s of Invited Talks Sketching for Data Streams and Numerical Linear Algebra

We survey recent developments related to the Minimum Circuit Size Problem.

We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.

We study the decomposition of multigraphs with a constant edge multiplicity into copies of a fixed star H=K"1","t: We present necessary and sufficient conditions for such a decomposition to exist where t=2 and prove NP-completeness of the corresponding decision problem for any t>=3. We also prove NP-completeness when the edge multiplicity function is not restricted either on the input G or on the fixed multistar H.

We study the Decomposition Conjecture posed by Bar\'at and Thomassen (2006), which states that for every tree $T$ there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(T)|$ divides $|E(G)|$, then $G$ admits a decomposition into copies of $T$. In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length $3$, and paths whose length is a power of $2$. We verify the Decomposition Conjecture for paths of length $5$.

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Bar a t and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. The same result has been independently obtained by Carsten Thomassen (2013). Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a $191$-edge-connected graph of size divisible by $4$, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant.

We study decision problems of the following form: Given an instance of a combinatorial problem, can it be solved by a greedy algorithm? We present algorithms for the recognition of greedy instances of certain problems, structural characterization of such instances for other problems, and proofs ofNP-hardness of the recognition problem for some other cases. Previous results of this type are also stated and reviewed.

We show that, for each natural number k>1, every graph (possibly with multiple edges but with no loops) of edge-connectivity at least 2k^2+k has an orientation with any prescribed outdegrees modulo k provided the prescribed outdegrees satisfy the obvious necessary conditions. For k=3 the edge-connectivity 8 suffices. This implies the weak 3-flow conjecture proposed in 1988 by Jaeger (a natural weakening of Tutte@?s 3-flow conjecture which is still open) and also a weakened version of the more general circular flow conjecture proposed by Jaeger in 1982. It also implies the tree-decomposition conjecture proposed in 2006 by Barat and Thomassen when restricted to stars. Finally, it is the currently strongest partial result on the (2+@e)-flow conjecture by Goddyn and Seymour.

We prove that a 171-edge-connected graph has an edgedecomposition into paths of length 3 if and only its size is divisible by 3. It is a long-standing problem whether 2-edge-connectedness is sufficient for planar triangle-free graphs, and whether 3-edge-connectedness suffices for graphs in general. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 286–292, 2008 Keyword: path-decompositions

We prove that a 171-edge-connected graph has an edge-decomposition into paths of length 3 if and only its size is divisible by 3. It is a long-standing problem whether 2-edge-connectedness is sufficient for planar triangle-free graphs, and whether 3-edge-connectedness suffices for graphs in general. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 286–292, 2008

We formally study two methods for data sanitation that have been used extensively in the database community: k-anonymity and l- diversity. We settle several open problems concerning the difficulty of applying these methods optimally, proving both positive and negative results: - 2-anonymity is in P. - The problem of partitioning the edges of a triangle-free graph into 4-stars (degree-three vertices) is NP-hard. This yields an alternative proof that 3-anonymity is NP-hard even when the database attributes are all binary. - 3-anonymity with only 27 attributes per record is MAX SNP-hard. - For databases with n rows, k-anonymity is in O(4n ċ poly(n)) time for all k > 1. - For databases with l attributes, alphabet size c, and n rows, k- Anonymity can be solved in 2O(k2(2c)l) + O(nl) time. - 3-diversity with binary attributes is NP-hard, with one sensitive attribute. - 2-diversity with binary attributes is NP-hard, with three sensitive attributes.

We consider the problem of switching cost in optical networks, where messages are sent along lightpaths. Given lightpaths, we have to assign them colors, so that at most glightpaths of the same color can share any edge (gis the grooming factor). The switching of the lightpaths is performed by electronic ADMs (Add-Drop-Multiplexers) at their endpoints and optical ADMs (OADMs) at their intermediate nodes. The saving in the switching components becomes possible when lightpaths of the same color can use the same switches. Whereas previous studies concentrated on the number of ADMs, we consider the cost function - incurred also by the number of OADMs - of f(i¾?) = i¾?|OADMs| + (1 i¾? i¾?)|ADMs|, where 0 ≤ i¾?≤ 1. We concentrate on chain networks, but our technique can be directly extended to ring networks. We show that finding a coloring which will minimize this cost function is NP-complete, even when the network is a chain and the grooming factor is g= 2, for any value of i¾?. We then present a general technique that, given an r-approximation algorithm working on particular instances of our problem, i.e. instances in which all requests share a common edge of the chain, builds a new algorithm for general instances having approximation ratio ri¾?logni¾?. This technique is used in order to obtain two polynomial time approximation algorithms for our problem: the first one minimizes the number of OADMs (the case of i¾?= 1), and its approximation ratio is 2 i¾?logni¾?; the second one minimizes the combined cost f(i¾?) for 0 ≤ i¾?< 1, and its approximation ratio is $2 \sqrt{g}~\lceil \log n \rceil$.

