Truthful Allocation in Graphs and Hypergraphs

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L^p-norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ∞, and extend the results to any p > 0.

Bidimensionality theory provides a general framework for developing subexponential fixed parameter algorithms for NP-hard problems. In this framework, to solve an optimization problem in a graph G, the branchwidth ${\mathop{\rm bw}}(G)$ is first computed or estimated. If ${\mathop{\rm bw}}(G)$ is small then the problem is solved by a branch-decomposition based algorithm which typically runs in polynomial time in the size of G but in exponential time in ${\mathop{\rm bw}}(G)$ . Otherwise, a large ${\mathop{\rm bw}}(G)$ implies a large grid minor of G and the problem is computed or estimated based on the grid minor. A representative example of such algorithms is the one for the longest path problem in planar graphs. Although many subexponential fixed parameter algorithms have been developed based on bidimensionality theory, little is known on the practical performance of these algorithms. We report a computational study on the practical performance of a bidimensionality theory based algorithm for the longest path problem in planar graphs. The results show that the algorithm is practical for computing/estimating the longest path in a planar graph. The tools developed and data obtained in this study may be useful in other bidimensional algorithm studies.

Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. By definition, every ‐degenerate graph is also weakly ‐degenerate. On the other hand, if is weakly ‐degenerate, then (and, moreover, the same bound holds for the list‐chromatic and even the DP‐chromatic number of ). It turns out that several upper bounds in graph coloring theory can be phrased in terms of weak degeneracy. For example, we show that planar graphs are weakly 4‐degenerate, which implies Thomassen's famous theorem that planar graphs are 5‐list‐colorable. We also prove a version of Brooks's theorem for weak degeneracy: a connected graph of maximum degree is weakly ‐degenerate unless . (By contrast, all ‐regular graphs have degeneracy .) We actually prove an even stronger result, namely that for every , there is such that if is a graph of weak degeneracy at least , then either contains a ‐clique or the maximum average degree of is at least . Finally, we show that graphs of maximum degree and either of girth at least 5 or of bounded chromatic number are weakly ‐degenerate, which is best possible up to the value of the implied constant.

We consider the problem of counting the number of copies of a fixed graph H within an input graph G . This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy . This is a rich family of graphs, containing all graphs without a fixed minor (e.g., planar graphs), as well as graphs generated by various random processes (e.g., preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from ’85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices and further proved that for every \( k \gt 5 \) there is a k -vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H . Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here, we show that this sufficient condition is also necessary, thus fully answering the Bera–Pashanasangi–Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.

A graph [Formula: see text] is [Formula: see text]-degenerate if every subgraph of [Formula: see text] has a vertex of degree at most [Formula: see text]. The well-known Brualdi-Solheid problem asks for the maximum spectral radius of a graph belonging to a specified class of graphs and the characterization of extremal graphs. In this paper, we first characterize [Formula: see text]-degenerate graphs with the minimum least eigenvalue. As an application, we give an answer to the Brualdi-Solheid problem for [Formula: see text]-degenerate bipartite graphs. In addition, we give a new proof to determine the maximum signless Laplacian spectral radius and characterize the corresponding extremal graphs among [Formula: see text]-degenerate graphs. We also raise a problem related to the second largest eigenvalues of [Formula: see text]-degenerate graphs and solve this problem for 1-degenerate graphs.

We consider the problem of makespan minimization on m unrelated machines in the context of algorithmic mechanism design, where the machines are the strategic players. This is a multidimensional scheduling domain, and the only known positive results for makespan minimization in such a domain are O(m)-approximation truthful mechanisms [22, 20]. We study a well-motivated special case of this problem, where the processing time of a job on each machine may either be or high, and the low and high values are public and job-dependent. This preserves the multidimensionality of the domain, and generalizes the restricted-machines (i.e., {pj,∞}) setting in scheduling. We give a general technique to convert anyc-approximation algorithm to a 3c-approximation truthful-in-expectation mechanism. This is one of the few known results that shows how to export approximation algorithms for a multidimensional problem into truthful mechanisms in a black-box fashion. When the low and high values are the same for all jobs, we devise a deterministic 2-approximation truthful mechanism. These are the first truthful mechanisms with non-trivial performance guarantees for a multidimensional scheduling domain.Our constructions are novel in two respects. First, we do not utilize or rely on explicit price definitions to prove truthfulness; instead we design algorithms that satisfy cycle monotonicity. Cycle monotonicity [23] is a necessary and sufficient condition for truthfulness, is a generalization of value monotonicity for multidimensional domains. However, whereas value monotonicity has been used extensively and successfully to design truthful mechanisms in single-dimensional domains, ours is the first work that leverages cycle monotonicity in the multidimensional setting. Second, our randomized mechanisms are obtained by first constructing a fractional truthful mechanism for a fractional relaxation of the problem, and then converting it into a truthful-in-expectation mechanism. This builds upon a technique of [16], and shows the usefulness of fractional mechanisms in truthful mechanism design.

