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We consider a model of stable edge sets (``matchings'') in a bipartite graph $G=(V,E)$ in which the preferences for vertices of one side (``firms'') are given via choice functions subject to standard axioms of consistency, substitutability and cardinal monotonicity, whereas the preferences for the vertices of the other side (``workers'') via linear orders. For such a model, we present a combinatorial description of the structure of rotations and develop an algorithm to construct the poset of rotations, in time $O(|E|^2)$ (including oracle calls). As consequences, one can obtain a ``compact'' affine representation of stable matchings and efficiently solve some related problems. Keywords: bipartite graph, choice function, linear preferences, stable matching, affine representation, sequential choice, Comment: 26 pages. This version is fully written in English

We consider one variant of stable assignment problems in a bipartite graph endowed with nonnegative capacities on the edges and quotas on the vertices. It can be viewed as a generalization of the stable allocation problem introduced by Ba\H{\i}ou and Balinsky, which arises when strong linear orders of preferences on the vertices in the latter are replaced by weak ones. At the same time, our stability problem can be stated in the framework of a theory by Alkan and Gale on stable schedule matchings generated by choice functions of a wide scope. In our case, the choice functions are of a special, so-called mixed, type. The main content of this paper is devoted to a study of rotations in our mixed model, functions on the edges determining ``elementary'' transformations between close stable assignments. These look more sophisticated compared with rotations in the stable allocation problem (which are generated by simple cycles). We efficiently construct a poset of rotations and show that the stable assignments are in bijection with the so-called closed functions for this poset; this gives rise to a ``compact'' affine representation for the lattice of stable assignments and leads to an efficient method to find a stable assignment of minimum cost., Comment: 39 pages, 1 figure. This is the improved and extended version

We consider the stable assignment problem on a graph with nonnegative real capacities on the edges and quotas on the vertices, in which the preferences of agents are given via diversifying choice functions. We prove that for any input of the problem, there exists exactly one stable assignment, and propose a polynomial time algorithm to find it., Comment: 10 pages, 1 figure. At the end of updated version, a generalization to hypergraphs is briefly discussed

The topic of stable matchings (marriages) in a bipartite graph has become widely popular, starting with the appearance of the classical work by Gale and Shapley. We give a detailed survey on selected known results in this field that demonstrate structural, polyhedral and algorithmic properties of such matchings and their sets, providing our description with relatively short proofs., Comment: 24 pages, in Russian language, 3 figures

As a generalization of weak Bruhat orders on permutations, in 1989 Manin and Schechtman introduced the notion of a higher Bruhat order on the $d$-element subsets of a set $[n]=\{1,2,\ldots,n\}$. Among other results in this field, they proved that the set of such orders for $n,d$ fixed, endowed with natural local transformations, constitutes a poset with one minimal and one maximal elements. In this paper we consider a wider model, involving the so-called convex order on certain path systems in an acyclic directed graph, introduce local transformations, or flips, on such orders and prove that the resulting structure gives a poset with one minimal and one maximal elements as well, yielding a generalization of the above-mentioned classical result., Comment: 16 pages, 3 figures. Compared with the initial version, the last section is re-written

We consider a hypergraph (I,C), with possible multiple (hyper)edges and loops, in which the vertices $i\in I$ are interpreted as agents, and the edges $c\in C$ as contracts that can be concluded between agents. The preferences of each agent i concerning the contracts where i takes part are given by use of a choice function $f_i$ possessing the so-called path independent property. In this general setup we introduce the notion of stable network of contracts. The paper contains two main results. The first one is that a general problem on stable systems of contracts for (I,C,f) is reduced to a set of special ones in which preferences of agents are described by use of so-called weak orders, or utility functions. However, for a special case of this sort, the stability may not exist. Trying to overcome this trouble when dealing with such special cases, we introduce a weaker notion of metastability for systems of contracts. Our second result is that a metastable system always exists., Comment: 18 pages

We propose versions of higher Bruhat orders for types $B$ and $C$. This is based on a theory of higher Bruhat orders of type~A and their geometric interpretations (due to Manin--Shekhtman, Voevodskii--Kapranov, and Ziegler), and on our study of the so-called symmetric cubillages of cyclic zonotopes., Comment: 22 pages, 12 figures, a reference is added, a misprint in commutative diagram is corrected

For a positive integer $n$, a collection $S$ of subsets of $[n]=\{1,\ldots,n\}$ is called symmetric if $X\in S$ implies $X^\ast\in S$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$ (the involution $\ast$ was introduced by Karpman). Leclerc and Zelevinsky showed that the set of maximal strongly (resp. weakly) separated collections in $2^{[n]}$ is connected via flips, or mutations, ``in the presence of six (resp. four) witnesses''. We give a symmetric analog of those results, by showing that each maximal symmetric strongly (weakly) separated collection in $2^{[n]}$ can be obtained from any other one by a series of special symmetric local transformations, so-called symmetric flips. Also we establish the connectedness via symmetric flips for the class of maximal symmetric $r$-separated collections in $2^{[n]}$ when $n,r$ are even (where sets $A,B\subseteq [n]$ are called $r$-separated if there are no elements $i_0

