Searcharxiv⌕ Search

arXiv subjects

Dirk Oliver Theis

Publications and source records attributed to Dirk Oliver Theis.

At least 37 records · Page 2Linked to original sources
arXiv2016

Short note on the number of 1-ascents in dispersed dyck paths

A dispersed Dyck path (DDP) of length n is a lattice path on $N\times N$ from (0,0) to (n,0) in which the following steps are allowed: "up" (x, y) $\to$ (x+1, y+1); "down" (x, y) $\to$ (x+1, y-1); and "right" (x,0) $\to$ (x+1,0). An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length n, A191386 in Sloane's OEIS. Previously, only implicit generating function relations and asymptotics were known.

math.CO↗
arXiv2014

Fooling sets and rank

An $n\times n$ matrix $M$ is called a \textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\ell} M_{\ell,k} = 0$ for every $k\ne \ell$. Dietzfelbinger, Hromkovi{č}, and Schnitger (1996) showed that $n \le (\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $\mbox{rk} M$ can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = \binom{\mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of matrices with $n= (1+o(1))(\mbox{rk} M)^2$.

math.CO↗
arXiv2013

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that $S(k,\ell) = \mbox{tr}(A_k^* B_\ell)$. The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern. In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.

math.CO↗
arXiv2013

An algorithm for random signed 3-SAT with Intervals

In signed k-SAT problems, one fixes a set M and a set $\mathcal S$ of subsets of M, and is given a formula consisting of a disjunction of m clauses, each of which is a conjunction of k literals. Each literal is of the form "$x \in S$", where $S \in \mathcal S$, and x is one of n variables. For Interval-SAT (iSAT), M is an ordered set and $\mathcal S$ the set of intervals in M. We propose an algorithm for 3-iSAT, and analyze it on uniformly random formulas. The algorithm follows the Unit Clause paradigm, enhanced by a (very limited) backtracking option. Using Wormald's ODE method, we prove that, if $m/n \le 2.3$, with high probability, our algorithm succeeds in finding an assignment of values to the variables satisfying the formula.

math.CO↗
arXiv2013

Fooling-sets and rank in nonzero characteristic (extended abstract)

An n\times n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k\ne \ell we have M_{k,\ell} M_{\ell,k} = 0. Dietzfelbinger, Hromkovič, and Schnitger (1996) showed that n \le (\rk M)^2, regardless of over which field the rank is computed, and asked whether the exponent on \rk M can be improved. We settle this question for nonzero characteristic by constructing a family of matrices for which the bound is asymptotically tight. The construction uses linear recurring sequences.

math.CO↗
arXiv2012

Symmetry Matters for Sizes of Extended Formulations

In 1991, Yannakakis (J. Comput. System Sci., 1991) proved that no symmetric extended formulation for the matching polytope of the complete graph K_n with n nodes has a number of variables and constraints that is bounded subexponentially in n. Here, symmetric means that the formulation remains invariant under all permutations of the nodes of K_n. It was also conjectured in the paper mentioned above that "asymmetry does not help much," but no corresponding result for general extended formulations has been found so far. In this paper we show that for the polytopes associated with the matchings in K_n with log(n) (rounded down) edges there are non-symmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulations of polynomial size exist. We furthermore prove similar statements for the polytopes associated with cycles of length log(n) (rounded down). Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abtract that has appeared in the Proceedings of IPCO XIV at Lausanne, this paper does not only contain proofs that had been ommitted there, but it also presents slightly generalized and sharpened lower bounds.

math.CO↗
arXiv2012

Good edge-labelings and graphs with girth at least five

A good edge-labeling of a graph [Araújo, Cohen, Giroire, Havet, Discrete Appl. Math., forthcoming] is an assignment of numbers to the edges such that for no pair of vertices, there exist two non-decreasing paths. In this paper, we study edge-labeling on graphs with girth at least 5. In particular we verify, under this additional hypothesis, a conjecture by Araújo et al. This conjecture states that if the average degree of G is less than 3 and G is minimal without an edge-labeling, then G \in {C_3,K_{2,3}}. (For the case when the girth is 4, we give a counterexample.)

math.CO↗
arXiv2012

Combinatorial Bounds on Nonnegative Rank and Extended Formulations

An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear programs of small sizes. The main known lower bounds on the minimum sizes of extended formulations for fixed polytope P (Yannakakis 1991) are closely related to the concept of nondeterministic communication complexity. We study the relative power and limitations of the bounds on several examples.

math.CO↗
arXiv2012

Compact Formulations of the Steiner Traveling Salesman Problem and Related Problems

