IA Scholar Query: Bounds on the Chromatic Polynomial and on the Number of Acyclic Orientations of a Graph.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 26 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Quantum-Inspired Perfect Matching under Vertex-Color Constraints
https://scholar.archive.org/work/mjy4fof2bffz7c7c6vblcytyey
We propose and study the graph-theoretical problem PM-VC: perfect matching under vertex-color constraints on graphs with bi-colored edges. PM-VC is of special interest because of its motivation from quantum-state identification and quantum-experiment design, as well as its rich expressiveness, i.e., PM-VC subsumes many constrained matching problems naturally, such as exact perfect matching. We give complexity and algorithmic results for PM-VC under two types of vertex color constraints: 1) symmetric constraints (PM-VC-Sym) and 2) decision-diagram constraints (PM-VC-DD). We prove that PM-VC-Sym is in RNC via a symbolic determinant algorithm, which can be derandomized on planar graphs. Moreover, PM-VC-Sym can be expressed in extended MSO, which encourages our design of an efficient dynamic programming algorithm for PM-VC-Sym on bounded-treewidth graphs. For PM-VC-DD, we reveal its NP-hardness by a graph-gadget technique. Our novel results for PM-VC provide insights to both constrained matching and scalable quantum experiment design.Moshe Y. Vardi, Zhiwei Zhangwork_mjy4fof2bffz7c7c6vblcytyeyMon, 26 Sep 2022 00:00:00 GMTList-avoiding orientations
https://scholar.archive.org/work/wumd5qdo3zavlkqticm5nog6gq
Given a graph G with a set F(v) of forbidden values at each v ∈ V(G), an F-avoiding orientation of G is an orientation in which deg^+(v) ∉F(v) for each vertex v. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if |F(v)| < 1/2 deg(v) for each v ∈ V(G), then G has an F-avoiding orientation, and they showed that this statement is true when 1/2 is replaced by 1/4. In this paper, we take a step toward this conjecture by proving that if |F(v)| < ⌊1/3 deg(v) ⌋ for each vertex v, then G has an F-avoiding orientation. Furthermore, we show that if the maximum degree of G is subexponential in terms of the minimum degree, then this coefficient of 1/3 can be increased to √(2) - 1 - o(1) ≈ 0.414. Our main tool is a new sufficient condition for the existence of an F-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.Peter Bradshaw, Yaobin Chen, Hao Ma, Bojan Mohar, Hehui Wuwork_wumd5qdo3zavlkqticm5nog6gqMon, 19 Sep 2022 00:00:00 GMTA (1.5+ϵ)-Approximation Algorithm for Weighted Connectivity Augmentation
https://scholar.archive.org/work/jievulsebfbl3mxmrihu5wnu7u
Connectivity augmentation problems are among the most elementary questions in Network Design. Many of these problems admit natural 2-approximation algorithms, often through various classic techniques, whereas it remains open whether approximation factors below 2 can be achieved. One of the most basic examples thereof is the Weighted Connectivity Augmentation Problem (WCAP). In WCAP, one is given an undirected graph together with a set of additional weighted candidate edges, and the task is to find a cheapest set of candidate edges whose addition to the graph increases its edge-connectivity. We present a (1.5+ε)-approximation algorithm for WCAP, showing for the first time that factors below 2 are achievable. On a high level, we design a well-chosen local search algorithm, inspired by recent advances for Weighted Tree Augmentation. To measure progress, we consider a directed weakening of WCAP and show that it has highly structured planar solutions. Interpreting a solution of the original problem as one of this directed weakening allows us to describe local exchange steps in a clean and algorithmically amenable way. Leveraging these insights, we show that we can efficiently search for good exchange steps within a component class for link sets that is closely related to bounded treewidth subgraphs of circle graphs. Moreover, we prove that an optimum solution can be decomposed into smaller components, at least one of which leads to a good local search step as long as we did not yet achieve the claimed approximation guarantee.Vera Traub, Rico Zenklusenwork_jievulsebfbl3mxmrihu5wnu7uFri, 16 Sep 2022 00:00:00 GMTMathematical programming for stable control and safe operation of gas transport networks
