool rather than in terms of contributing to the development of adaptive perspectives from a uniquely Himalayan vantage point. The contributions to this special issue suggest some of Himalayan cultural ecology's new themes as it more directly assumes a truly processual approach that incorporates the individual and domestic dimensions of adaptation within historical and social contexts .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44482/1/10745_2004_Article_BF00889710.pd