We study the PSPACE-complete $k$-Canadian Traveller Problem, where a weighted graph $G=(V,E,\omega)$ with a source $s\in V$ and a target $t\in V$ are given. This problem also has a hidden input $E_* \subsetneq E$ of cardinality at most $k$ representing blocked edges. The objective is to travel from $s$ to $t$ with the minimum distance. At the beginning of the walk, the blockages $E_*$ are unknown: the traveller discovers that an edge is blocked when visiting one of its endpoints. Online algorithms, also called strategies, have been proposed for this problem and assessed with the competitive ratio, i.e. the ratio between the distance actually traversed by the traveller divided by the distance we would have traversed knowing the blockages in advance. Even though the optimal competitive ratio is $2k+1$ even on unit-weighted planar graphs of treewidth 2, we design a polynomial-time strategy achieving competitive ratio $9$ on unit-weighted outerplanar graphs. This value $9$ also stands as a lower bound for this family of graphs as we prove that, for any $\varepsilon > 0$, no strategy can achieve a competitive ratio $9-\varepsilon$. Finally, we show that it is not possible to achieve a constant competitive ratio (independent of $G$ and $k$) on weighted outerplanar graphs.
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Given a graph $G=(V,E)$ and an integer $k\in \mathbb{N}$, we study {\sc 2-Eigenvalue Vertex Deletion} (2-EVD), where the goal is to remove at most $k$ vertices such that the adjacency matrix of the resulting graph has at most 2 eigenvalues. It is known that the adjacency matrix of a graph has at most 2 eigenvalues if and only if the graph is a collection of equal sized cliques. So {\sc 2-Eigenvalue Vertex Deletion} amounts to removing a set of at most $k$ vertices such that the resulting graph is a collection of equal sized cliques. The {\sc 2-Eigenvalue Edge Editing} (2-EEE), {\sc 2-Eigenvalue Edge Deletion} (2-EED) and {\sc 2-Eigenvalue Edge Addition} (2-EEA) problems are defined analogously. We provide a kernel of size $\mathcal{O}(k^{3})$ for {\sc $2$-EVD}. For the problems {\sc $2$-EEE} and {\sc $2$-EED}, we provide kernels of size $\mathcal{O}(k^{2})$. Finally, we provide a linear kernel of size $6k$ for {\sc $2$-EEA}. We thereby resolve three open questions listed by Misra et al. (ISAAC 2023) concerning the complexity of these problems parameterized by the solution size.
In this paper, we consider the minimization of a nonsmooth nonconvex objective function $f(x)$ over a closed convex subset $\mathcal{X}$ of $\mathbb{R}^n$, with additional nonsmooth nonconvex constraints $c(x) = 0$. We develop a unified framework for developing Lagrangian-based methods, which takes a single-step update to the primal variables by some subgradient methods in each iteration. These subgradient methods are ``embedded'' into our framework, in the sense that they are incorporated as black-box updates to the primal variables. We prove that our proposed framework inherits the global convergence guarantees from these embedded subgradient methods under mild conditions. In addition, we show that our framework can be extended to solve constrained optimization problems with expectation constraints. Based on the proposed framework, we show that a wide range of existing stochastic subgradient methods, including the proximal SGD, proximal momentum SGD, and proximal ADAM, can be embedded into Lagrangian-based methods. Preliminary numerical experiments on deep learning tasks illustrate that our proposed framework yields efficient variants of Lagrangian-based methods with convergence guarantees for nonconvex nonsmooth constrained optimization problems.
Given a metric space $(X,d_X)$, a $(\beta,s,\Delta)$-sparse cover is a collection of clusters $\mathcal{C}\subseteq P(X)$ with diameter at most $\Delta$, such that for every point $x\in X$, the ball $B_X(x,\frac\Delta\beta)$ is fully contained in some cluster $C\in \mathcal{C}$, and $x$ belongs to at most $s$ clusters in $\mathcal{C}$. Our main contribution is to show that the shortest path metric of every $K_r$-minor free graphs admits $(O(r),O(r^2),\Delta)$-sparse cover, and for every $\epsilon>0$, $(4+\epsilon,O(\frac1\epsilon)^r,\Delta)$-sparse cover (for arbitrary $\Delta>0$). We then use this sparse cover to show that every $K_r$-minor free graph embeds into $\ell_\infty^{\tilde{O}(\frac1\epsilon)^{r+1}\cdot\log n}$ with distortion $3+\eps$ (resp. into $\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$ with distortion $O(r)$). Further, we provide applications of these sparse covers into padded decompositions, sparse partitions, universal TSP / Steiner tree, oblivious buy at bulk, name independent routing, and path reporting distance oracles.
The classical Heawood inequality states that if the complete graph $K_n$ on $n$ vertices is embeddable in the sphere with $g$ handles, then $g \ge\dfrac{(n-3)(n-4)}{12}$. A higher-dimensional analogue of the Heawood inequality is the K\"uhnel conjecture. In a simplified form it states that for every integer $k>0$ there is $c_k>0$ such that if the union of $k$-faces of $n$-simplex embeds into the connected sum of $g$ copies of the Cartesian product $S^k\times S^k$ of two $k$-dimensional spheres, then $g\ge c_k n^{k+1}$. For $k>1$ only linear estimates were known. We present a quadratic estimate $g\ge c_k n^2$. The proof is based on beautiful and fruitful interplay between geometric topology, combinatorics and linear algebra.