https://scholar.archive.org/work/jbrpzy7fcrf4hccjg7q4udzdzm
The fight against climate change makes extreme but inevitable changes in the energy sector necessary. These in turn lead to novel and complex challenges for the transmission system operators (TSOs) of gas transport networks. In this thesis, we consider four different planning problems emerging from real-world operations and present mathematical programming models and solution approaches for all of them. Due to regulatory requirements and side effects of renewable energy production, controlling today's gas networks with their involved topologies is becoming increasingly difficult. Based on the network station modeling concept for approximating the technical capabilities of complex subnetworks, e.g., compressor stations, we introduce a tri-level MIP model to determine important global control decisions. Its goal is to avoid changes in the network elements' settings while deviations from future inflow pressures as well as supplies and demands are minimized. A sequential linear programming inspired post-processing routine is run to derive physically accurate solutions w.r.t. the transient gas flow in pipelines. Computational experiments based on real-world data show that meaningful solutions are quickly and reliably determined. Therefore, the algorithmic approach is used within KOMPASS, a decision support system for the transient network control that we developed together with the Open Grid Europe GmbH (OGE), one of Europe's largest natural gas TSOs. Anticipating future use cases, we adapt the aforementioned algorithmic approach for hydrogen transport. We investigate whether the natural gas infrastructure can be repurposed and how the network control changes when energy-equivalent amounts of hydrogen are transported. Besides proving the need for purpose-built compressors, we observe that, due to the reduced linepack, the network control becomes more dynamic, compression energy increases by 440% on average, and stricter regulatory rules regarding the balancing of supply and demand become necessary. Extreme load flows [...]Kai Hoppmann-Baum, Technische Universität Berlin, Thorsten Kochwork_jbrpzy7fcrf4hccjg7q4udzdzmWed, 14 Sep 2022 00:00:00 GMTWhat is a combinatorial interpretation?
https://scholar.archive.org/work/4cjduawoc5ahxk4n2iojrx32ey
In this survey we discuss the notion of combinatorial interpretation in the context of Algebraic Combinatorics and related areas. We approach the subject from the Computational Complexity perspective. We review many examples, state a workable definition, discuss many open problems, and present recent results on the subject.Igor Pakwork_4cjduawoc5ahxk4n2iojrx32eyTue, 13 Sep 2022 00:00:00 GMTProving a directed analogue of the Gyárfás-Sumner conjecture for orientations of P_4
https://scholar.archive.org/work/c364c4f7gfgzteavs2ud67zl6u
An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph D is H-free if D does not contain H as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest F, there is some function f such that every F-free graph G with clique number ω(G) has chromatic number at most f(ω(G)). Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph D is the minimum number of colors required to color the vertex set of D so that no directed cycle in D is monochromatic. Aboulker, Charbit, and Naserasr's χ-boundedness conjecture states that for every oriented forest F, there is some function f such that every F-free oriented graph D has dichromatic number at most f(ω(D)), where ω(D) is the size of a maximum clique in the graph underlying D. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr's χ-boundedness conjecture by showing that it holds when F is any orientation of a path on four vertices.Linda Cook, Tomáš Masařík, Marcin Pilipczuk, Amadeus Reinald, Uéverton S. Souzawork_c364c4f7gfgzteavs2ud67zl6uTue, 13 Sep 2022 00:00:00 GMTOnline Algorithms with Lookaround
https://scholar.archive.org/work/dosd4c67l5hqzohusswxgfkusm
