For graphs $G$ and $H$, we say that $G$ is $H$-free if it does not contain $H$ as an induced subgraph. Already in the early 1980s Alekseev observed that if $H$ is connected, then the \textsc{Max Weight Independent Set} problem (MWIS) remains \textsc{NP}-hard in $H$-free graphs, unless $H$ is a path or a subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory. A general belief is that the problem is polynomial-time solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in $H$-free graphs, where $H$ is any fixed path. If $H$ is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al., SODA 2019]. In this paper we make an important step towards solving the problem by showing that for any subdivided claw $H$, MWIS is polynomial-time solvable in $H$-free graphs of bounded degree.
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Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal $O(n \log n)$ sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using $L^p$-norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS'13). The non-linearity of $L^p$-norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the $L^p$-analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph $G(n,d/n)$, where the previously known algorithms run in time $n^{O(\log d)}$ or applied only to large $d$. We refine these algorithmic bounds significantly, and develop fast $n^{1+o(1)}$ algorithms based on Glauber dynamics that apply to all $d$, throughout the uniqueness regime.
A partial orientation $\vec{H}$ of a graph $G$ is a weak $r$-guidance system if for any two vertices at distance at most $r$ in $G$, there exists a shortest path $P$ between them such that $\vec{H}$ directs all but one edge in $P$ towards this edge. In case $\vec{H}$ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most $r$ in $G$. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.
We describe a polynomial-time algorithm which, given a graph $G$ with treewidth $t$, approximates the pathwidth of $G$ to within a ratio of $O(t\sqrt{\log t})$. This is the first algorithm to achieve an $f(t)$-approximation for some function $f$. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least $th+2$ has treewidth at least $t$ or contains a subdivision of a complete binary tree of height $h+1$. The bound $th+2$ is best possible up to a multiplicative constant. This result was motivated by, and implies (with $c=2$), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant $c$ such that every graph with pathwidth $\Omega(k^c)$ has treewidth at least $k$ or contains a subdivision of a complete binary tree of height $k$. Our main technical algorithm takes a graph $G$ and some (not necessarily optimal) tree decomposition of $G$ of width $t'$ in the input, and it computes in polynomial time an integer $h$, a certificate that $G$ has pathwidth at least $h$, and a path decomposition of $G$ of width at most $(t'+1)h+1$. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height $h$. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth.
In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with $m$ edges and $n$ vertices in $O(mn)$ time with space complexity $O(n)$.
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to $H$-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.
For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most $d$ if $d\leq 2$ and NP-complete if $d\geq 3$. We prove the same dichotomy for graphs of bounded radius. For a graph $H$, a graph is $H$-free if it does not contain $H$ as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for $P_6$-free graphs, extending a recent result of Feghali for $P_5$-free graphs. We then extend our result to hold even for $(sP_3+P_6)$-free graphs for every $s\geq 0$ and initiate a complexity classification of Matching Cut for $H$-free graphs.
Computing a maximum independent set (MaxIS) is a fundamental NP-hard problem in graph theory, which has important applications in a wide spectrum of fields. Since graphs in many applications are changing frequently over time, the problem of maintaining a MaxIS over dynamic graphs has attracted increasing attention over the past few years. Due to the intractability of maintaining an exact MaxIS, this paper aims to develop efficient algorithms that can maintain an approximate MaxIS with an accuracy guarantee theoretically. In particular, we propose a framework that maintains a $(\frac{\Delta}{2} + 1)$-approximate MaxIS over dynamic graphs and prove that it achieves a constant approximation ratio in many real-world networks. To the best of our knowledge, this is the first non-trivial approximability result for the dynamic MaxIS problem. Following the framework, we implement an efficient linear-time dynamic algorithm and a more effective dynamic algorithm with near-linear expected time complexity. Our thorough experiments over real and synthetic graphs demonstrate the effectiveness and efficiency of the proposed algorithms, especially when the graph is highly dynamic.
Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.
In the storied Colonel Blotto game, two colonels allocate $a$ and $b$ troops, respectively, to $k$ distinct battlefields. A colonel wins a battle if they assign more troops to that particular battle, and each colonel seeks to maximize their total number of victories. Despite the problem's formulation in 1921, the first polynomial-time algorithm to compute Nash equilibrium (NE) strategies for this game was discovered only quite recently. In 2016, \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} formulated a breakthrough algorithm to compute NE strategies for the Colonel Blotto game\footnote{To the best of our knowledge, the algorithm from \citep{ahmadinejad_dehghani_hajiaghayi_lucier_mahini_seddighin_2019} has computational complexity $O(k^{14}\max\{a,b\}^{13})$}, receiving substantial media coverage (e.g. \citep{Insider}, \citep{NSF}, \citep{ScienceDaily}). In this work, we present the first known $\epsilon$-approximation algorithm to compute NE strategies in the two-player Colonel Blotto game in runtime $\widetilde{O}(\epsilon^{-4} k^8 \max\{a,b\}^2)$ for arbitrary settings of these parameters. Moreover, this algorithm computes approximate coarse correlated equilibrium strategies in the multiplayer (continuous and discrete) Colonel Blotto game (when there are $\ell > 2$ colonels) with runtime $\widetilde{O}(\ell \epsilon^{-4} k^8 n^2 + \ell^2 \epsilon^{-2} k^3 n (n+k))$, where $n$ is the maximum troop count. Before this work, no polynomial-time algorithm was known to compute exact or approximate equilibrium (in any sense) strategies for multiplayer Colonel Blotto with arbitrary parameters. Our algorithm computes these approximate equilibria by a novel (to the author's knowledge) sampling technique with which we implicitly perform multiplicative weights update over the exponentially many strategies available to each player.
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