Finding Cycles and Trees in Sublinear Time Oded Goldreich Weizmann Institute of Science Joint work with Artur Czumaj, Dana Ron, C. Seshadhri, Asaf Shapira, and Christian Sohler General perspective – find small substructures in case the object is far from lacking them.

Preliminaries (boring, but needed…) Consider algorithms in the bounded-degree graph model (for a fixed degree bound, d): The algorithms use queries of the form (v,i) (where id) that are answered with the ith neighbor of v. Distances are measured as fractions of the maximum possible number of edges (i.e., dN/2). For simplicity, far = being (1)-far. (The results extend to the case that the algorithm is given a proximity parameter , but then the complexity depends on .) All algorithms that I will discuss operate by queries to an input graph, viewed as an oracle. cycle = simple cycle

Our results at a glance – take 1 (naïve) In the bounded-degree graph model cycle = simple cycle THM 1: An Õ(N1/2)-time algorithm for finding (small) cycles in N-vertex graphs that are far from being cycle-free. THM 2: For every fixed k>3, an Õ(N1/2)-time algorithm for finding (small) cycles of length at least k in N-vertex graphs that are far from lacking cycles of such length. Optimality: No o(N1/2)-query algorithm can find such cycles. THM 3: For every fixed k>1, an O(1)-time algorithm for finding trees with at least k leaves in graphs that are far from lacking such trees. All algorithms run in time that is polynomial in the reciprocal of the proximity parameter. small = polylog(N)-size

Our results at a glance – take 2 (minors) In the bounded-degree graph model cycle = simple cycle Ck = k-vertex cycle Def: A graph G has an H-minor if H can be obtained from G by vertex and edge removal and edge contraction. THM 1: An Õ(N1/2)-time algorithm for finding (small) C3-minors in N-vertex graphs that are far from being C3-minor free. THM 2: For every fixed k>3, an Õ(N1/2)-time algorithm for finding (small) Ck-minors in N-vertex graphs that are far from being Ck-minor free. Optimality: For any H that contains a cycle, no o(N1/2)-query algorithm can find H-minors in a N-vertex graphs that is far from being H-minor free. THM 3: For every fixed k>1, an O(1)-time algorithm for finding Tk-minors in graphs that are far from Tk–minor free, where Tk denotes the k-vertex star. THM 4: For any cycle-free H, an O(1)-time algorithm for finding H-minors in graphs that are far from H–minor free. The algorithms of Thms 1-3 run in time that is polynomial in the reciprocal of the proximity parameter. Dichotomy: H with/w.o. cycles.

Property Testing: One-sided versus two-sided error (Aux. slide) Property Testing: One-sided versus two-sided error Specialized to testing graph properties (in the bounded degree model) A (two-sided error) tester is a probabilistic oracle machine T that is given input n (#vertices) and oracle access to a n-vertex graph G and satisfies: If G has the property, then Prob[TG(n)=1] ≥ 2/3 (1 for one-sided error). If G is far from having the property, then Prob[TG(n)=1] ≤ 1/3. The definition extend to the case that the tester is also given a proximity parameter  (and input), and then “far” = “e-far”. Definitions of one/two-sided error testers are provided here.

One-sided error testing and finding structures Observation: When a one-sided error tester for a property  rejects a graph G, it must be the case that the subgraph viewed by the tester is inconsistent with any graph in . In some cases, this subgraph has a natural appeal. E.g., if  is being bipartite, then the subgraph must be a non-bipartite graph; if  is being H-minor free, then the subgraph must be an H-minor. Thus, all our results can be stated in terms of results regarding one-sided error testers (see next slide…). Recall that two-sided error testers of O(1)-time are known for H-minor freeness (cf. [BSS] vastly extending [GR]). N.B.: These testers do not yield algorithms for finding minors. Definitions of one/two-sided error testers should be provided. Dichotomy: one/two-sided error.