In this work, we give a unifying view of locality in four settings: distributed algorithms, sequential greedy algorithms, dynamic algorithms, and online algorithms. We introduce a new model of computing, called the online-LOCAL model: the adversary reveals the nodes of the input graph one by one, in the same way as in classical online algorithms, but for each new node the algorithm can also inspect its radius-T neighborhood before choosing the output. Instead of looking ahead in time, we have the power of looking around in space. We compare the online-LOCAL model with three other models: the LOCAL model of distributed computing, where each node produces its output based on its radius-T neighborhood, its sequential counterpart SLOCAL, and the dynamic-LOCAL model, where changes in the dynamic input graph only influence the radius-T neighborhood of the point of change. SLOCAL and dynamic-LOCAL models are sandwiched between LOCAL and online-LOCAL models, with LOCAL being the weakest and online-LOCAL the strongest model. In this work, we seek to answer the following question: is the online-LOCAL model strictly stronger than the LOCAL model when we look at graph algorithms for solving locally checkable labeling problems (LCLs)? First, we show that for LCL problems in paths, cycles, and rooted trees, all four models are roughly equivalent: the locality of any LCL problem falls in the same broad class - O(log^* n), Θ(log n), or n^Θ(1) - in all four models. In particular, prior work on the LOCAL model directly generalizes to all four models. Second, we show that this equivalence does not hold in two-dimensional grids. We show that the locality of the 3-coloring problem is O(log n) in the online-LOCAL model, while it is known to be Ω(√(n)) in the LOCAL model.Amirreza Akbari, Navid Eslami, Henrik Lievonen, Darya Melnyk, Joona Särkijärvi, Jukka Suomelawork_dosd4c67l5hqzohusswxgfkusmMon, 12 Sep 2022 00:00:00 GMTLocal-to-global functional inequalities in simplicial complexes
https://scholar.archive.org/work/7jqg2yfv7naqhc3ewwvmiwdgce
A study of random walks over simplicial complexes with a particular emphasis on matroids. A framework is developed that yields results on the entropy contraction and modified log-Sobolev constant of the exchange walks over the levels of a simplicial complex, on the basis of entropy contraction properties of some local walks. This provides a general method for analyzing a variety of Markov chains by analyzing some of their lower-dimensional instances.Giorgos Mousa, University Of Edinburgh, Heng Guo, Mary Cryanwork_7jqg2yfv7naqhc3ewwvmiwdgceTue, 06 Sep 2022 00:00:00 GMTColoring Mixed and Directional Interval Graphs
https://scholar.archive.org/work/oaparejc3rf7tm4j5zux3444i4
A mixed graph has a set of vertices, a set of undirected egdes, and a set of directed arcs. A proper coloring of a mixed graph G is a function c that assigns to each vertex in G a positive integer such that, for each edge uv in G, c(u) c(v) and, for each arc uv in G, c(u) < c(v). For a mixed graph G, the chromatic number χ(G) is the smallest number of colors in any proper coloring of G. A directional interval graph is a mixed graph whose vertices correspond to intervals on the real line. Such a graph has an edge between every two intervals where one is contained in the other and an arc between every two overlapping intervals, directed towards the interval that starts and ends to the right. Coloring such graphs has applications in routing edges in layered orthogonal graph drawing according to the Sugiyama framework; the colors correspond to the tracks for routing the edges. We show how to recognize directional interval graphs, and how to compute their chromatic number efficiently. On the other hand, for mixed interval graphs, i.e., graphs where two intersecting intervals can be connected by an edge or by an arc in either direction arbitrarily, we prove that computing the chromatic number is NP-hard.Grzegorz Gutowski and Florian Mittelstädt and Ignaz Rutter and Joachim Spoerhase and Alexander Wolff and Johannes Zinkwork_oaparejc3rf7tm4j5zux3444i4Fri, 02 Sep 2022 00:00:00 GMTLower bound for constant-size local certification
https://scholar.archive.org/work/lwjeqta3nfcxvlcpdxzrm5ixzm
