Independent set reconfiguration on directed graphs

ABSTRACT. Directed Token Sliding asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set with one of its out-neighbors, while keeping the nonadjacency. It can be seen as a reconfiguration process where a token is placed on each vertex in the current set, and the local operation slides a token along an arc respecting its direction. Previously, such a problem was extensively studied on undirected graphs, where the edges have no directions and thus the local operation is symmetric. Directed Token Sliding is a generalization of its undirected variant since an undirected edge can be simulated by two arcs of opposite directions.
In this paper, we initiate the algorithmic study of Directed Token Sliding. We first observe that the problem is PSPACE-complete even if we forbid parallel arcs in opposite directions and that the problem on directed acyclic graphs is NP-complete and W[1]-hard parameterized by the size of the sets in consideration. We then show our main result: a linear-time algorithm for the problem on directed graphs whose underlying undirected graphs are trees, which are called polytrees. Such a result is also known for the undirected variant of the problem on trees [Demaine et al. TCS 2015], but the techniques used here are quite different because of the asymmetric nature of the directed problem. We present a characterization of yes-instances based on the existence of a certain set of directed paths, and then derive simple equivalent conditions from it by some observations, which admits an efficient algorithm. For the polytree case, we also present a quadratic-time algorithm that outputs, if the input is a yes-instance, one of the shortest reconfiguration sequences.

Fixed-point cycles and approximate EFX allocations

ABSTRACT. We study edge-labelings of the complete bidirected graph $K_n$ with functions that map the set $[d] = \{1, \dots, d\}$ to itself. We call a directed cycle in $K_n$ a {fixed-point cycle} if composing the labels of its edges in order results in a map that has a fixed point, and we say that a labeling is {fixed-point-free} if no fixed-point cycle exists. For a given $d$, we ask for the largest value of $n$, denoted $R_f(d)$, for which there exists a fixed-point-free labeling of $K_n$. Determining $R_f(d)$ for all $d>0$ is a natural Ramsey-type question, generalizing some well-studied zero-sum problems in extremal combinatorics. The problem was recently introduced by Chaudhury, Garg, Mehlhorn, Mehta, and Misra [EC 2021], who proved that $d \leq R_f(d) \leq d^4+d$ and showed that the problem has close connections to \emph{EFX allocations}, a central problem of fair allocation in social choice theory.

In this paper we show the improved bound $R_f(d) \leq d^{2 + o(1)}$, yielding an efficient ${(1-\varepsilon)}$-EFX allocation with $n$ agents and $O(n^{0.67})$ unallocated goods for any constant $\varepsilon \in (0,1/2]$; this improves the bound of $O(n^{0.8})$ of Chaudhury, Garg, Mehlhorn, Mehta, and Misra.

Additionally, we prove the stronger upper bound $2d-2$, in the case where all edge-labels are {permutations}. A very special case of this problem, that of finding {zero-sum cycles} in digraphs whose edges are labeled with elements of $\mathbb{Z}_d$, was recently considered by Alon and Krivelevich [JGT 2021] and by M\'{e}sz\'{a}ros and Steiner [EJC 2021]. Our result improves the bounds obtained by these authors and extends them to labelings with elements of an arbitrary (not necessarily commutative) group, while also simplifying the proof.

On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph

ABSTRACT. We consider the problem of characterizing degree sequences that can be realized by a bipartite graph. If a partition of the sequence into the two sides of the bipartite graph is given as part of the input, then a complete characterization has been established over 60 years ago. However, the general question, in which a partition and a realizing graph need to be determined, is still open.

We investigate the role of an important class of special partitions, called {\em High-Low partitions}, which separate the degrees of a sequence into two groups, the high degrees and the low degrees. We show that when the High-Low partition exists and satisfies some natural properties, analyzing the High-Low partition resolves the bigraphic realization problem. For sequences that are known to be not realizable by a bipartite graph or that are undecided, we provide approximate realizations based on the High-Low partition.

