Section: New Results

Graph and Combinatorial Algorithms

Induced Matching algorithms

In [21] we study the maximum induced matching problem on a graph $G$. Induced matchings correspond to independent sets in $L^2\left(G\right)$, the square of the line graph of $G$. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families with forbidden vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. That is, given a graph class $𝒢$ characterized by a vertex ordering, and a graph $G=(V,E)\in 𝒢$ with a corresponding vertex ordering $\sigma$ of $V$, one can produce (in linear time in the size of $G$) an ordering on the vertices of $L^2\left(G\right)$, that shows that $L^2\left(G\right)\in 𝒢$. This result gives alternate closure proofs for the $L^2(•)$ closure operation. Furthermore, these orderings on $L^2\left(G\right)$ can be exploited algorithmically to compute a maximum induced matching for graphs belonging to $𝒢$ faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, and trapezoid graphs, and we present the first $O\left(mn\right)$ time algorithm to compute a maximum weighted induced matching on G; an improvement from the best known $O\left(n^4\right)$ time algorithm for the unweighted case.

The LexBFS cycle on cocomparability graphs

Since its introduction to recognize chordal graphs by Rose, Tarjan, and Lueker, Lexicographic Breadth First Search (LexBFS) has been used to come up with simple, often linear time, algorithms on various classes of graphs. These algorithms, called multi-sweep algorithms, compute a number of LexBFS orderings ${\sigma }_1,...,{\sigma }_k$, where ${\sigma }_i$ is used to break ties for ${\sigma }_{i+1}$, we write $LexBF{S}^{+}\left({\sigma }_i\right)={\sigma }_{i+1}$. For instance, Corneil et al. gave a linear time multi-sweep algorithm to recognize interval graphs [SODA 1998], Kratsch et al. gave a certifying recognition algorithm for interval and permutation graphs [SODA 2003]. Since the number of LexBFS orderings for a graph is finite, after some fixed number of ${}^{+}$ sweeps, we will eventually loop in a sequence of ${\sigma }_1,...,{\sigma }_k$ vertex orderings such that ${\sigma }_{i+1}=LexBF{S}^{+}\left({\sigma }_i\right)$ modulo $k$.

In [13] we introduce and study this new graph invariant, LexCycle($G$), defined as the maximum length of a cycle of vertex orderings obtained via a sequence of LexBFS${}^{+}$. In this work, we focus on graph classes with small LexCycle. We give evidence that a small LexCycle often leads to linear structure that has been exploited algorithmically on a number of graph classes. In particular, we show that for proper interval, interval, co-bipartite, domino-free cocomparability graphs, as well as trees, there exists two orderings $\sigma$ and $\tau$ such that $\sigma =LexBF{S}^{+}\left(\tau \right)$ and $\tau =LexBF{S}^{+}\left(\sigma \right)$. One of the consequences of these results is the simplest algorithm to compute a transitive orientation for these graph classes.

It was conjectured by Stacho [2015] that LexCycle is at most the asteroidal number of the graph class, we disprove this conjecture by giving a construction for which the LexCycle($G$) grows polynomially in the asteroidal number of $G$.

Approximation Strategies for Generalized Binary Search in Weighted Trees

In [15], we have considered the following generalization of the binary search problem. A search strategy is required to locate an unknown target node $t$ in a given tree $T$. Upon querying a node $v$ of the tree, the strategy receives as a reply an indication of the connected component of $T\setminus \left\{v\right\}$ containing the target $t$. The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total query cost for a worst-case choice of the target.

Designing an optimal strategy for a weighted tree search instance is known to be strongly NP-hard, in contrast to the unweighted variant of the problem which can be solved optimally in linear time. Here, we show that weighted tree search admits a quasi-polynomial time approximation scheme: for any $0<\epsilon <1$, there exists a $(1+\epsilon )$-approximation strategy with a computation time of $n^{O(logn/{\epsilon }^2)}$. Thus, the problem is not APX-hard, unless $NP\subseteq DTIME\left(n^{O(logn)}\right)$. By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time $O\left(\sqrt{logn}\right)$-approximation. This improves previous $\widehat{O}(logn)$-approximation approaches, where the $\widehat{O}$-notation disregards $O(\mathrm{poly}loglogn)$-factors.

The Dependent Doors Problem: An Investigation into Sequential Decisions without Feedback

In [24] we introduce the dependent doors problem as an abstraction for situations in which one must perform a sequence of possibly dependent decisions, without receiving feedback information on the effectiveness of previously made actions. Informally, the problem considers a set of $d$ doors that are initially closed, and the aim is to open all of them as fast as possible. To open a door, the algorithm knocks on it and it might open or not according to some probability distribution. This distribution may depend on which other doors are currently open, as well as on which other doors were open during each of the previous knocks on that door. The algorithm aims to minimize the expected time until all doors open. Crucially, it must act at any time without knowing whether or which other doors have already opened. In this work, we focus on scenarios where dependencies between doors are both positively correlated and acyclic.

The fundamental distribution of a door describes the probability it opens in the best of conditions (with respect to other doors being open or closed). We show that if in two configurations of $d$ doors corresponding doors share the same fundamental distribution, then these configurations have the same optimal running time up to a universal constant, no matter what are the dependencies between doors and what are the distributions. We also identify algorithms that are optimal up to a universal constant factor. For the case in which all doors share the same fundamental distribution we additionally provide a simpler algorithm, and a formula to calculate its running time. We furthermore analyse the price of lacking feedback for several configurations governed by standard fundamental distributions. In particular, we show that the price is logarithmic in $d$ for memoryless doors, but can potentially grow to be linear in $d$ for other distributions.

We then turn our attention to investigate precise bounds. Even for the case of two doors, identifying the optimal sequence is an intriguing combinatorial question. Here, we study the case of two cascading memoryless doors. That is, the first door opens on each knock independently with probability $p_1$. The second door can only open if the first door is open, in which case it will open on each knock independently with probability $p_2$. We solve this problem almost completely by identifying algorithms that are optimal up to an additive term of 1.