generalized linear programming
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Generalized Linear Programming
Jiří Matoušek
Charles University, Prague
The cool slides in this presentation are included by the courtesy of Tibor Szabó.
Linear Programming
• Minimize cx subject to Ax b.
• Geometry: Minimize a linear function over the intersection of n halfspaces in Rd (=convex polyhedron).
LP Algorithms
• Simplex method [Dantzig 1947] – very fast in practice– very good “average case” – exponential-time examples for almost all pivot
rules
• Ellipsoid method [Khachyian], interior-point methods [Karmakar],…– weakly polynomial but no (worst-case) bound
in terms of n and d alone
Combinatorial LP algorithms
• wanted: time f(d,n) for all inputs
• computations “coordinate independent”; use only combinatorial structure of the feasible set (polyhedron) or of the arrangement of bounding hyperplanes
Combinatorial LP algorithms
Computational geometry: research started with d fixed (and small)– [Megiddo] exp(exp(d)).n– [Clarkson] randomization; d2n+dd/2 log n– [Seidel] simple randomized; d! n– [Chazelle, M.] exp(O(d)).n deterministic– parallel [Alon, Megiddo] [Ajtai, Megiddo]
A subexponential algorithm
Theory of convex polytopes (Hirsch conjecture):
[Kalai] 1992
Computational geometry:
[Sharir, Welzl],
[M., Sharir, Welzl] 1992
exp((d log d)).n (randomized expected)– known as RANDOM FACET :
In the current vertex of the feasible polytope, choose a random improving facet, recursively find its optimum, and repeat
– still the best known running time!
Abstract frameworks
• systems of axioms capturing some of the properties of linear programming
• running time of algorithms counted in terms of certain primitive operations
• to apply to a specific problem, need to implement them …
• … and then algorithms become available (such as Kalai/MSW, Clarkson)
Abstract frameworks
Abstract objective functions [Adler, Saigal 1976], [Wiliamson Hoke 1988], [Kalai 1988]– P a (convex) polytope– f : V(P) → R is an abstract objective function if
a local minimum of any face F is also the unique global minimum of F
– every generic linear function induces an AOF
– but there are nonrealizable AOF on the 3-dimensional cube!
Abstract frameworks
Acyclic Unique Sink Orientations (AUSO)– acyclic orientation of the graph of the
considered polytope such that every nonempty face has exactly one sink (sink = all edges incoming)
– same as abstract objective functions
Abstract frameworks
LP-type problems [Sharir, Welzl]– also called Generalized Linear Programs
[Amenta]– encompass many geometric optimization
problems [MSW,Amenta,Halman…]• smallest enclosing ball of n points in Rd
• smallest enclosing ellipsoid of n points in Rd
• distance of two (convex) polyhedra in Rd
• ………
– plus some non-geometric (games on graphs)
LP-type problems• H a finite set of constraints• (W,) a linearly ordered set (such as the reals)• w: 2H W a value function; intuitively: w(G) is
the minimum value of a solution attainable under the constraints in G
• Axiom M (monotonicity): If F G, then w(F) w(G).
• Axiom L (locality): If F G and w(F) = w(G) =w(F{h}), then w(G)=w(G{h}).
Example: Smallest enclosing ball• H a finite set of points in the plane• w(G) = radius of the smallest disk containing G
a
e
c
d
b
monotonicity trivial
locality depends onuniqeness of the smallestenclosing ball!
LP-type problems: more notions
• basis for G: inclusion-minimal B G with w(B)=w(G)
• dimension d of (H,w): maximum cardinality of a basis
• computational primitives (B a given basis)
– violation test: value(B{h})>value(G)?
– pivoting: compute a basis for B{h}
Abstract frameworks
Abstract Optimization Problems [Gärtner]– only one parameter: dimension d=|H| (no n)– a linear ordering of 2H
– primitive operation: Is G optimal among all sets containing F? If not, give a better G’
– nice randomized algorithm: exp(O(d)) [Gärtner]– allows a (rather) efficient implementation of
“primitives” in Kalai/MSW, e.g., for the smallest enclosing ball problem
Algorithms in the abstract frameworks
• several algorithms (Kalai/MSW = RANDOM FACET; Clarkson) work for AOF’s, same analysis– AUSO given by oracle: returns edge orientations for a
given vertex– yields n.exp(O(d)) randomized algorithm – analysis tight in this abstract setting [M.]