Our results at a glance – take 3 (1-sided-error-testers) In the bounded-degree graph model cycle = simple cycle Ck = k-vertex cycle Def: A graph G has an H-minor if H can be obtained from G by vertex and edge removal and edge contraction. THM 1: An Õ(N1/2)-time one-sided error tester for C3-minor freeness (a.k.a cycle-freeness). THM 2: For every fixed k>3, an Õ(N1/2)-time one-sided error tester for Ck-minor freeness. Optimality: For any H that contains a cycle, no o(N1/2)-query one-sided error tester for H-minor freeness. Yet, an O(1)-time two-sided error tester exists (cf. [BSS])! THM 4: For any cycle-free H, an O(1)-time one-sided error tester for H–minor freeness. The algorithms of Thms 1-3 run in time that is polynomial in the reciprocal of the proximity parameter. Two dichotomies: H with/without cycles, one/two-sided error

Techniques: testing cycle-freeness THM 1: An Õ(N1/2)-time one-sided error tester for cycle-freeness. In the bounded-degree graph model cycle = simple cycle The two-sided error tester just compares the # of edges to the # of cc. Idea: randomly reduce testing cycle-freeness to testing bipartiteness, by replacing each edge with a 2-path w.p. ½ (and leaving it intact otherwise). A cycle-free graph is always mapped to a bipartite graph, whereas each cycle is mapped with probability ½ to an odd cycle. CLM: A graph that is -far from being cycle-free is mapped, w.v.h.p, to a graph that is ()-far from being bipartite. Details: Local implementation of the reduction. Operations of the bipartite-tester are emulated via queries to the original graph. Before presenting the one-sided error tester, recall that the two-sided error tester does not even try to find cycles.

Techniques: testing Ck-minor freeness Two triangles sharing an edge contain a cycle of length four. A C4-minor free graph is a “tree” of triangles and edges. THM 2: An Õ(N1/2)-time one-sided error tester for Ck-minor freeness. In the bounded-degree graph model cycle = simple cycle Ck = k-vertex cycle Idea: (deterministically) reduce testing Ck-minor freeness to testing cycle-freeness, by replacing cycles of length < k with adequate gadgets. E.g., for k=4, replace each triangle by a 3-star. A C4-minor free graph is always mapped to a cycle-free graph, whereas any C4-minor is mapped to a cycle. CLM: A graph that is -far from being C4-minor free is mapped to a graph that is ()-far from being cycle-free. Details: Local implementation of the reduction. Operations of the bipartite-tester are emulated via queries to the original graph. The claim is illustrated in the upper-right rectangle and in the next (aux.) slide. The reduction blows-up the size of the graph by a poly(d^k) factor! (See next slide)

(Aux. slide) Testing C4-minor freeness (via a reduction): Replacing triangles by 3-stars The reduction replaces red edges by blue edges. (Black edges remain intact.) Two triangles sharing an edge contain a cycle of length four. They are replaced by edges that contain a 4-cycle. A C4-minor free graph is a “tree” of triangles and edges. These triangles disappear (in replacement) and the tree remains. Triangles are replaced by 3-stars; the effect on BAD and GOOD graphs is shown.

Techniques: testing Tk-minor freeness THM 3: An poly(k/)-time one-sided error tester for Tk-minor freeness. In the bounded-degree graph model Tk = the k-vertex star The tester performs a BFS from a randomly chosen start vertex till either encountering k vertices in a layer or visiting 4k/ layers. Accept iff the explored subgraph is Tk-minor free. Call a vertex v bad if it is contained in a set S such that the subgraph induced by S contains a Tk-minor and has radius at most 4k/ from v. Observe that if the graph has few bad vertices, then it is close to being Tk-minor free (by isolating all bad vertices and omitting the edges that separate each 4k/-depth BFS from the rest of the graph). The algorithms of Thms 1-3 run in time that is polynomial in the reciprocal of the proximity parameter.

A few open problems Main: Sublinear-time one-sided error tester of H-minor freeness for every fixed H. Recall: we only handle cycles and trees (actually forests). For sake of curiosity: Deterministic (local) reduction of testing cycle-freeness to testing bipartiteness? Recall: Our reduction was randomized. One-sided error tester of sublinear-time for Ck-minor freeness also for d > sqrt(n)? Our tester (via reduction) has a hidden poly(dk) factor. One-sided error tester of query complexity poly(k/e) for Pk-minor freeness (i.e., absence of k-paths). There is an obvious exp(k)/e-time tester. Re last item: Finding a longest path *may* be NPC even if the graph is far from lacking it.

The End The slides of this talk are available at http://www.wisdom.weizmann.ac.il/~oded/T/minor.ppt The paper itself is available at http://www.wisdom.weizmann.ac.il/~oded/p_minor.html