For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated collections of subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced by Leclerc and Zelevinsky. In this paper, extending results due to Leclerc-Zelevinsky, we develop a geometric approach to establish a number of nice combinatorial properties of maximal weakly r-separated collections. As a supplement, we also discuss an analogous concept when $r$ is even., Comment: 28 pages, 3 figures

We consider a planar graph $G$ in which the edges have nonnegative integer lengths such that the length of every cycle of $G$ is even, and three faces are distinguished, called holes in $G$. It is known that there exists a packing of cuts and (2,3)-metrics with nonnegative integer weights in $G$ which realizes the distances within each hole. We develop a strongly polynomial purely combinatorial algorithm to find such a packing., Comment: 25 pages, 10 figures

In ArXiv:1604.00338[math.QA] we gave a complete combinatorial characterization of homogeneous quadratic identities for minors of quantum matrices. It was obtained as a consequence of results on minors of matrices of a special sort, the so-called path matrices $Path_G$ generated by paths in special planar directed graphs $G$. In this paper we prove two assertions that were stated but left unproved in ArXiv:1604.00338[math.QA]. The first one says that any minor of $Path_G$ is determined by a system of disjoint paths, called a flow, in $G$ (generalizing a similar result of Lindstr\"om's type for the path matrices of Cauchon graphs by Casteels). The second, more sophisticated, assertion concerns certain transformations of pairs of flows in $G$., Comment: 28 pages

We consider an undirected multi(commodity)flow demand problem in which a supply graph is planar, each source-sink pair is located on one of three specified faces of the graph, and the capacities and demands are integer-valued and Eulerian. It is known that such a problem has a solution if the cut and (2,3)-metric conditions hold, and that the solvability implies the existence of an integer solution. We develop a purely combinatorial strongly polynomial solution algorithm., Comment: 16 pages, 4 figures

In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated collections of subsets of the ordered $n$-element set $[n]$ (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains $D\subseteq 2^{[n]}$ (in particular, of the hypercube $2^{[n]}$ itself, and the hyper-simplex $\{X\subseteq[n]\colon |X|=m\}$ for $m$ fixed), where $D$ is called pure if all maximal weakly separated collections in $D$ have the same cardinality. These conjectures have been answered affirmatively. In this paper, generalizing those earlier results, we reveal wider classes of pure domains in $2^{[n]}$. This is obtained as a consequence of our study of a novel geometric--combinatorial model for weakly separated set-systems, so-called \emph{combined (polygonal) tilings} on a zonogon, which yields a new insight in the area., Comment: 30 pages. Revised version. To appear in Selecta Mathematica

Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its maximal connected subcrystals with colors $1,...,n-1$ and with colors $2,...,n$ and characterize the interlacing structure for all pairs of these subcrystals. This is used to give a recursive description of the combinatorial structure of $K$ and develop an efficient procedure of assembling $K$., Comment: 24 pages. This is an improved version of the first part of ArXiv:1201.4549v3[math.CO]

Let $G = (VG, AG)$ be a directed graph with a set $S \subseteq VG$ of terminals and nonnegative integer arc capacities $c$. A feasible multiflow is a nonnegative real function $F(P)$ of "flows" on paths $P$ connecting distinct terminals such that the sum of flows through each arc $a$ does not exceed $c(a)$. Given $\mu \colon S \times S \to \R_+$, the \emph{$\mu$-value} of $F$ is $\sum_P F(P) \mu(s_P, t_P)$, where $s_P$ and $t_P$ are the start and end vertices of a path $P$, respectively. Using a sophisticated topological approach, Hirai and Koichi showed that the maximum $\mu$-value multiflow problem has an integer optimal solution when $\mu$ is the distance generated by subtrees of a weighted directed tree and $(G,S,c)$ satisfies certain Eulerian conditions. We give a combinatorial proof of that result and devise a strongly polynomial combinatorial algorithm., Comment: 12 pages

Adapting Lindstr\"om's well-known construction, we consider a wide class of functions which are generated by flows in a planar acyclic directed graph whose vertices (or edges) take weights in an arbitrary commutative semiring. We give a combinatorial description for the set of "universal" quadratic relations valid for such functions. Their specializations to particular semirings involve plenty of known quadratic relations for minors of matrices (e.g., Pl\"ucker relations) and the tropical counterparts of such relations. Also some applications and related topics are discussed., Comment: 35 pages. This is the revised version accepted for publication in J. Algebraic Comb. (The final publication is available at springerlink.com.) Also in the Appendix we add the assertion that the function of minors of any matrix over a field is generated (using Lindstr\"om method) by flows in a weighted planar acyclic directed graph

