Extremal results in sparse pseudorandom graphs

Szemer\'edi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemer\'edi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.Comment: 70 pages, accepted for publication in Adv. Mat

Finding a Maximum Restricted tt-Matching via Boolean Edge-CSP

The problem of finding a maximum $2$-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum $t$-matching which excludes specified complete $t$-partite subgraphs, where $t$ is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete $t$-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most $2t-1$. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum $2$-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most $t+1$. Our algorithms are obtained from exploiting the discrete structure of restricted $t$-matchings and employing an algorithm for the Boolean edge-CSP.Comment: 20 pages, 2 figure

Streaming Verification of Graph Properties

Streaming interactive proofs (SIPs) are a framework for outsourced computation. A computationally limited streaming client (the verifier) hands over a large data set to an untrusted server (the prover) in the cloud and the two parties run a protocol to confirm the correctness of result with high probability. SIPs are particularly interesting for problems that are hard to solve (or even approximate) well in a streaming setting. The most notable of these problems is finding maximum matchings, which has received intense interest in recent years but has strong lower bounds even for constant factor approximations. In this paper, we present efficient streaming interactive proofs that can verify maximum matchings exactly. Our results cover all flavors of matchings (bipartite/non-bipartite and weighted). In addition, we also present streaming verifiers for approximate metric TSP. In particular, these are the first efficient results for weighted matchings and for metric TSP in any streaming verification model.Comment: 26 pages, 2 figure, 1 tabl

Finding a Maximum 2-Matching Excluding Prescribed Cycles in Bipartite Graphs

We introduce a new framework of restricted 2-matchings close to Hamilton cycles. For an undirected graph (V,E) and a family U of vertex subsets, a 2-matching F is called U-feasible if, for each setU in U, F contains at most |setU|-1 edges in the subgraph induced by U. Our framework includes C_{= 5. For instance, in bipartite graphs in which every cycle of length six has at least two chords, our algorithm solves the maximum C_{<=6}-free 2-matching problem in O(n^2 m) time, where n and m are the numbers of vertices and edges, respectively

Dynamic programming on bipartite tree decompositions

We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that $K_t$-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by bipartite treewidth. We provide the following complexity dichotomy when $H$ is a 2-connected graph, for each of $H$-Subgraph-Packing, $H$-Induced-Packing, $H$-Scattered-Packing, and $H$-Odd-Minor-Packing problem: if $H$ is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if $H$ is non-bipartite, then it is solvable in XP-time. We define 1-${\cal H}$-treewidth by replacing the bipartite graph class by any class ${\cal H}$. Most of the technology developed here works for this more general parameter.Comment: Presented in IPEC 202

Structure of conflict graphs in constraint alignment problems and algorithms

We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if the vertex pairs are adjacent in their respective graphs. %In general terms The goal is to provide a one-to-one mapping between the vertices of the input graphs in order to maximize edge conservation. However the allowed mappings are restricted since each vertex from $V_1$ (resp. $V_2$) is allowed to be mapped to at most $m_1$ (resp. $m_2$) specified vertices in $V_2$ (resp. $V_1$). Most of results in this paper deal with the case $m_2=1$ which attracted most attention in the related literature. We formulate the problem as a maximum independent set problem in a related {\em conflict graph} and investigate structural properties of this graph in terms of forbidden subgraphs. We are interested, in particular, in excluding certain wheals, fans, cliques or claws (all terms are defined in the paper), which corresponds in excluding certain cycles, paths, cliques or independent sets in the neighborhood of each vertex. Then, we investigate algorithmic consequences of some of these properties, which illustrates the potential of this approach and raises new horizons for further works. In particular this approach allows us to reinterpret a known polynomial case in terms of conflict graph and to improve known approximation and fixed-parameter tractability results through efficiently solving the maximum independent set problem in conflict graphs. Some of our new approximation results involve approximation ratios that are function of the optimal value, in particular its square root; this kind of results cannot be achieved for maximum independent set in general graphs.Comment: 22 pages, 6 figure