Given a network property or a data structure, a local certification is a labeling that allows to efficiently check that the property is satisfied, or that the structure is correct. The quality of a certification is measured by the size of its labels: the smaller, the better.This notion plays a central role in self-stabilization, because the size of the certification is a lower bound (and often an upper bound) on the memory needed for silent self-stabilizing construction of distributed data structures. From the point of view of distributed computing in general, it is also a measure of the locality of a property (e.g. properties of the network itself, such as planarity). When it comes to the size of the certification labels, one can identify three important regimes: the properties for which the optimal size is polynomial in the number of vertices of the graph, the ones that require only polylogarithmic size, and the ones that can be certified with a constant number of bits. The first two regimes are well studied, with several upper and lower bounds, specific techniques, and active research questions. On the other hand, the constant regime has never been really explored, at least on the lower bound side. The main contribution of this paper is the first non-trivial lower bound for this low regime. More precisely, we show that by using certification on just one bit (a binary certification), one cannot certify k-colorability for k≥ 3. To do so, we develop a new technique, based on the notion of score, and both local symmetry arguments and a global parity argument. We hope that this technique will be useful for establishing stronger results. We complement this result by a discussion of the implication of lower bounds for this constant-size regime, and with an upper bound for a related problem, illustrating that in some cases one can do better than the natural upper bound.Virgina Ardévol Martínez, Marco Caoduro, Laurent Feuilloley, Jonathan Narboni, Pegah Pournajafi, Jean-Florent Raymondwork_lwjeqta3nfcxvlcpdxzrm5ixzmTue, 30 Aug 2022 00:00:00 GMTDagstuhl Reports, Volume 12, Issue 2, February 2022, Complete Issue
https://scholar.archive.org/work/scntyrlsivecdcac4psbic4qzy
Dagstuhl Reports, Volume 12, Issue 2, February 2022, Complete Issuework_scntyrlsivecdcac4psbic4qzyTue, 23 Aug 2022 00:00:00 GMTLogic and Random Discrete Structures (Dagstuhl Seminar 22061)
https://scholar.archive.org/work/uk7dhmhnfvahbkrzv56wkjcn3a
This report documents the program and the outcomes of Dagstuhl Seminar 22061 "Logic and Random Discrete Structures". The main topic of this seminar has been the analysis of large random discrete structures, such as trees, graphs, or permutations, from the perspective of mathematical logic. It has brought together both experts and junior researchers from a number of different areas where logic and random structures play a role, with the goal to establish new connections between such areas and to encourage interactions between foundational research and different application areas, including probabilistic databases.Erich Grädel, Phokion G. Kolaitis, Marc Noywork_uk7dhmhnfvahbkrzv56wkjcn3aTue, 23 Aug 2022 00:00:00 GMTChromatic quasisymmetric functions and noncommutative P-symmetric functions
https://scholar.archive.org/work/xkrfgxeaenguba3v72mjof3btm
For a natural unit interval order P, we describe proper colorings of the incomparability graph of P in the language of heaps. After then, we introduce a combinatorial operation, called a local flip, on the heaps. This operation defines an equivalence relation on the proper colorings, and the equivalent relation refines the ascent statistic introduced by Shareshian and Wachs. We also define an analogue of noncommutative symmetric functions introduced by Fomin and Greene, with respect to P. We establish a duality between the chromatic quasisymmetric function of P and these noncommutative symmetric functions. This duality leads us to positive expansions of the chromatic quasisymmetric functions into several symmetric function bases. Also, we present some partial results for the e-positivity conjecture.Byung-Hak Hwangwork_xkrfgxeaenguba3v72mjof3btmSun, 21 Aug 2022 00:00:00 GMTVarious bounds on the minimum number of arcs in a k-dicritical digraph