The Steiner Traveling Salesman Problem (STSP) is a variant of the Traveling Salesman Problem (TSP) that is particularly suitable when dealing with sparse networks, such as road networks. The standard integer programming formulation of the STSP has an exponential number of constraints, just like the standard formulation of the TSP. On the other hand, there exist several known {\em compact} formulations of the TSP, i.e., formulations with a polynomial number of both variables and constraints. In this paper, we show that some of these compact formulations can be adapted to the STSP. We also briefly discuss the adaptation of our formulations to some closely-related problems.

math.OC↗
arXiv2012

Small Minors in Dense Graphs

A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe functions $f$ and $h$ such that every graph with $n$ vertices and average degree at least $f(t)$ contains a $K_t$-model with at most $h(t)\cdot\log n$ vertices. The logarithmic dependence on $n$ is best possible (for fixed $t$). In general, we prove that $f(t)\leq 2^{t-1}+\eps$. For $t\leq 4$, we determine the least value of $f(t)$; in particular $f(3)=2+\eps$ and $f(4)=4+\eps$. For $t\leq4$, we establish similar results for graphs embedded on surfaces, where the size of the $K_t$-model is bounded (for fixed $t$).

math.CO↗
arXiv2011

On the satisfiability of random regular signed SAT formulas

Regular signed SAT is a variant of the well-known satisfiability problem in which the variables can take values in a fixed set V \subset [0,1], and the `literals' have the form "x \le a" or "x \ge a". We answer some open question regarding random regular signed k-SAT formulas: the probability that a random formula is satisfiable increases with |V|; there is a constant upper bound on the ratio m/n of clauses m over variables n, beyond which a random formula is asypmtotically almost never satisfied; for k=2 and V=[0,1], there is a phase transition at m/n=2.

cs.DM↗
arXiv2011

On some lower bounds on the number of bicliques needed to cover a bipartite graph

The biclique covering number of a bipartite graph G is the minimum number of complete bipartite subgraphs (bicliques) whose union contains every edge of G. In this little note we compare three lower bounds on the biclique covering number: A bound jk(G) proposed by Jukna & Kulikov (Discrete Math. 2009); the well-known fooling set bound fool(G); the "tensor-power" fooling set bound fool^\infty(G). We show jk \le fool le fool^\infty \le min_Q (rk Q)^2, where the minimum is taken over all matrices with a certain zero/nonzero-pattern. Only the first inequality is really novel, the third one generalizes a result of Dietzfelbinger, Hromkovič, Schnitger (1994). We also give examples for which fool \ge (rk)^{log_4 6} improving on Dietzfelbinger et al.

math.CO↗
arXiv2011

On a class of metrics related to graph layout problems

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the literature, and also to a class of combinatorial optimization problems known as graph layout problems. We prove several results about the structure of these metrics. In particular, it is shown that their convex hull is not closed in general. We then show that certain linear inequalities define facets of the closure of the convex hull. Finally, we characterise the unbounded edges of the convex hull and of its closure.

math.CO↗
arXiv2011

A note on the relationship between the Graphical Traveling Salesman Polyhedron, the Symmetric Traveling Salesman Polytope, and the Metric Cone

In this short communication, we observe that the Graphical Traveling Salesman Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric Traveling Salesman Polytope and the polar of the metric cone. This follows almost trivially from known facts. There are two reasons why we find this observation worth communicating none-the-less: It is very surprising; it helps to understand the relationship between these two important families of polyhedra.

math.CO↗
arXiv2011

Random lifts of $K_5\setminus e$ are 3-colourable

Amit, Linial, and Matou\vsek (Random lifts of graphs III: independence and chromatic number, Random Struct. Algorithms, 2001) have raised the following question: Is the chromatic number of random $h$-lifts of $K_5$ asymptotically (for $h\to\infty$) almost surely equal to a single number? In this paper, we offer the following partial result: The chromatic number of a random lift of $K_5\setminus e$ is asymptotically almost surely three.

math.CO↗
arXiv2011

A note on the Cops & Robber game on graphs embedded in non-orientable surfaces

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.

math.CO↗
arXiv2009

On the facial structure of Symmetric and Graphical Traveling Salesman Polyhedra

The Symmetric Traveling Salesman Polytope $S_n$ for a fixed number $n$ of cities is a face of the corresponding Graphical Traveling Salesman Polyhedron $P_n$. This has been used to study facets of $S_n$ using $P_n$ as a tool. In this paper, we study the operation of "rotating" (or "lifting") valid inequalities for $S_n$ to obtain a valid inequalities for $P_n$. As an application, we describe a surprising relationship between (a) the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope and (b) a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron.

math.CO↗