We consider the problem of discovering overlapping communities in networks which we model as generalizations of Graph Packing problems with overlap. We seek a collection $\mathcal{S}' \subseteq \mathcal{S}$ consisting of at least $k$ sets subject to certain disjointness restrictions. In the $r$-Set Packing with $t$-Membership, each element of $\mathcal{U}$ belongs to at most $t$ sets of $\mathcal{S'}$ while in $t$-Overlap each pair of sets in $\mathcal{S'}$ overlaps in at most $t$ elements. Each set of $\mathcal{S}$ has at most $r$ elements. Similarly, both of our graph packing problems seek a collection $\mathcal{K}$ of at least $k$ subgraphs in a graph $G$ each isomorphic to a graph $H \in \mathcal{H}$. In $\mathcal{H}$-Packing with $t$-Membership, each vertex of $G$ belongs to at most $t$ subgraphs of $\mathcal{K}$ while in $t$-Overlap each pair of subgraphs in $\mathcal{K}$ overlaps in at most $t$ vertices. Each member of $\mathcal{H}$ has at most $r$ vertices and $m$ edges. We show NP-Completeness results for all of our packing problems and we give a dichotomy result for the $\mathcal{H}$-Packing with $t$-Membership problem analogous to the Kirkpatrick and Hell \cite{Kirk78}. We reduce the $r$-Set Packing with $t$-Membership to a problem kernel with $O((r+1)^r k^{r})$ elements while we achieve a kernel with $O(r^r k^{r-t-1})$ elements for the $r$-Set Packing with $t$-Overlap. In addition, we reduce the $\mathcal{H}$-Packing with $t$-Membership and its edge version to problem kernels with $O((r+1)^r k^{r})$ and $O((m+1)^{m} k^{{m}})$ vertices, respectively. On the other hand, we achieve kernels with $O(r^r k^{r-t-1})$ and $O(m^{m} k^{m-t-1})$ vertices for the $\mathcal{H}$-Packing with $t$-Overlap and its edge version, respectively. In all cases, $k$ is the input parameter while $t$, $r$, and $m$ are constants.

We consider the problem of discovering overlapping communities in networks that we model as generalizations of the Set and Graph Packing problems with overlap. As usual for Set Packing problems, we seek a collection <i>S^′</i> ⊆ <i>S</i> consisting of at least <i>k</i> sets subject to certain disjointness restrictions. In the <i>r</i>-Set Packing with <i>t</i>-Membership, each element of <i>U</i> belongs to at most <i>t</i> sets of <i>S^′</i>, while in <i>r</i>-Set Packing with <i>t</i>-Overlap, each pair of sets in <i>S^′</i> overlaps in at most <i>t</i> elements. For both problems, each set of <i>S</i> has at most <i>r</i> elements. Similarly, both of our Graph Packing problems seek a collection <i>K</i> of at least <i>k</i> subgraphs in a graph <i>G</i>, each isomorphic to a graph <i>H</i> ∈ <i>H</i>. In <i>H</i>-Packing with <i>t</i>-Membership, each vertex of <i>G</i> belongs to at most <i>t</i> subgraphs of <i>K</i>, while in <i>H</i>-Packing with <i>t</i>-Overlap, each pair of subgraphs in <i>K</i> overlaps in at most <i>t</i> vertices. For both problems, each member of <i>H</i> has at most <i>r</i> vertices and <i>m</i> edges, where <i>t</i>, <i>r</i>, and <i>m</i> are constants. Here, we show NP-completeness results for all of our packing problems. Furthermore, we give a dichotomy result for the <i>H</i>-Packing with <i>t</i>-Membership problem analogous to the Kirkpatrick and Hell dichotomy [Kirkpatrick and Hell 1978]. Using polynomial parameter transformations, we reduce the <i>r</i>-Set Packing with <i>t</i>-Membership to a problem kernel with <i>O</i>((<i>r</i> + 1)^<i>r</i><i>k^r</i>) elements and the <i>H</i>-Packing with <i>t</i>-Membership and its edge version to problem kernels with <i>O</i>((<i>r</i> + 1)^<i>r</i><i>k^r</i>) and <i>O</i>((<i>m</i> + 1)^<i>m</i><i>k^m</i>) vertices, respectively. On the other hand, by generalizing [Fellows et al. 2008; Moser 2009], we achieve a kernel with <i>O</i>(<i>r^rk</i>^{<i>r</i> − <i>t</i> − 1}) elements for the <i>r</i>-Set Packing with <i>t</i>-Overlap and kernels with <i>O</i>(<i>r^rk</i>^{<i>r</i> − <i>t</i> − 1}) and <i>O</i>(<i>m^mk</i>^{<i>m</i> − <i>t</i> − 1}) vertices for the <i>H</i>-Packing with <i>t</i>-Overlap and its edge version, respectively. In all cases, <i>k</i> is the input parameter, while <i>t</i>, <i>r</i>, and <i>m</i> are constants.