https://scholar.archive.org/work/azczn6sup5dgpdyc7y2tlqmuai
The dichromatic number χ⃗(G) of a digraph G is the least integer k such that G can be partitioned into k acyclic digraphs. A digraph is k-dicritical if χ⃗(G) = k and each proper subgraph H of G satisfies χ⃗(H) ≤ k-1. cycle of length 2. We prove various bounds on the minimum number of arcs in a k-dicritical digraph, a structural result on k-dicritical digraphs and a result on list-dicolouring. We characterise 3-dicritical digraphs G with (k-1)|V(G)| + 1 arcs. For k ≥ 4, we characterise k-dicritical digraphs G on at least k+1 vertices and with (k-1)|V(G)| + k-3 arcs, generalising a result of Dirac. We prove that, for k ≥ 5, every k-dicritical digraph G has at least (k-1/2 - 1/(k-1)) |V(G)| - k(1/2 - 1/(k-1)) arcs, which is the best known lower bound. We prove that the number of connected components induced by the vertices of degree 2(k-1) of a k-dicritical digraph is at most the number of connected components in the rest of the digraph, generalising a result of Stiebitz. Finally, we generalise a Theorem of Thomassen on list-chromatic number of undirected graphs to list-dichromatic number of digraphs.Pierre Aboulker, Quentin Vermandework_azczn6sup5dgpdyc7y2tlqmuaiWed, 17 Aug 2022 00:00:00 GMTPlethysms of Chromatic and Tutte Symmetric Functions
https://scholar.archive.org/work/aeyoi3gj3bbpzno65s6amgflui
Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question. In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities.Logan Crew, Sophie Spirklwork_aeyoi3gj3bbpzno65s6amgfluiTue, 02 Aug 2022 00:00:00 GMTMultipath cohomology of directed graphs
https://scholar.archive.org/work/kdae3jiuqfad3k64ettpofjepq
This work is part of a series of papers focusing on multipath cohomology of directed graphs. Multipath cohomology is defined as the (poset) homology of the path poset -- i.e., the poset of disjoint simple paths in a graph -- with respect to a certain functor. This construction is essentially equivalent, albeit more computable, to taking the higher limits of said functor on (a certain modification of) the path poset. We investigate the functorial properties of multipath cohomology. We provide a number of sample computations, show that the multipath cohomology does not vanish on trees, and that, when evaluated at the coherently oriented polygon, it recovers Hochschild homology. Finally, we use the same techniques employed to study the functoriality to investigate the connection with the chromatic homology of (undirected) graphs introduced by L. Helme-Guizon and Y. Rong.Luigi Caputi, Carlo Collari, Sabino Di Traniwork_kdae3jiuqfad3k64ettpofjepqMon, 11 Jul 2022 00:00:00 GMTVerification and search algorithms for causal DAGs
https://scholar.archive.org/work/iev55ekcabh5bo72ocynxktfgm
We study two problems related to recovering causal graphs from interventional data: (i) verification, where the task is to check if a purported causal graph is correct, and (ii) search, where the task is to recover the correct causal graph. For both, we wish to minimize the number of interventions performed. For the first problem, we give a characterization of a minimal sized set of atomic interventions that is necessary and sufficient to check the correctness of a claimed causal graph. Our characterization uses the notion of covered edges, which enables us to obtain simple proofs and also easily reason about earlier results. We also generalize our results to the settings of bounded size interventions and node-dependent interventional costs. For all the above settings, we provide the first known provable algorithms for efficiently computing (near)-optimal verifying sets on general graphs. For the second problem, we give a simple adaptive algorithm based on graph separators that produces an atomic intervention set which fully orients any essential graph while using 𝒪(log n) times the optimal number of interventions needed to verify (verifying size) the underlying DAG on n vertices. This approximation is tight as any search algorithm on an essential line graph has worst case approximation ratio of Ω(log n) with respect to the verifying size. With bounded size interventions, each of size ≤ k, our algorithm gives an 𝒪(log n ·loglog k) factor approximation. Our result is the first known algorithm that gives a non-trivial approximation guarantee to the verifying size on general unweighted graphs and with bounded size interventions.Davin Choo, Kirankumar Shiragur, Arnab Bhattacharyyawork_iev55ekcabh5bo72ocynxktfgmThu, 30 Jun 2022 00:00:00 GMTFully-Dynamic α + 2 Arboricity Decompositions and Implicit Colouring