• for LP-type problems they work too (but…)– O(n) algorithms for fixed d usually immediate– but primitives “depend on d” … may be hard– sometimes Gärtner’s algorithm helps
Algorithms in the abstract frameworks
RANDOM EDGE
• the simplex algorithm that selects an improving edge uniformly at random
• for AUSO: random outgoing edge
• great expectations: perhaps always quadratic??? [Williamson Hoke 1988]
RANDOM EDGEExpected running time
– on the d-dimensional simplex: (log d) [Liebling]
– on d-dimensional polytopes with d+2 facets: (log2d) [Gärtner et al. 2001]
– on the d-dimensional Klee-Minty cube:• O(d2) Williamson Hoke (1988)(d2/log d) Gärtner, Henk, Ziegler (1995)(d2) Balogh, Pemantle (2004)
RANDOM EDGE can be (mildly) exponential
There exists an AUSO of the d-dimensional cube such that RANDOM EDGE, started at a random vertex, makes at least exp(c.d1/3) steps before reaching the sink, with probability at least 1- exp(-c.d1/3).
[M., Szabó, FOCS 2004]
The Klee-Minty cube
reversed KMm-1
KMm-1
KMm
A blowup construction
Hypersink reorientation
A simpler construction
Let A be a d-dimensional cube on which RANDOM EDGE is slow (constructed recursively)
– take the blowup of A with random KMm‘s whose sink is in the same copy of A, m=d
– reorient the hypersink by placing a random copy of A
– thus, a step from d to d+d
A
A
A
rand A
A simpler construction
A typical RANDOM EDGE move
• Move in the frame:– RANDOM EDGE move in KMm
– stay put in A
• Move within a hypervertex:– RANDOM EDGE move in A– move to a random vertex of
KMm on the same level
A
rand A
A
A
v
Random walk with reshuffles on KMmRANDOM EDGE on A
Walk with reshuffles on KMm
• Start at a random v(0) of KMm
• v(i) is chosen as follows:– with probability pi,step make a step of RANDOM
EDGE from v(i-1);
– with probability pi,resh randomly permute (reshuffle) the coordinates of v(i-1) to obtain v(i)
– with probability 1- pi,step - pi,resh, v(i) = v(i-1).
Walk with reshuffles on KMm is slow
Proposition. Suppose that
Then with probability at least
the random walk with reshuffles makes
at least steps (α and β are constants).
stepireshi pp ,, max11min me 1
me
Reaching the hypersink
• Either we reach the sink by reaching the sink of a copy of A and then perform RANDOM EDGE on KMm. This takes at least T(d) time.
• Or we reach the hypersink without entering the sink of any copy of A. That is, the random walk with reshuffles reaches the sink of KMm . This takes at least exp(m) T(d) time.
The recursion
• RANDOM EDGE arrives to the hypersink at a random vertex. Then it needs T(d) more steps.
So passing from dimension d to d+d the expected running time of RANDOM EDGE doubles.
• Iterating d - times gives T(2d) 2d T(d).• In order to guarantee that reshuffles are frequent
enough we need a more complicated construction and that is why we are only able to prove a running time of exp(c.d1/3).
Open questions
• Obtain any reasonable upper bound on the running time of RANDOM EDGE
• Can one modify the construction such that the cube is realizable? (Probably not …)
• Or at least it satisfies the Holt-Klee condition?
• Or at least each three-dimensional subcube satisfies the Holt-Klee condition?
More open questions
• Find an algorithm for AOF on the d-cube better than exp(d)
• The model of unique sink orientations of cubes (possibly with cycles) include LP on an arbitrary polytope.
Find a subexponential algorithm!
THE END
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