On Revealed Preference and Indivisibilities

We consider a market model in which all commodities are inherently indivisible and thus are traded in integer quantities. We ask whether a finite set of price-quantity observations satisfying the Generalized Axiom of Revealed Preference (GARP) is consistent with utility maximization. Although familiar conditions such as non-satiation become meaningless in the current discrete model, by refining the standard notion of demand set we show that Afriat's celebrated theorem still holds true. Exploring network structure and a new and easy-to-use variant of GARP, we propose an elementary, simple, intuitive, combinatorial, and constructive proof for the result.Afriat's theorem, GARP, indivisibilities, revealed preference.

A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions

This paper presents the first combinatorial polynomial-time algorithm for minimizing submodular set functions, answering an open question posed in 1981 by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the largest length of the function value. The paper also presents a strongly polynomial-time version that runs in time bounded by a polynomial in the size of the underlying set independent of the function value.Comment: 17 page

Decentralized Market Processes to Stable Job Matchings with Competitive Salaries

We analyze a decentralized trading process in a basic labor market where heterogeneous firms and workers meet directly and randomly, and negotiate salaries with each other over time. Firms and workers may not have a complete picture of the entire market and can thus behave myopically in the process. Our main result establishes that, starting from an arbitrary initial market state, there exists a finite sequence of successive myopic (firm-worker) pair improvements, or bilateral trades, leading to a stable matching between firms and workers with a scheme of competitive salary offers. An important implication of this result is that a general random process where every possible bilateral trade is chosen with a positive probability converges with probability one to a competitive equilibrium of the market.Decentralized market, job matching, random path, competitive salary, stability

A note on polylinking flow networks

This is a supplementary note on M. X. Goemans, S. Iwata, and R. Zenklusen’s paper that proposes a flow model based on polylinking systems. Their flow model is a series (or tandem) connection of polylinking systems. We can consider an apparently more general model of a polylinking flow network which consists of an ordinary arc-capacitated network endowed with polylinking systems on the vertex set, one for each vertex of the network. This is a natural, apparent generalization of polymatroidal flow model of E. L. Lawler and C. U. Martel and of generalized-polymatroidal flow model of R. Hassin. We give a max-flow min-cut formula for the polylinking network flow problem and discuss some acyclic flow property of polylinking flows

A note on submodular function minimization by Chubanov’s LP algorithm

Recently Dadush et al. (2017) have devised a polynomial submodular function minimization (SFM) algorithm based on their LP algorithm. In the present note we also show a weakly polynomial algorithm for SFM based on the recently developed linear programming feasibility algorithm of Chubanov (2017) to stimulate further research on SFM

A Note on Integrality of Convex Polyhedra Represented by Linear Inequalities with {0,±1}-coefficients

ファイルを差し替え（2021/10/21）We consider a polyhedron P represented by linear inequalities with {0, ±1}-coefficients. We show a condition that guarantees existence of an integral vector in P, which also turns out to be an extreme point of P. We reveal how our polyhedral and geometric approach shows the recent interesting integrality results of Murota and Tamura about subdifferentials of integrally convex functions. Their proofs are algebraic, based on the Fourier-Motzkin elimination for the relevant systems of linear inequalities. Our approach provides further insight into subdifferentials of integrally convex functions to fully appreciate the integrality results of Murota and Tamura from a polyhedral and geometric point of view

Expression of MsLEC1 transgenes in alfalfa plants causes symbiotic abnormalities

Legume lectins have been proposed to have important symbiotic roles during Rhizobium-legume symbioses. To test this hypothesis, the symbiotic responses of transgenic alfalfa plants that express a portion of the putative alfalfa lectin gene MsLEC1 or MsLEC2 in either the antisense or sense orientation were analyzed following inoculation with wild-type Sinorhizobium meliloti 1021. MsLEC1-antisense (LEC1AS) plants were stunted, exhibited hypernodulation, and developed not only abnormally large nodules but also numerous small nodules, both of which senesced prematurely. MsLEC2-antisense plants were intermediate in growth and nodule number compared with LEC1AS and vector control plants. The symbiotic abnormalities of MsLEC1-sense transgene plants were similar to but milder than the responses shown by the LEC1AS plants, whereas MsLEC2-sense transgene plants exhibited symbiotic responses that were identical to those of vector and nontransgenic control plants. MsLEC1 mRNA accumulation was not detected in nodule RNA by Northern blot analysis but was localized to alfalfa nodule meristems and the adjacent cells of the invasion zone by in situ hybridization; transcripts were also detected in root meristems. A similar spatial pattern of MsLEC2 expression was found by using a whole-mount in situ hybridization procedure. Moreover, mRNAs for an orthologous lectin gene (MaLEC) were detected in white sweetclover (Melilotus alba) nodules and root tips

Sugar-binding activity of pea lectin enhances heterologous infection of transgenic alfalfa plants by Rhizobium leguminosarum biovar viciae

Transgenic alfalfa (Medicago sativa L. cv Regen) roots carrying genes encoding soybean lectin or pea (Pisum sativum) seed lectin (PSL) were inoculated with Bradyrhizobium japonicum or Rhizobium leguminosarum bv viciae, respectively, and their responses were compared with those of comparably inoculated control plants. We found that nodule-like structures formed on alfalfa roots only when the rhizobial strains produced Nod factor from the alfalfa-nodulating strain, Sinorhizobium meliloti. Uninfected nodule-like structures developed on the soybean lectin-transgenic plant roots at very low inoculum concentrations, but bona fide infection threads were not detected even when B. japonicum produced the appropriate S. meliloti Nod factor. In contrast, the PSL-transgenic plants were not only well nodulated but also exhibited infection thread formation in response to R. leguminosarum bv viciae, but only when the bacteria expressed the complete set of S. meliloti nod genes. A few nodules from the PSL-transgenic plant roots were even found to be colonized by R. leguminosarum bv viciae expressing S. meliloti nod genes, but the plants were yellow and senescent, indicating that nitrogen fixation did not take place. Exopolysaccharide appears to be absolutely required for both nodule development and infection thread formation because neither occurred in PSL-transgenic plant roots following inoculation with an Exo- R. leguminosarum bv viciae strain that produced S. meliloti Nod factor

Investigations of Rhizobium biofilm formation

The development of nitrogen-fixing nodules of the Rhizobium-legume symbiosis, especially the early stages of root hair deformation and curling, infection thread formation, and nodule initiation, has been well studied from a genetic standpoint. In contrast, the factors important for the colonization of surfaces by rhizobia, including roots-an important prerequisite for nodule formation-have not been as thoroughly investigated. We developed conditions for analyzing the ability of two fast-growing rhizobia, Sinorhizobium meliloti and Rhizobium leguminosarum bv. viciae, to produce biofilms on abiotic surfaces such as glass, plastic microtiter plates, sand and soil as a prelude to characterizing the genes important for aggregation and attachment. Factors involved in adherence to abiotic surfaces are likely to be used in rhizobial attachment to legume root cells. In this report, we show that S. meliloti exopolysaccharide-deficient mutants as well as exopolysaccharide overproducers exhibit reduced biofilm phenotypes that show parallels with their nodulation abilities. We also investigated two flagella-less S. meliloti mutants and found them to have reduced biofilming capabilities. To investigate whether there was a symbiotic phenotype, we tested one of the Fla- mutants on two different S. meliloti hosts, alfalfa and white sweetclover, and found that nodule formation was significantly delayed on the latter