Beyond Value Iteration for Parity Games: Strategy Iteration with Universal Trees

ABSTRACT. Parity games have witnessed several new quasi-polynomial algorithms since the breakthrough result of Calude et al. (STOC 2017). The combinatorial object underlying these approaches is a universal tree, as identified by Czerwiński et al. (SODA 2019). By proving a quasi-polynomial lower bound on the size of a universal tree, they have highlighted a barrier that must be overcome by all existing approaches to attain polynomial running time. This is due to the existence of worst case instances which force these algorithms to explore a large portion of the tree.

As an attempt to overcome this barrier, we propose a strategy iteration framework which can be applied on any universal tree. It is at least as fast as its value iteration counterparts, while allowing one to take bigger leaps in the universal tree. Our main technical contribution is an efficient method for computing the least fixed point of 1-player games. This is achieved via a careful adaptation of shortest path algorithms to the setting of ordered trees. By plugging in the universal tree of Jurdziński and Lazić (LICS 2017), or the Strahler universal tree of Daviaud et al. (ICALP 2020), we obtain instantiations of the general framework that take time O(mn^2 log n log d) and O(mn^2 log^3 n log d) respectively per iteration.

Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

ABSTRACT. Cai and Hemachandra used iterative constant-setting to prove that Few \subseteq ParityP (and thus that FewP \subseteq ParityP). In this paper, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed "nongappy"-ness) of the easy-to-find "targets" used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant's unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra-Pomerance-Wagstaff Conjecture implies that all O(loglogn)-ambiguity NP sets are in the restricted counting class RC_{PRIMES}.

On the Number of Quantifiers as a Complexity Measure

ABSTRACT. In 1981, Neil Immerman described a two-player game, which he called the "separability game", that captures the number of quantifiers needed to describe a property in first order logic. Immerman's paper laid the groundwork for studying the number of quantifiers needed to express properties in first order logic, but the game seemed to be too complicated to study, and the arguments of the paper almost exclusively used quantifier rank as a lower bound on the total number of quantifiers. However, last year Fagin, Lenchner, Regan and Vyas rediscovered the games, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results. First we carefully distinguish minimum number of quantifiers from the more usual descriptive complexity measures, minimum quantifier rank and minimum number of variables. Then, for each positive integer k, we give an explicit example of a property of finite structures (in particular, of finite graphs) that can be expressed with a sentence of quantifier rank k, but where the same property needs 2^Omega (k^2) quantifiers to be expressed.We next give the precise number of quantifiers needed to distinguish two rooted trees of different depths. Finally, we give a new upper bound on the number of quantifiers needed to express s-t connectivity, improving the previous known bound by a constant factor.

Oracle with P = NP ∩ coNP, but no Many-One Completeness in UP, DisjNP, and DisjCoNP

ABSTRACT. We construct an oracle relative to which P = NP ∩ coNP, but there are no many-one complete sets in UP, no many-one complete disjoint NP-pairs, and no many-one complete disjoint coNP-pairs. This contributes to a research program initiated by Pudlák, which studies incompleteness in the finite domain and which mentions the construction of such oracles as open problem. The oracle shows that the hypothesis NP∩coNP is indispensable in the list of hypotheses studied by Pudlák. Hence one should consider stronger hypotheses, in order to find a universal one.

Non-Determinism in Lindenmayer Systems and Global Transformations

ABSTRACT. Global transformations provide a categorical framework for capturing synchronous rewriting systems, generalizing cellular automata to dynamical systems over dynamic spaces.
Originally developed for addressing deterministic dynamical systems, the presented work raises the question of non-determinism in global transformations.
While a usual approach is to develop a general non-deterministic setting where deterministic systems can be retrieved as a specific case, we show here that by choosing the right parametrization, global transformations can already be used to tackle the issue of non-determinism.
Context-free Lindenmayer systems, already shown to be captured by global transformation in the deterministic case, are used to illustrate the approach.
From this concrete example, the formal obstructions are exhibited, leading to a solution involving a 2-categorical monad and its associated Kleisli construction.