We consider dynamic subgraph connectivity problems for planar undirected graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either off (failed) or (recovered). We wish to answer connectivity queries with respect to the subgraph. The model has two natural variants, one in which there are d edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed. We present a d-failure connectivity oracle for planar graphs that processes any d edge/vertex failures in sort(d,n) time so that connectivity queries can be answered in pred(d,n) time. (Here sort and pred are the time for integer sorting and integer predecessor search over a subset of [n] of size d.) Our algorithm has two discrete parts. The first is an algorithm tailored to triconnected planar graphs. It makes use of Barnette's theorem, which states that every triconnected planar graph contains a degree-3 spanning tree. The second part is a generic reduction from general (planar) graphs to triconnected (planar) graphs. Our algorithm is, moreover, provably optimal. An implication of Pǎtrascu and Thorup's lower bound on predecessor search is that no d-failure connectivity oracle (even on trees) can beat pred(d,n) query time. We extend our algorithms to the subgraph connectivity model where edge/vertex failures (but no recoveries) are intermixed with connectivity queries. In triconnected planar graphs each failure and query is handled in O(logn) time (amortized), whereas in general planar graphs both bounds become O(log2n).

On the algorithmic complexity of twelve covering and independence parameters of graphs

Given a graph , a decomposition of is a partition of its edges. A graph is ‐decomposable if its edge set can be partitioned into a ‐degenerate graph and a graph with maximum degree at most . For , we are interested in the minimum integer such that every planar graph is ‐decomposable. It was known that , , and . This paper proves that , and .

We study methods for finding strict upper bounds on the fractional chromatic number $\chi_f(G)$ of a graph $G$. We illustrate these methods by providing short proofs of known inequalities in connection with Grötzsch's 3-color theorem and the 5-color theorem for planar graphs. We also apply it to $d$-degenerate graphs and conclude that every $K_{d+1}$-free $d$-degenerate graph with $n$ vertices has independence number $<n/(d+1)$. We show that for each surface $S$ and every $\varepsilon>0$, the fractional chromatic number of any graph embedded on $S$ of sufficiently large width (depending only on $S$ and $\varepsilon$) is at most $4+\varepsilon$. In the same spirit we prove that Eulerian triangulations or triangle-free graphs of large width have $\chi_f\le 3+\varepsilon$, and quadrangulations of large width have $\chi_f\le 2+\varepsilon$. While the $\varepsilon$ is needed in the latter two results, we conjecture that in the first result $4+\varepsilon$ can be replaced by 4. The upper bounds $\chi_f\le 4+\varepsilon$, $\chi_f\le 3+\varepsilon$, $\chi_f\le 2+\varepsilon$, respectively, are already known for graphs on orientable surfaces, but our results are also valid for graphs on nonorientable surfaces. Surprisingly, a strict lower bound on the fractional chromatic number may imply an upper bound on the chromatic number: Grötzsch's theorem implies that every 4-chromatic planar graph $G$ has fractional chromatic number $\chi_f(G)\ge 3$. We conjecture that this inequality is always strict and observe that this implies the 4-color theorem for planar graphs.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80's, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C2k and C2k+1 in n-vertex graphs of bounded degeneracy in time , where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time . Conversely, one can transform any algorithm for computing the number of homomorphic copies of C2k or of C2k+1 in n-vertex graphs of bounded degeneracy, into an time algorithm for detecting directed copies of Ck in general m-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of Ck-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

A dominating set $D$ of a graph $G$ is a set of vertices such that any vertex in $G$ is in $D$ or its neighbor is in $D$. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a $k$-degenerate graph in $O(k)$ time per solution using $O(n + m)$ space, where $n$ and $m$ are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, $H$-minor free graphs with some fixed $H$. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.

Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C 2k and C 2k+1 in n -vertex graphs of bounded degeneracy in time Õ( n d k ), where the fastest known algorithm for detecting directed copies of C k in general m -edge digraphs runs in time Õ( m d k ). Conversely, one can transform any O(n b k ) algorithm for computing the number of homomorphic copies of C 2k or of C 2k+1 in n -vertex graphs of bounded degeneracy, into an Õ( m b k ) time algorithm for detecting directed copies of C k in general m -edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of C k -homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k -cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.

This paper introduces the concept of weak $(d,h)$-decomposition of a graph $G$, which is defined as a partition of $E(G)$ into two subsets $E_1,E_2$, such that $E_1$ induces a $d$-degenerate graph $H_1$ and $E_2$ induces a subgraph $H_2$ with $\alpha(H_1[N_{H_2}(v)]) \le h$ for any vertex $v$. We prove that each planar graph admits a weak $(2,3)$-decomposition. As a consequence, every planar graph $G$ has a subgraph $H$ such that $G-E(H)$ is $3$-paintable and any proper coloring of $G-E(H)$ is a $3$-defective coloring of $G$. This improves the result in [G. Gutowski, M. Han, T. Krawczyk, and X. Zhu, Defective $3$-paintability of planar graphs, Electron. J. Combin., 25(2):\#P2.34, 2018] that every planar graph is 3-defective $3$-paintable.

We considered routing problems for plane graphs to solve control problems of cutting machines in the industry. According to the cutting plan, we form its homeomorphic image in the form of a plane graph G. We determine the appropriate type of route for the given graph: OE-route represents an ordered sequence of chains satisfying the requirement that the part of the route that is not passed does not intersect the interior of its passed part, AOE-chain represents OE-chain consecutive edges which are incident to vertex v and they are neighbours in the cyclic order O±(v), NOE-route represents the non-intersecting OE-route, PPOE-route represents the Pierce Point NOE-route with allowable pierce points that are start points of OE-chains forming this route. We analyse the solvability of the listed routing problems in graph G. We developed the polynomial algorithms for obtaining listed routes with the minimum number of covering paths and the minimum length of transitions between the ending of the current path and the beginning of the next path. The solutions proposed in the article can improve the quality of technological preparation of cutting processes in CAD/CAM systems.