A Condorcet domain (CD) is a collection of linear orders on a set of candidates satisfying the following property: for any choice of preferences of voters from this collection, a simple majority rule does not yield cycles. We propose a method of constructing "large" CDs by use of rhombus tiling diagrams and explain that this method unifies several constructions of CDs known earlier. Finally, we show that three conjectures on the maximal sizes of those CDs are, in fact, equivalent and provide a counterexample to them., Comment: 16 pages. To appear in Discrete Applied Mathematics

It is well known, due to Lindstr\"om, that the minors of a (real or complex) matrix can be expressed in terms of weights of flows in a planar directed graph. Another classical fact is that there are plenty of homogeneous quadratic relations involving flag minors, or Pl\"ucker coordinates of the corresponding flag manifold. Generalizing and unifying these facts and their tropical counterparts, we consider a wide class of functions on $2^{[n]}$ that are generated by flows in a planar graph and take values in an arbitrary commutative semiring, where $[n]=\{1,2,\ldots,n\}$. We show that the ``universal'' homogeneous quadratic relations fulfilled by such functions can be described in terms of certain matchings, and as a consequence, give combinatorial necessary and sufficient conditions on the collections of subsets of $[n]$ determining these relations., Comment: 21 pges

We consider an undirected graph $G = (VG, EG)$ with a set $T \subseteq VG$ of terminals, and with nonnegative integer capacities $c(v)$ and costs $a(v)$ of nodes $v\in VG$. A path in $G$ is a \emph{$T$-path} if its ends are distinct terminals. By a \emph{multiflow} we mean a function $F$ assigning to each $T$-path $P$ a nonnegative rational \emph{weight} $F(P)$, and a multiflow is called \emph{feasible} if the sum of weights of $T$-paths through each node $v$ does not exceed $c(v)$. The \emph{value} of $F$ is the sum of weights $F(P)$, and the \emph{cost} of $F$ is the sum of $F(P)$ times the cost of $P$ w.r.t. $a$, over all $T$-paths $P$. Generalizing known results on edge-capacitated multiflows, we show that the problem of finding a minimum cost multiflow among the feasible multiflows of maximum possible value admits \emph{half-integer} optimal primal and dual solutions. Moreover, we devise a strongly polynomial algorithm for finding such optimal solutions., Comment: 26 pages; improved presentation; replaced skew-symmetric graphs with bidirected ones in the main content; accepted to Journal of Combinatorial Optimization

For a permutation $\omega\in S_n$, Leclerc and Zelevinsky \cite{LZ} introduced a concept of $\omega$-{\em chamber weakly separated collection} of subsets of $\{1,2,...,n\}$ and conjectured that all inclusion-wise maximal collections of this sort have the same cardinality $\ell(\omega)+n+1$, where $\ell(\omega)$ is the length of $\omega$. We answer affirmatively this conjecture and present a generalization and additional results., Comment: 36 pages. This is the final version for J. Algebr. Comb

For the ordered set $[n]$ of $n$ elements, we consider the class $\Bscr_n$ of bases $B$ of tropical Pl\"ucker functions on $2^{[n]}$ such that $B$ can be obtained by a series of mutations (flips) from the basis formed by the intervals in $[n]$. We show that these bases are representable by special wiring diagrams and by certain arrangements generalizing rhombus tilings on the $n$-zonogon. Based on the generalized tiling representation, we then prove that each weakly separated set-system in $2^{[n]}$ having maximum possible size belongs to $\Bscr_n$, thus answering affirmatively a conjecture due to Leclerc and Zelevinsky. We also prove an analogous result for a hyper-simplex $\Delta_n^m=\{S\subseteq[n]\colon |S|=m\}$., Comment: 47 pages. In this revision we add an Appendix containing results on weakly separated set-systems in a hyper-simplex and related subjects

We consider functions $f:B\to\Rset$ that obey tropical analogs of classical Pl\"ucker relations on minors of a matrix. The most general set $B$ that we deal with in this paper is of the form $\{x\in \Zset^n\colon 0\le x\le a, m\le x_1+...+x_n\le m'\}$ (a rectangular integer box ``truncated from below and above''). We construct a basis for the set $\Tscr$ of tropical Pl\"ucker functions on $B$, a subset $\Bscr\subseteq B$ such that the restriction map $\Tscr\to\Rset^\Bscr$ is bijective. Also we characterize, in terms of the restriction to the basis, the classes of submodular, so-called skew-submodular, and discrete concave functions in $\Tscr$, discuss a tropical analogue of the Laurentness property, and present other results., Comment: 44 pages. This is a revision of the original version, where some improvements are done and new results are added (in particular, the classes of submodular and discrete concave tropical Pl\"ucker functions are characterized, and a tropical analogue of the Laurent phenomenon is shown

