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Kanstantsin Pashkovich

Publications and source records attributed to Kanstantsin Pashkovich.

At least 37 records · Page 2Linked to original sources
arXiv2018

Three-dimensional Stable Matching with Cyclic Preferences

We consider the three-dimensional stable matching problem with cyclic preferences, a problem originally proposed by Knuth. Despite extensive study of the problem by experts from different areas, the question of whether every instance of this problem admits a stable matching remains unanswered. In 2004, Boros, Gurvich, Jaslar and Krasner showed that a stable matching always exists when the number of agents in each of the groups is three. In 2006, Eriksson, Sjöstrand and Strimling showed that a stable matching exists also when the number of agents in each group is four. In this paper, we demonstrate that a stable matching exists when each group has five agents. Furthermore, we show that there are at least two distinct stable matchings in that setting.

cs.GT↗
arXiv2018

Bounds on the number of 2-level polytopes, cones and configurations

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones and d-configurations to faces of the correlation cone. We complement this with a $2^{Ω(d^2)}$ lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite graphs.

math.CO↗
arXiv2018

On the approximability of the stable marriage problem with one-sided ties

The classical stable marriage problem asks for a matching between a set of men and a set of women with no blocking pairs, which are pairs formed by a man and a woman who would both prefer switching from their current status to be paired up together. When both men and women have strict preferences over the opposite group, all stable matchings have the same cardinality, and the famous Gale-Shapley algorithm can be used to find one. Differently, if we allow ties in the preference lists, finding a stable matching of maximum cardinality is an NP-hard problem, already when the ties are one-sided, that is, they appear only in the preferences of one group. For this reason, many researchers have focused on developing approximation algorithm for this problem. In this paper, we give a refined analysis of an approximation algorithm given by Huang and Telikepalli (IPCO14) for the stable marriage problem with one-sided ties, which shows an improved 13/9 -approximation factor for the problem. Interestingly, our analysis is tight.

cs.DS↗
arXiv2017

On the Circuit Diameter of some Combinatorial Polytopes

The combinatorial diameter of a polytope $P$ is the maximum value of a shortest path between two vertices of $P$, where the path uses the edges of $P$ only. In contrast to the combinatorial diameter, the circuit diameter of $P$ is defined as the maximum value of a shortest path between two vertices of $P$, where the path uses potential edge directions of $P$ i.e., all edge directions that can arise by translating some of the facets of $P$. In this paper, we study the circuit diameter of polytopes corresponding to classical combinatorial optimization problems, such as the Matching polytope, the Traveling Salesman polytope and the Fractional Stable Set polytope.

math.OC↗
arXiv2017

On the Integrality Gap of the Prize-Collecting Steiner Forest LP

In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph $G=(V,E)$, edge costs $\{c_e\geq 0\}_{e\in E}$, terminal pairs $\{(s_i,t_i)\}_{i=1}^k$, and penalties $\{π_i\}_{i=1}^k$ for each terminal pair; the goal is to find a forest $F$ to minimize $c(F)+\sum_{i: (s_i,t_i)\text{ not connected in }F}π_i$. The Steiner forest problem can be viewed as the special case where $π_i=\infty$ for all $i$. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least $9/4$. This holds even for planar graphs. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than $4$. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP- approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most $1/3$ and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than $3$ using a direct iterative rounding method.

cs.DM↗
arXiv2017

Enumeration of $2$-level polytopes

A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytope $P_0$, we enumerate all $d$-dimensional $2$-level polytopes $P$ that have $P_0$ as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet $P_0$, we obtain all $2$-level polytopes in dimension $d$.

math.CO↗
arXiv2016

Every rational polyhedron has finite split rank: new proof

Split rank of a rational polyhedron is finite. The well known proof of this is based on the fact that split closure is stronger than the Chvátal closure, and the Chvátal rank of a rational polyhedron is finite due to the result of Chvátal and Schrijver. In this note we provide an independent proof for the fact that every rational polyhedron has finite split rank. In principal, we construct a nonnegative potential function which decreases by at least one with "every" second split closure unless the integer hull of the polyhedron is reached.