https://scholar.archive.org/work/ymyqvcyiojcmlpliukdxboygpu
The arboricity α of a graph is the smallest number of forests necessary to cover its edges, and an arboricity decomposition of a graph is a decomposition of its edges into forests. The best near-linear time algorithm for arboricity decomposition guarantees at most α +2 forests if the graph has arboricity α (Blumenstock and Fischer [Markus Blumenstock and Frank Fischer, 2020]). In this paper, we study arboricity decomposition for dynamic graphs, that is, graphs that are subject to insertions and deletions of edges. We give an algorithm that, provided the arboricity of the dynamic graph never exceeds α, maintains an α+2 arboricity decomposition of the graph in poly(log n,α) update time, thus matching the number of forests currently obtainable in near-linear time for static (non-changing) graphs. Our construction goes via dynamic bounded out-degree orientations, and we present a fully-dynamic algorithm that explicitly orients the edges of the dynamic graph, such that no vertex has an out-degree exceeding ⌊ (1+ε)α ⌋ + 2. Our algorithm is deterministic and has a worst-case update time of O(ε^{-6}α² log³ n). The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a β⋅ α + log_β n out-orientation in O(β²α²+βαlog_β n) time [Tsvi Kopelowitz et al., 2014]. As a consequence, we get an algorithm that maintains an implicit vertex colouring with 4⋅ 2^α colours, in amortised poly-log n update time, and with O(α log n) worst-case query time. Thus, at the expense of log n-factors in the update time, we improve on the number of colours from 2^O(α) to O(2^α) compared to the state-of-the-art for implicit dynamic colouring [Monika Henzinger et al., 2020].Aleksander B. G. Christiansen, Eva Rotenberg, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruffwork_ymyqvcyiojcmlpliukdxboygpuTue, 28 Jun 2022 00:00:00 GMTRelation between the correspondence chromatic number and the Alon–Tarsi number
https://scholar.archive.org/work/p33edyskirhfrops2aq4clzdrm
We study the relation between the correspondence chromatic number and the Alon--Tarsi number, both upper bounds on the list chromatic number of a graph. There are many graphs with Alon--Tarsi number greater than the correspondence chromatic number. We present here a family of graphs with arbitrary Alon--Tarsi number, with correspondence chromatic number one larger. Keywords: correspondence coloring, Alon--Tarsi number AMS Mathematics Subject Classification: 05C15Eric Culver, Stephen Hartkework_p33edyskirhfrops2aq4clzdrmThu, 23 Jun 2022 00:00:00 GMTMaximum likelihood thresholds via graph rigidity
https://scholar.archive.org/work/bfqzx3gxhnes7mq7xizz2eanvm
The maximum likelihood threshold (MLT) of a graph G is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of G and use this characterization to give new combinatorial lower bounds on the MLT of any graph. We use the new lower bounds to give high-probability guarantees on the maximum likelihood thresholds of sparse Erdös-Rényi random graphs in terms of their average density. These examples show that the new lower bounds are within a polylog factor of tight, where, on the same graph families, all known lower bounds are trivial. Based on computational experiments made possible by our methods, we conjecture that the MLT of an Erdös-Rényi random graph is equal to its generic completion rank with high probability. Using structural results on rigid graphs in low dimension, we can prove the conjecture for graphs with MLT at most 4 and describe the threshold probability for the MLT to switch from 3 to 4. We also give a geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.Daniel Irving Bernstein, Sean Dewar, Steven J. Gortler, Anthony Nixon, Meera Sitharam, Louis Theranwork_bfqzx3gxhnes7mq7xizz2eanvmSun, 19 Jun 2022 00:00:00 GMT