The problem of finding, in an edge-weighted bidirected graph $G=(V,E)$, a cycle with minimum mean weight of its edges generalizes similar problems for both directed and undirected graphs. (The problem is considered in two variants: for the cycles without repeated edges and for the cycles without repeated nodes.) In this note we develop an algorithm to solve this problem in $O(V^2 \min(V^2, E\log V))$-time (to compare: the complexity of an improved version of Barahona's algorithm for undirected cycles is $O(V^4)$). Our algorithm is based on a certain general approach to minimum mean problems and uses, as a subroutine, Gabow's algorithm for the minimum weight 2-factor problem in a graph. The problem admits a reformulation in terms of regular cycles in a skew-symmetric graph., Comment: 10 pages, 2 figures

A convex triangular grid is represented by a planar digraph $G$ embedded in the plane so that (a) each bounded face is surrounded by three edges and forms an equilateral triangle, and (b) the union $\Rscr$ of bounded faces is a convex polygon. A real-valued function $h$ on the edges of $G$ is called a concave cocirculation if $h(e)=g(v)-g(u)$ for each edge $e=(u,v)$, where $g$ is a concave function on $\Rscr$ which is affinely linear within each bounded face of $G$. Knutson and Tao [J. Amer. Math. Soc. 12 (4) (1999) 1055--1090] proved an integrality theorem for so-called honeycombs, which is equivalent to the assertion that an integer-valued function on the boundary edges of $G$ is extendable to an integer concave cocirculation if it is extendable to a concave cocirculation at all. In this paper we show a sharper property: for any concave cocirculation $h$ in $G$, there exists an integer concave cocirculation $h'$ satisfying $h'(e)=h(e)$ for each boundary edge $e$ with $h(e)$ integer and for each edge $e$ contained in a bounded face where $h$ takes integer values on all edges. On the other hand, we explain that for a 3-side grid $G$ of size $n$, the polytope of concave cocirculations with fixed integer values on two sides of $G$ can have a vertex $h$ whose entries are integers on the third side but $h(e)$ has denominator $\Omega(n)$ for some interior edge $e$. Also some algorithmic aspects and related results on honeycombs are discussed., Comment: 20 pages, 12 figures, submitted to IPCO2004

Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\Rscr$ of these faces forms a convex polygon. The question is: given a function $\sigma$ on the boundary edges of $G$, does there exist a concave function $f$ on $\Rscr$ which is affinely linear within each bounded face and satisfies $f(v)-f(u)=\sigma(e)$ for each boundary edge $e=(u,v)$? The functions $\sigma$ admitting such an $f$ form a polyhedral cone $C$, and when the region $\Rscr$ is a triangle, $C$ turns out to be exactly the cone of boundary data of honeycombs. Studing honeycombs in connection with a problem on spectra of triples of zero-sum Hermitian matrices, Knutson, Tao, and Woodward \cite{KTW} showed that $C$ is described by linear inequalities of Horn's type with respect to so-called {\em puzzles}, along with obvious linear constraints. The purpose of this paper is to give an alternative proof of that result, working in terms of discrete concave finctions, rather than honeycombs, and using only linear programming and combinatorial tools. Moreover, we extend the result to an arbitrary convex polygon $\Rscr$., Comment: 21 pages

The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented., Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original version

Let $\mu$ be a metric on a set T, and let c be a nonnegative function on the unordered pairs of elements of a superset $V\supseteq T$. We consider the problem of minimizing the inner product $c\cdot m$ over all semimetrics m on V such that m coincides with $\mu$ within T and each element of V is at zero distance from T (a variant of the {\em multifacility location problem}). In particular, this generalizes the well-known multiterminal multiway) cut problem. Two cases of metrics $\mu$ have been known for which the problem can be solved in polynomial time: (a) $\mu$ is a modular metric whose underlying graph $H(\mu)$ is hereditary modular and orientable (in a certain sense); and (b) $\mu$ is a median metric. In the latter case an optimal solution can be found by use of a cut uncrossing method. \Xcomment{We give a common generalization for both cases by proving that the problem is in P for any modular metric $\mu$ whose all orbit graphs are hereditary modular and orientable. To this aim, we show the existence of a retraction of the Cartesian product of the orbit graphs to $H(\mu)$, which enables us to elaborate an analog of the cut uncrossing method for such metrics $\mu$.} In this paper we generalize the idea of cut uncrossing to show the polynomial solvability for a wider class of metrics $\mu$, which includes the median metrics as a special case. The metric uncrossing method that we develop relies on the existence of retractions of certain modular graphs. On the negative side, we prove that for $\mu$ fixed, the problem is NP-hard if $\mu$ is non-modular or $H(\mu)$ is non-orientable., Comment: 25 pages