math.OC↗
arXiv2016

An Elementary Integrality Proof of Rothblum's Stable Matching Formulation

In this paper we provide a short new proof for the integrality of Rothblum's linear description of the convex hull of incidence vectors of stable matchings in bipartite graphs. In the spirit of iterative rounding proofs, the key feature of our proof is to show that extreme points of the formulation must have a 0, 1-component.

math.CO↗
arXiv2016

Fast Approximation Algorithms for the Generalized Survivable Network Design Problem

In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(δ(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the $f$-connectivity problem with general proper functions $f$. Implementing Jain's algorithm, this yields a strongly polynomial time $(2+ε)$-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation).

math.OC↗
arXiv2015

Cut dominants and forbidden minors

The cut dominant of a graph is the unbounded polyhedron whose points are all those that dominate some convex combination of proper cuts. Minimizing a nonnegative linear function over the cut dominant is equivalent to finding a minimum weight cut in the graph. We give a forbidden-minor characterization of the graphs whose cut dominant can be defined by inequalities with integer coefficients and right-hand side at most 2. Our result is related to the forbidden-minor characterization of TSP-perfect graphs by Fonlupt and Naddef (Math. Prog., 1992). We prove that our result implies theirs, with a shorter proof. Furthermore, we establish general properties of forbidden minors for right-hand sides larger than 2.

math.CO↗
arXiv2015

Small Extended Formulations for Cyclic Polytopes

We provide an extended formulation of size O(log n)^{\lfloor d/2 \rfloor} for the cyclic polytope with dimension d and n vertices (i,i^2,\ldots,i^d), i in [n]. First, we find an extended formulation of size log(n) for d= 2. Then, we use this as base case to construct small-rank nonnegative factorizations of the slack matrices of higher-dimensional cyclic polytopes, by iterated tensor products. Through Yannakakis's factorization theorem, these factorizations yield small-size extended formulations for cyclic polytopes of dimension d>2.

math.OC↗
arXiv2015

Hidden Vertices in Extensions of Polytopes

Some widely known compact extended formulations have the property that each vertex of the corresponding extension polytope is projected onto a vertex of the target polytope. In this paper, we prove that for heptagons with vertices in general position none of the minimum size extensions has this property. Additionally, for any d >= 2 we construct a family of d-polytopes such that at least 1/9 of all vertices of any of their minimum size extensions is not projected onto vertices.

math.CO↗
arXiv2015

Stable Sets and Graphs with no Even Holes

We develop decomposition/composition tools for efficiently solving maximum weight stable sets problems as well as for describing them as polynomially sized linear programs (using "compact systems"). Some of these are well-known but need some extra work to yield polynomial "decomposition schemes". We apply the tools to graphs with no even hole and no cap. A hole is a chordless cycle of length greater than three and a cap is a hole together with an additional node that is adjacent to two adjacent nodes of the hole and that has no other neighbors on the hole.

math.CO↗
arXiv2014

The Projected Faces Property and Polyhedral Relations

Margot (1994) in his doctoral dissertation studied extended formulations of combinatorial polytopes that arise from "smaller" polytopes via some composition rule. He introduced the "projected faces property" of a polytope and showed that this property suffices to iteratively build extended formulations of composed polytopes. For the composed polytopes, we show that an extended formulation of the type studied in this paper is always possible only if the smaller polytopes have the projected faces property. Therefore, this produces a characterization of the projected faces property. Affinely generated polyhedral relations were introduced by Kaibel and Pashkovich (2011) to construct extended formulations for the convex hull of the images of a point under the action of some finite group of reflections. In this paper we prove that the projected faces property and affinely generated polyhedral relation are equivalent conditions.

math.CO↗
arXiv2013

Uncapacitated Flow-based Extended Formulations

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages. Finally, we state a few open problems.

math.OC↗
arXiv2013

Which Nonnegative Matrices Are Slack Matrices?

In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown.

math.OC↗