We consider the following classes of quantified boolean formulas. Fix a finite set of basic boolean functions. Take conjunctions of these basic functions applied to variables and constants in arbitrary ways. Finally quantify existentially or universally some of the variables. We prove the following dichotomy theorem: For any set of basic boolean functions, the resulting set of formulas is either polynomially learnable from equivalence queries alone or else it is not PAC-predictable even with membership queries under cryptographic assumptions. Furthermore, we identify precisely which sets of basic functions are in which of the two cases.

We consider an optimization problem arising in the design of optical networks. We are given a bipartite graph G = (L,R,E) over the node set L ∪ R where the edge set is E ⊆ {[u, v] : u ∈ L, v ∈ R}, and implicitly a collection of all four-nodes cycles in the complete graph over V. The goal is to find a minimum size sub-collection of graphs G1,G2, . . . , Gp where for each i Gi is isomorphic to a cycle over four nodes, and such that the edge set E is contained in the union (over all i) of the edge sets of Gi. Noting that every four edge cycle can be a part of the solution, this covering problem is a special case of the unweighted 4-set cover problem. This specialization allows us to obtain an improved approximation guarantee. Whereas the currently best known approximation algorithm for the general unweighted 4-set cover problem has an approximation ratio of H4 - 196/390 ≅ 1.58077 (where Hp denotes the p-th harmonic number), we show that for every Ɛ > 0 there is a polynomial time (13/10 + Ɛ)-approximation algorithm for our problem. Our analysis of the greedy algorithm shows that when applied to covering a bipartite graph using copies of Kq,q bicliques, it returns a feasible solution whose cost is at most (Hq2 -Hq +1)OPT +1 where OPT denotes the optimal cost, thus improving the approximation bound by a factor of almost 2.

We conjecture that, for each tree T , there exists a natural number kT such that the following holds: If G is a kT -edge-connected graph such that |E(T )| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T . We prove that for T = K1,3 (the claw), this holds if and only if there exists a (smallest) natural number kt such that every kt-edge-connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte’s 3-flow conjecture says that kt = 4. We prove the weaker statement that every 4dlog ne-edge-connected graph with n vertices has an edge-decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge-decomposition into claws.

We begin a systematic study of how Graph Decomposition problems may be represented using propositional formulas, and hence solved using SAT-solver technology. By making use of symmetry breaking techniques we are able to obtain solutions to several previously unknown cases and to significantly reduce the time needed to compute decompositions. However some fairly small instances remain unsolved, and thus provide an interesting challenge to SAT-solver technology.

This thesis contains various new results in the areas of design theory and edge decompositions of graphs and hypergraphs. Most notably, we give a new proof of the existence conjecture, dating back to the 19th century. For \(r\)-graphs \(F\) and \(G\), an \(F\)-decomposition of G is a collection of edge-disjoint copies of F in G covering all edges of \(G\). In a recent breakthrough, Keevash proved that every sufficiently large quasirandom \(r\)-graph G has a \(K\)\(_f\)\(^{(r)}\) -decomposition (subject to necessary divisibility conditions), thus proving the existence conjecture. We strengthen Keevash's result in two major directions: Firstly, our main result applies to decompositions into any \(r\)-graph \(F\), which generalises a fundamental theorem of Wilson to hypergraphs. Secondly, our proof framework applies beyond quasirandomness, enabling us e.g. to deduce a minimum degree version. For graphs, we investigate the minimum degree setting further. In particular, we determine the decomposition threshold' of every bipartite graph, and show that the threshold of cliques is equal to its fractional analogue. We also present theorems concerning optimal path and cycle decompositions of quasirandom graphs. This thesis is based on joint work with Daniela Kuhn and Deryk Osthus, Allan Lo and Richard Montgomery.

This paper builds a general mathematical and algorithmic theory for balloon-twisting structures by modeling their underlying edge skeleta, evolving classic balloon animals into the new world of balloon polyhedra. What if Euler were a clown?