We investigate the consequence of two Lip$(\gamma)$ functions, in the sense of Stein, being close throughout a subset of their domain. A particular consequence of our results is the following. Given $K_0 > \varepsilon > 0$ and $\gamma > \eta > 0$ there is a constant $\delta = \delta(\gamma,\eta,\varepsilon,K_0) > 0$ for which the following is true. Let $\Sigma \subset \mathbb{R}^d$ be closed and $f , h : \Sigma \to \mathbb{R}$ be Lip$(\gamma)$ functions whose Lip$(\gamma)$ norms are both bounded above by $K_0$. Suppose $B \subset \Sigma$ is closed and that $f$ and $h$ coincide throughout $B$. Then over the set of points in $\Sigma$ whose distance to $B$ is at most $\delta$ we have that the Lip$(\eta)$ norm of the difference $f-h$ is bounded above by $\varepsilon$. More generally, we establish that this phenomenon remains valid in a less restrictive Banach space setting under the weaker hypothesis that the two Lip$(\gamma)$ functions $f$ and $h$ are only close in a pointwise sense throughout the closed subset $B$. We require only that the subset $\Sigma$ be closed; in particular, the case that $\Sigma$ is finite is covered by our results. The restriction that $\eta < \gamma$ is sharp in the sense that our result is false for $\eta := \gamma$.
A subset $S$ of vertices in a graph $G$ is a secure total dominating set of $G$ if $S$ is a total dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup \{u\}$ is also a total dominating set of $G$. We show that if $G$ is a maximal outerplanar graph of order $n$, then $G$ has a total secure dominating set of size at most $\lfloor 2n/3 \rfloor$. Moreover, if an outerplanar graph $G$ of order $n$, then each secure total dominating set has at least $\lceil (n+2)/3 \rceil$ vertices. We show that these bounds are best possible.
We study the problem of finding a maximum-cardinality set of $r$-cliques in an undirected graph of fixed maximum degree $\Delta$, subject to the cliques in that set being either vertex-disjoint or edge-disjoint. It is known for $r=3$ that the vertex-disjoint (edge-disjoint) problem is solvable in linear time if $\Delta=3$ ($\Delta=4$) but APX-hard if $\Delta \geq 4$ ($\Delta \geq 5$). We generalise these results to an arbitrary but fixed $r \geq 3$, and provide a complete complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree $\Delta$. Specifically, we show that the vertex-disjoint problem is solvable in linear time if $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta < 5r/3 - 1$, and APX-hard if $\Delta \geq \lceil 5r/3 \rceil - 1$. We also show that if $r\geq 6$ then the above implications also hold for the edge-disjoint problem. If $r \leq 5$, then the edge-disjoint problem is solvable in linear time if $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta \leq 2r - 2$, and APX-hard if $\Delta > 2r - 2$.
We study the algorithmic task of finding large independent sets in Erdos-Renyi $r$-uniform hypergraphs on $n$ vertices having average degree $d$. Krivelevich and Sudakov showed that the maximum independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$. We show that the class of low-degree polynomial algorithms can find independent sets of density $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$ but no larger. This extends and generalizes earlier results of Gamarnik and Sudan, Rahman and Virag, and Wein on graphs, and answers a question of Bal and Bennett. We conjecture that this statistical-computational gap holds for this problem. Additionally, we explore the universality of this gap by examining $r$-partite hypergraphs. A hypergraph $H=(V,E)$ is $r$-partite if there is a partition $V=V_1\cup\cdots\cup V_r$ such that each edge contains exactly one vertex from each set $V_i$. We consider the problem of finding large balanced independent sets (independent sets containing the same number of vertices in each partition) in random $r$-partite hypergraphs with $n$ vertices in each partition and average degree $d$. We prove that the maximum balanced independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$ asymptotically. Furthermore, we prove an analogous low-degree computational threshold of $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$. Our results recover and generalize recent work of Perkins and the second author on bipartite graphs. While the graph case has been extensively studied, this work is the first to consider statistical-computational gaps of optimization problems on random hypergraphs. Our results suggest that these gaps persist for larger uniformities as well as across many models. A somewhat surprising aspect of the gap for balanced independent sets is that the algorithm achieving the lower bound is a simple degree-1 polynomial.
Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \to L^2(0,T)$, known as the modified Hilbert transform. This operator has shown its significance in space-time formulations related to the heat and wave equations. In this paper, we establish a direct connection between the modified Hilbert transform $\mathcal{H}_T$ and the canonical Hilbert transform $\mathcal{H}$. Specifically, we prove the relationship $\mathcal{H}_T \varphi = -\mathcal{H} \tilde{\varphi}$, where $\varphi \in L^2(0,T)$ and $\tilde{\varphi}$ is a suitable extension of $\varphi$ over the entire $\mathbb{R}$. By leveraging this crucial result, we derive some properties of $\mathcal{H}_T$, including a new inversion formula, that emerge as immediate consequences of well-established findings on $\mathcal{H}$.
In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N , \mathcal{C}_{ < 5} ) = \Omega_r ( N^{3/2 - o(1)} )$ for $r \in \{4,5,6 \}$. We address an unproved claim from [31] asserting a technique of Ruzsa can be used to show that this lower bound holds for all $r \geq 3$. We carefully explain one of the main obstacles that was overlooked at the time the claim from [31] was made, and show that this obstacle can be overcome when $r\in \{4,5,6\}$. We use constructions from coding theory to prove nontrivial lower bounds that hold for all $r \geq 3$. Finally, we use a recent result of Conlon, Fox, Sudakov, and Zhao to show that the sphere packing bound from coding theory may be improved when upper bounding the size of linear $q$-ary codes of distance $6$.
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